Pandiagonal and semi-pandiagonal magic tori of order 4 connected by plus-and-minus links |

General information about the 255 magic tori of order 4, and about the 880 magic squares that those tori display, can be found in a previous article, published on the 15th June 2023, and entitled *"440 Torus-Opposite Pairs of the 880 Frénicle Magic Squares of Order-4"*.

The 137 complete-torus, same-integer, plus-or-minus groups of magic tori of order 4 have been presented in a previous article, published on the 24th April 2024, and entitled *"Plus or Minus Groups of Magic Tori of Order 4"*.

Further information about the magic tori can be found in the lists of references that are appended to the papers in each of the above-mentioned articles.

As we are looking for connections between complete-torus, same-integer, plus-or-minus groups of magic tori, it seems logical to search for links with 2-or-more-integer plus-and-minus operations. The closest links between the tori will be those where only 4 out of the 16 numbers are changed, and these will be for example 3/4 & 1/4: (±0, ±1), or 3/4 & 1/4: (±0, ±2) etc., depending on the differences between the cell entries.

A variant of these links will be those where 12 out of the 16 numbers are changed, and these can be written, for example, as 3/4 & 1/4: (±1, ±0), or as 3/4 & 1/4: (±2, ±0) etc., depending on the differences between the cell entries. However, a change of only 1/4 of the cell entries seems less disruptive than a change of 3/4 of these, and we can therefore suppose that the previously-mentioned 3/4 & 1/4: (±0, ±1), or 3/4 & 1/4: (±0, ±2) are closer links.

Close links also include those where only 8 out of the 16 numbers are changed, and these will be, for example, 1/2 & 1/2: (±0, ±1), or 1/2 & 1/2: (±0, ±2) etc., depending on the differences between the torus cell entries.

In some cases 2-integer plus-and-minus operations cannot be found, but more-complex 3-integer links may exist, such as 1/2, 3/8 & 1/8: (±2, ±0, ±1) or 1/2, 1/4 & 1/4: (±4, ±0, ±8). 3-integer links like these are frequent between the paired complementary and self-complementary torus groups.

Sometimes even 3-integer plus-and-minus operations are unavailable, but there exist 4-integer links, such as 1/4, 1/4, 1/4, & 1/4: (±0, ±2, ±4, ±6), or 3/8, 1/4, 1/4 & 1/8: (±3, ±0, ±6, ±9), which, if not particularly close, show interesting patterns. Links like these are common within the paired complementary torus groups that are to be found in the pages that follow.

Because of its A3 format, this enclosed PDF file can at first seem unwieldy, especially when it is displayed on small-screen mobile phones. However the large display size allows same-page illustrations of complete sets of ± groups, which can be comprised of up to 24 examples. Please bear in mind that, in order to reduce space requirements and facilitate reading, most of the 137 ± groups are represented by single magic square viewpoints of only one of their magic tori. But occasionally, a same group is represented more than once, so as to facilitate the comprehension of multiple links. To find our which other magic tori are within a ± group, please refer to the details given in the paper "Plus or Minus Groups of Magic Tori of Order 4", already mentioned above.

In order to simplify the annotation of the pages of the paper enclosed below, the links between the ± groups are written as ± operations. This is slightly ambiguous, as there must always be a same number of plusses as minuses of any particular integer. Here therefore, the ± sign in front of an integer will mean an equal number of plusses and minuses of that integer. This will become evident as soon as we compare any pair of magic square viewpoints which are connected by the links in question. For full details, please refer to the paper below.

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As the positive integer entries of normal magic tori, and of their displayed magic squares, are always arranged in same-sum orthogonal arrays, it would seem logical to compensate any additions or subtractions to those entries, by making corresponding subtractions or additions. How can we develop this simple observation?

A normal magic square is an N x N array of same-sum rows, columns and main diagonals. If the magic diagonal condition is not satisfied, the square is deemed to be semi-magic. The torus, upon which a magic or semi-magic square is displayed, can be visualised by joining the opposite edges of the magic or semi-magic square in question.

There are N x N square viewpoints on the surface of a torus of order N. A normal magic torus has N x N arrays of same-sum latitudes and longitudes, and at least one magic intersection of magic diagonals on its surface. In his paper "Conformal Tiling on a Torus" published by Bridges in 2011, John M. Sullivan shows that, *on a square torus T1, 0, the diagonal grid lines “form (1, ±1) diagonals on the torus, each of which is a round (Villarceau) circle in space.”*

Semi-magic tori can also have magic diagonals, but the latter can never produce the magic intersections required for magic squares, either because the magic diagonals are single or parallel, or because they do not intersect correctly. Additional information about magic and semi-magic tori, with further explanations of their diagonals, can be found in the references at the end of the enclosed paper.

In order 4, starting with any magic torus, and examining cases where half of the torus numbers are subjected to equal additions, and the other half are subjected to same-integer subtractions, we notice the following: From its initial state (which we can call 0), each magic square can be transformed by four plus or minus operations that will either produce alternative magic or semi-magic square viewpoints of the same torus, or square viewpoints of other essentially different magic or semi-magic tori. Here is a simple example of two transformations:

3 Magic Tori of Order 4 linked by by Complete-Torus, Same-Integer, Plus or Minus Operations |

Please note that, in order 4, the orthogonal totals of magic and semi-magic tori are always 34. Therefore, only the diagonal totals, which can vary, are announced here. These totals are interesting because of their symmetries, and in the example illustrated above, there are pairs of diagonals that always sum to 68 (twice the magic sum). Each of the 255 magic tori of order 4 is similarly examined, and the results are given in tables and observations at the end of the following paper: *"Groups of Magic Tori of Order 4
Assembled by Complete-Torus, Same-Integer, Plus or Minus Operations"*

The same method cannot always be applied to higher even orders. A quick look at some examples from even orders N = 6, N = 8, and N = 10 shows that certain magic tori have either no solutions whatsoever, or only one that can be used in a complete-torus same-integer plus or minus matrix to produce another orthogonally magic torus. In singly-even orders, even divisors with odd quotients have to be ruled out. Each case will need to be tested, and systematic computer checks will be necessary for these higher even orders.

An adaptation is of course required for odd orders, as their odd square totals do not have even integer divisors. But partial-torus same-integer divisions produce plus or minus solutions, and there are a variety of approaches.

The following paper, entitled *"Examples of Partial Groups of Magic Tori of Orders N > 4
Assembled by Complete-Torus (or Near-Complete in Odd Orders) Same-Integer, Plus or Minus Operations"* shows how the method used for order 4 gives some good results in higher orders:

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The following diagram, which* *illustrates the array of *n²* essentially-different square viewpoints of a basic magic torus of order-*n*, starting here with the 4x4 magic square that has the Frénicle index n° 1, shows how a torus-opposite magic square of order-4 can be identified:

Array of 16 essentially-different square viewpoints of a basic magic torus of order-4 |

This basic magic torus, now designated as type n° T4.05.1.01, can be seen to display a torus-opposite pair of Frénicle-indexed magic squares n° 1-458 (with Dudeney pattern VI), as well as 7 torus-opposite pairs of semi-magic squares that the discerning reader will easily be able to spot.

The relative positions of the numbers of torus-opposite squares can be expressed by a simple plus or minus vector. There are always two equal shortest paths towards the far side of the torus: These are east or west along the latitudes, and north or south along the longitudes of the doubly-curved 2D surface. So, if the even-order is *n*, then:

**v** = ( ± n/2, ± n/2 )

"Tesseract Torus" by Tilman Piesk CC-BY-4.0 https://commons.wikimedia.org/w/index.php?curid=101975795 |

Torus-opposite squares are not just limited to basic magic tori, as they also exist on pandiagonal, semi-pandiagonal, partially pandiagonal, and even semi-magic tori of even-orders. Therefore, for order-4, we can either say that there are 880 magic squares, or we can announce that there are 440 torus-opposite pairs of magic squares. Similarly, the 67,808 semi-magic squares of order-4 can be expressed as 33,904 torus-opposite pairs of semi-magic squares, etc.

A torus-opposite magic square always exists in even-orders because an even-order magic square has magic diagonals that produce a first magic intersection at a centre *between numbers, *and another magic intersection at a second centre *between numbers* on the far side of the torus (at the meeting point of the four corners of the first magic square viewpoint). However, in odd-orders, where a magic square has magic diagonals that produce a magic intersection *over a number*, then a sterile *non-magic intersection* always occurs *between numbers* on the far side of the torus (at the meeting point of the four corners of the initial magic square). This is why torus-opposite pairs of magic squares cannot exist in the odd-orders.

The enumeration of the 255 magic tori of order-4 was first published in French on the 28th October 2011, before being translated into English on the 9th January 2012: "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus." In this previous article, the corresponding 880 fourth-order magic squares were listed by their Frénicle index numbers, but not always illustrated. The intention of the enclosed paper is therefore to facilitate the understanding of the magic tori of order-4, by portraying each case.

Here, Frénicle index numbers continue to be used, as they have the double advantage of being both well known and commonly accepted for cross-reference purposes. But please also note, that now, in order to simplify the visualisation of the magic tori, their displayed magic squares are not systematically presented in Frénicle standard form.

In the illustrations of the 255 magic tori of order-4, listed by type with the presentation of their magic squares, the latter are labelled from left to right with their Frénicle index number, followed by, in brackets, the Frénicle index number of their torus-opposite magic square, and finally by the Roman numeral of their Dudeney complementary number pattern. Therefore, for the magic square of order-4 with Frénicle index n° 1 (that forms a torus-opposite pair of magic squares with Frénicle index n° 458; both squares having the same Dudeney pattern VI), its label is 1 (458) VI.

To find full details of the "255 magic tori of order-4, listed by type, with details of the 440 torus-opposite pairs of the 880 Frénicle magic squares of order-4" please consult the following PDF:

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Bernard Frénicle de Bessy was the first to determine that there were 880 essentially different magic squares of order-4, and his findings were published posthumously in the "Table Générale des Quarrez Magiques de Quatre," in 1693.

Title of Frénicle's study of magic squares published in 1693 |

The "Table Générale des Quarrez Magiques de Quatre" published in 1693 |

It was more than 300 years later, that it was discovered that these 880 magic squares of order-4 were displayed on 255 magic tori of order-4. At first listed by type in 2011, the 255 magic tori of order-4 were later given additional index numbers in a "Table of Fourth-Order Magic Tori" in 2012. Then in 2018, it was found that the 255 magic tori came from 82 multiplicative magic tori of order-4. In 2019 some of the magic tori of order-4 were found to be extra-magic, with nodal intersections of 4 or more magic lines and / or knight move magic diagonals. Other studies of the magic tori of order-4 have included sub-magic 2 x 2 squares (in 2013), magic torus complementary number patterns (in 2017), and even and odd number patterns (in 2019).

It has become increasingly important to provide an easily accessible document that recapitulates these different findings. I have therefore compiled the following "List of the 880 Frénicle Indexed Magic Squares of Order-4," with their corresponding Dudeney types and also full details of the magic tori:

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A magic torus can be found with any magic square, as has already been demonstrated in the article "From the Magic Square to the Magic Torus". In fact, there are n^2 essentially different semi-magic or magic squares, displayed by every magic torus of order-n. Particularly interesting to observe with pandiagonal (or panmagic) examples, a magic torus can easily be represented by repeating the number cells of one of its magic square viewpoints outside its limits. However, once we begin to look at *area *magic squares, it becomes much less evident to visualise and construct the corresponding *area *magic tori using repeatable* area* cells, especially when the latter have to be irregular quadrilaterals... The following illustration shows a sketch of an area magic torus of order-3 that I created back in January 2017. I call it a sketch because it may be necessary to use consecutive areas starting from 2 or from 3, should the construction of an area magic torus of order-3, using consecutive areas from 1 to 9, prove to be impossible. And while it can be seen that such a torus is theoretically constructible, many calculations will be necessary to ensure that the areas are accurate, and that the irregular quadrilateral cells can be assembled with precision:

At the time discouraged by the complications of such geometries, I decided to suspend the research of area magic tori. But since the invention of area magic squares, other authors have introduced some very interesting polyomino versions that open new perspectives:

On the 20th May 2021, Morita Mizusumashi (盛田みずすまし
@nosiika) tweeted a nice polyomino area magic square of order-3 constructed with 9 assemblies of *5* to *13* monominoes. On the 21st May 2021, Yoshiaki Araki (面積魔方陣がテセレーションみたいな件
@alytile) tweeted several order-4 and order-3 solutions in a polyomino area magic square thread. These included (amongst others) an order-4 example constructed with 16 assemblies of 5 to 20 dominoes. On the 24th May 2021, Yoshiaki Araki then tweeted an order-3 polyomino area magic square constructed using 9 assemblies of 1 to 9 same-shaped pentominoes! Edo Timmermans, the author of this beautiful square, had apparently been inspired by Yoshiaki Araki's previous posts! Since the 22nd June 2021, Inder Taneja has also published a paper entitled "Creative Magic Squares: Area Representations" in which he studies polyomino area magic using perfect square magic sums.

The intention of the present article is to explore the use of polyominoes for area magic torus construction, with the objective of facilitating the calculation and verification of the cell areas, while avoiding the geometric constraints of irregular quadrilateral assemblies. Here, it is useful to give a definition of a polyomino area magic torus:

1/ In the diagram of the torus, the entries of the cells of each column, row, and of at least two intersecting diagonals, will add up to the same magic sum. The intersecting magic diagonals can be offset or broken, as the area magic torus has a limitless surface, and can therefore display semi-magic square viewpoints.2/ Each cell will have an area in proportion to its number. The different areas will be represented by tiling with same-shaped holeless polyominoes.3/ The cells can be of any regular or irregular rectangular shape that results from their holeless tiling.4/ Depending on the order-n of the area magic torus, each cell will have continuous edge connections with contiguous cells (and these connections can be wrap-around, because the torus diagram represents a limitless curved surface).5/ The vertex meeting points of four cells can only take place at four convex (i.e. 270° exterior angled) vertices of each of the cells.

Magic Torus index n° T3, of order-3. Magic sums = 15.Please note that this is not a Polyomino Area Magic Torus,but it is the Agrippa "Saturn" magic square, after a rotation of +90°, in Frénicle standard form. |

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 15. Tetrominoes. Consecutively numbered areas 1 to 9, in an irregular rectangular shape of 180 units. |

Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 18. Trominoes. Consecutively numbered areas 2 to 10, in an irregular rectangular shape of 162 units. |

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 24. Trominoes. Consecutively numbered areas 4 to 12, in an oblong 12 ⋅ 18 = 216 units. |

Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 36. Trominoes Version 1. Square 18 ⋅ 18 = 324 units. |

Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 36. Trominoes Version 2. Square 18 ⋅ 18 = 324 units. |

Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 60. Pentominoes Version 1. Square 30 ⋅ 30 = 900 units. |

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 24. Monominoes. Consecutively numbered areas 4 to 12, in an irregular rectangular shape of 72 units. |

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 27. Monominoes Version 1. Consecutively numbered areas 5 to 13, in a square 9 ⋅ 9 = 81 units. |

Polyomino Area Magic Torus (PAMT) of order-3. Magic sums = 27. Monominoes Version 2. Consecutively numbered areas 5 to 13, in a square 9 ⋅ 9 = 81 units. |

Magic Torus index n° T4.198, of order-4. Magic sums = 34.Please note that this is not a Polyomino Area Magic Torus,but it is a pandiagonal torus represented by a pandiagonal square that has Frénicle index n° 107. |

The pandiagonal torus above displays 16 Frénicle indexed magic squares n° 107, 109, 171, 204, 292, 294, 355, 396, 469, 532, 560, 621, 691, 744, 788, and 839. It is entirely covered by 16 sub-magic 2x2 squares. The torus is self-complementary and has the magic torus complementary number pattern I. The even-odd number pattern is P4.1. This torus is extra-magic with 16 extra-magic nodal intersections of 4 magic lines.
It displays pandiagonal Dudeney I Nasik magic squares. It is classified with a Magic Torus index n° T4.198,
and is of Magic Torus type n° T4.01.2. Also, when compared with its two pandiagonal torus cousins of order-4, the *unique* Magic Torus T4.198 of the Multiplicative Magic Torus MMT4.01.1 is distinguished by its total *self-complementarity*.

Pandiagonal Polyomino Area Magic Torus (PAMT) of order-4. Magic sums = 34. Pentominoes. Index PAMT4.198, Version 1, Viewpoint 1/16, displaying Frénicle magic square index n° 107. Consecutively numbered areas 1 to 16, in an oblong 34 ⋅ 20 = 680 units. |

Pandiagonal Polyomino Area Magic Torus (PAMT) of order-4. Magic sums = 50. Dominoes. Version 1, Viewpoint 1/16. Consecutively numbered areas 5 to 20, in a square 20 ⋅ 20 = 400 units. |

Pandiagonal Polyomino Area Magic Torus (PAMT) of Order-4. Sums = 50. Dominoes. Version 1, Viewpoint 16/16. Consecutively numbered areas 5 to 20, in an irregular rectangular shape of 400 units. |

Pandiagonal Torus type n° T5.01.00X of order-5. Magic sums = 65.Please note that this is not a Polyomino Area Magic Torus. |

This pandiagonal torus of order-5 displays 25 pandiagonal magic squares. It is a direct descendant of the T3 magic torus of order-3, as demonstrated in page 49 of "Magic Torus Coordinate and Vector Symmetries" (MTCVS). In "Extra-Magic Tori and Knight Move Magic Diagonals" it is shown to be an Extra-Magic Pandiagonal Torus Type T5.01 with 6 Knight Move Magic Diagonals. Note that when centred on the number 13, the magic square viewpoint becomes associative. The torus is classed under type n° T5.01.00X (provisional number), and is one of 144 pandiagonal or panmagic tori type 1 of order-5 that display 3,600 pandiagonal or panmagic squares. On page 72 of "Multiplicative Magic Tori" it is present within the type MMT5.01.00x.

Pandiagonal Polyomino Area Magic Torus (PAMT) of order-5. Magic sums = 65. Hexominoes. Index PAMT5.01.00X, Version 1, Viewpoint 1/25. Consecutively numbered areas 1 to 25, in an oblong 65 ⋅ 30 = 1950 units. |

Pandiagonal Polyomino Area Magic Torus (PAMT) of Order-5. Magic sums = 125. Monominoes. Version 1, Viewpoint 1/25. Consecutively numbered areas 13 to 37, in a square 25 ⋅ 25 = 625 units. |

Pandiagonal Polyomino Area Magic Torus (PAMT) of Order-5. Magic sums = 125. Monominoes. Version 2, Viewpoint 1/25. Consecutively numbered areas 13 to 37, in a square 25 ⋅ 25 = 625 units. |

Partially Pandiagonal Torus type n° T6 of order-6. Magic sums = 111.Please note that this is not a Polyomino Area Magic Torus. |

Harry White has kindly authorised me to use this order-6 magic square viewpoint. With a supplementary broken magic diagonal (24, 19, 31, 3, 5, 29), this partially pandiagonal torus displays 4 partially pandiagonal squares and 32 semi-magic squares. In "Extra-Magic Tori and Knight Move Magic Diagonals" it is shown to be an Extra-Magic Partially Pandiagonal Torus of Order-6 with 6 Knight Move Magic Diagonals. This is one of 2627518340149999905600 magic and semi-magic tori of order-6 (total deduced from findings by Artem Ripatti - see OEIS A271104 "Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2").

Partially Pandiagonal Polyomino Area Magic Torus (PAMT) of order-6. Magic sums = 111. Heptominoes, Version 1, Viewpoint 1/36. Consecutively numbered areas 1 to 36, in an oblong 111 ⋅ 42 = 4662 units. |

Partially Pandiagonal Polyomino Area Magic Torus (PAMT) of order-6. Sums = 147. Dominoes, Version 1, Viewpoint 1/36. Consecutively numbered areas 7 to 42, in a square 42 ⋅ 42 = 1764 units. |

As they are the first of their kind, these Polyomino Area Magic Tori (PAMT) can most likely be improved: The examples illustrated above are all constructed with their cells aligned horizontally or vertically; and though it is convenient to do so, because it allows their representation as oblongs or squares, this method of constructing PAMT is not obligatory. Representations of PAMT that have irregular rectangular contours may well give better results, with less-elongated cells and simpler cell connections.

While the use of polyominoes has the immense advantage of allowing the construction of area magic tori with easily quantifiable units, it also introduces the constraint of the tiling of the cells. It has been seen in the examples above that the PAMT can be represented as oblongs or as squares, while other irregular rectangular solutions also exist. A normal magic square of order-3 displays the numbers 1 to 9 and has a total of 45, which is not a perfect square. As the smallest addition to each of the nine numbers 1 to 9, in order to reach a perfect square total is four (45 + 9 ⋅ 4 = 81), this implies that when searching for a *square* PAMT with consecutive areas of 1 to 9, *in theory *the smallest polyominoes for this purpose will* *be pentominoes.

But to date, in the various shaped examples of PAMT shown above, the smallest cell area used to represent the area 1 is a tetromino, as this gives sufficient flexibility for the connections of a nine-cell PAMT of order-3 with consecutive areas of 1 to 9. Edo Timmermans has already constructed a Polyomino Area Magic Square of order-3 using pentominoes for the consecutive areas of 1 to 9, but it seems that such polyominoes cannot be used for the construction of a same-sized and shaped PAMT of order-3. Straight polyominoes are always used in the examples given above, as these facilitate long connections, but other polyomino shapes will in some cases be possible.

We should keep in mind that the PAMT are theoretical, in that, per se, they cannot tile a torus: As a consequence of Carl Friedrich Gauss's *"Theorema Egregium"*, and because the Gaussian curvature of the torus is not always zero, there is no local isometry between the torus and a flat surface: We can't flatten a torus without distortion, which therefore makes a perfect map of that torus impossible. Although we can create conformal maps that preserve angles, these do not
necessarily preserve lengths, and are not ideal for our purpose. And while two topological spheres are conformally equivalent, different
topologies of tori can make these conformally distinct and lead to further mapping complications. For those wishing to know more, the paper by Professor John M. Sullivan, entitled "Conformal Tiling on a Torus", makes excellent reading.

Notwithstanding their theoreticality, the PAMT nevertheless offer an interesting field of research that transcends the complications of tiling doubly-curved torus surfaces, while suggesting interesting patterns for planar tiling: For those who are not convinced by 9-colour tiling, 2-colour pandiagonal tiling can also be a good choice for geeky living spaces:

Tiling with irregular rectangular shaped PAMT of order-3. Monominoes. S=24. |

Tiling with irregular rectangular shaped PAMT of order-3. Tetrominoes. S=15. |

Tiling with irregular rectangular shaped PAMT of order-3. Trominoes. S=18. |

Tiling with oblong PAMT of order-3. Trominoes. S=24. |

Tiling with oblong pandiagonal PAMT of order-4. Pentominoes. S=34. |

Tiling with oblong pandiagonal PAMT of order-5. Hexominoes. S=65. |

Tiling with square pandiagonal PAMT of order-5. Monominoes. S=125. |

Tiling with oblong partially pandiagonal PAMT of order-6. Heptominoes. S=111. |

Tiling with square partially pandiagonal PAMT of order-6. Dominoes. S=147. |

There are still plenty of other interesting PAMT that remain to be found, and I hope you will authorise me to publish or relay your future discoveries and suggestions!

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The title of this post may at first seem rather strange, especially when we
know that the main subjects of these pages are "Magic Squares, Spheres and
Tori." However, the famous
"Melencolia I"
engraving by
Albrecht Dürer
does depict, amongst other symbols, a magic square of order-4 (already
examined in
"A Magic Square Tribute to Dürer, 500+ Years After Melencolia I,"
and
"Pan-Zigzag Magic Tori Magnify the "Dürer" Magic Square").

In the section reserved for correspondence at the end of the post
"A Magic Square Tribute to Dürer, 500+ Years After Melencolia I,"
I have recently received some interesting comments from
Rob Sellars. Rob looks at Dürer's engraving from a Judaic point of view and describes
the bat-like animal (at the top left) as a flying chimera which has a
combination of the
Tinshemet features
of the "flying waterfowl and the earth mole." Rob's description has made me
look harder at this beast, and in doing so, I have noticed some aspects that
explain the very essence of "Melencolia I."

Detail of the bat-like beast at the top left-hand side of "Melencolia I" which was engraved by Albrecht Dürer in 1514 |

At first sight, the cartouche at the top left-hand side of Dürer's "Melencolia
I" seems to be a flying bat, bearing the title of the engraving on its open
wings. The length and thickness of the tail both look oversized, but we can
suppose that Dürer was using his artistic licence to amplify the visual impact
of the swooping beast. Nearly all species of bats have tails, even if most (if
not all) of these, are shorter and thinner than the one that Dürer has
depicted.

But looking again with more attention, we can see that, quite weirdly,
the body of the animal is placed above its wings, which is impossible unless
the bat is flying upside-down! Closer examination suggests that this is not
the case, as the mouth and eyes of the beast are clearly those of an animal
with an upright head. All the same, we might well ask where the hind feet are,
and how the creature can possibly make a safe landing without these!

Looking once again more closely, we can see another, even more troubling
detail, in that the “wings,” which carry the title of the engraving, are in
fact two large strips of ragged skin, ripped outwards from the belly, as if
the animal has disembowelled itself!

Judging from the thickness of its tail and the form of its head, the
airborne creature was initially a rat before it began its painful
metamorphosis. It has since carried out an auto-mutilation, and is now showing
its inner melancholy to the outer world, but at the same time flying free with
its hard-earned wings!

Symbolically, the cartouche is telling us that ""Melencolia I" is a painful
metamorphosis which precedes a liberating *"Renaissance!"*"

During 1514 CE the artist's mother,
Barbara Dürer
(née Holper), passed away, or “died hard” as he described it, and we can
therefore suppose that Dürer’s grief would have been a strong catalyst of the
very melancholic atmosphere depicted in his “Melencolia I.” The melancholy,
referred to in the title of the engraving, is illustrated by an extraordinary
collection of symbols that fill the scene. Some of these are tools associated
with craft and carpentry. Others are objects and instruments that refer to
alchemy, geometry or mathematics. In addition to the bat-like beast, the sky
also contains what might be a
moonbow
and a comet.
Further symbols include a
putto seated
on a millstone, and a robust winged person, also seated, which could well be
an allegorical self-portrait. These, and many other symbols, are the object of
multiple interpretations by various authors. Some scholars consider the
engraving to be an
allegory,
which can be interpreted through the correct comprehension of the symbols,
while others think that the ambiguity is intentional, and designed to resist
complete interpretation. I tend to agree with the latter point of view, and
think that the confusion symbolises the unfinished studies and works of the
main melancholic figure; an apprentice
angel, who
believes that despite his worldly efforts, he lacks inspiration, and is not
making sufficient progress.

Notwithstanding the melancholy that reigns, there is still hope: The 4 x 4
magic square, for example, has the same dimension as
Agrippa's Jupiter square, a talisman that supposedly counters melancholy. The intent expression of
the main winged person suggests a determination to overcome his doubts, and
transcend the obstacles that continue to block his progression. Positive
symbols of a resurrection or *"Renaissance"* are also plainly visible,
not only in the hard-earned wings of the flying creature, but also in the
growing wings that Dürer gives himself in his portrayal as the apprentice
angel.

"Melencolia I" engraved by Albrecht Dürer in 1514 |

On page 171 of his book entitled "The Life and Art of Albrecht Dürer,"
Erwin Panofsky
considers that "Melencolia I" is the spiritual self-portrait of the artist.
There is indeed much resemblance between the features of the apprentice angel,
and those of the engraver in previous self-portraits.

Dürer had already adopted a striking religious pose in his last
*declared* self-portrait of 1500 CE, giving himself
a strong resemblance to Christ
by respecting the iconic pictorial conventions of the time. In other presumed
self-portraits, (but not declared as such), Dürer had also presented himself
in a Christic manner; in his c.1493
"Christ as a Man of Sorrows;"
and in his 1503
"Head of the Dead Christ."
What is more, Dürer inserted his self-portraits in altarpieces; in 1506 for
the San Bartolomeo church in Venice ("Feast of Rose Garlands"); in 1509 for the Dominican Church in Frankfurt ("Heller Altarpiece"); and in 1511 for a Chapel in Nuremberg's "House of Twelve Brothers" ("Landauer Altarpiece"
or "The Adoration of the Trinity"). Thus Dürer was already a master of
religious
self-portraiture
when he engraved "Melencolia I" in 1514, and he might well have continued in
the same manner. But this time, probably because the theological,
philosophical and humanistic ideas of the Renaissance were not only
spiritually, but also intellectually inspiring, he went even further, and gave
himself wings!

Passages of "The Historical Context" are inspired by the writings of Bonnie James, in her excellent article "Albrecht Dürer: The Search for the Beautiful In a Time of Trials" (Fidelio Volume 14, Number 3, Fall 2005), a publication of the Schiller Institute.

After reading this article,
Miguel Angel Amela
(who like me, is not only interested in magic squares, but also in
"Melencolia I") sent me his thanks by email, and enclosed *"a paper of 2020
about a painful love triangle..."* His paper is entitled
"A Hidden Love Story"
and interprets the "portrait of a young woman with her hair done up," which was first
painted
by
Albrecht Dürer
in 1497, and then reproduced in an
engraving
by
Wenceslaus Hollar, almost 150 years later in 1646. Miguel's story is captivating, and I wish to
thank him for kindly authorising me to publish it here.

On the 6th July 2020, enclosing notes written by Joachim Brügge, Awani Kumar sent an e-mail to our circle of magic square enthusiasts, asking whether a bimagic queen's tour might exist on an 8 x 8 or larger board, and invited us to "settle the question." I was intrigued by this interesting challenge, and after a first analysis of Joachim Brügge's approach, and exchanging e-mails with him, I decided to search for a solution.

A standard chess board is an 8 x 8 square grid upon which the queen can move any number of squares, either orthogonally, or along ± 1 / ± 1 diagonals. A queen's tour is a sequence of queen's moves in which she visits each of the 64 squares once. If the queen ends her tour on a square which is at a single queen's move from the starting square, the tour is closed; otherwise the queen's tour is open. Here it is useful to clarify the definition of a *bimagic queen's tour*, which is *different to that of a bimagic square*: In chess literature, as confirmed by George Jelliss, the term *magic* is commonly used for all tours in which the successively numbered positions of the chess piece in each rank and file add up to the same orthogonal total (the magic constant S1), and should the entries on the two long diagonals also add up to the same magic constant, the tour is deemed to be diagonally magic. (In fact, it is now known, thanks to the work of G. Stertenbrink, J-C. Meyrignac and H. Mackay, who have completed the list of all of the 140 *magic knight tours* on an 8 x 8 board, that none of these have two long magic diagonals). Extending the above definition of a magic tour to that of a *bimagic* queen's tour, not only the successively numbered positions of the queen in each rank and file should add up to a same orthogonal total *(the magic constant S1)*, but also the *squared entries* of each rank and file should add up to an additional total which is *the bimagic constant S2*.

For a N x N board, when N = 8, the magic constant S1 = 260.

For a N x N board, when N = 8, *the bimagic constant S2 = 11180*.

More information about how the magic and bimagic constants are calculated can be found in the website of Christian Boyer.

Before beginning the search for a solution to the bimagic queen's tour question, the following hypotheses were considered:

Firstly, it seemed logical to privilege diagonal queen moves that would be less likely to perturb the orthogonal bimagic series of the ranks and files.

Secondly, it seemed pertinent to make selections from the 240 immutable bimagic series of order-8; series named this way by Dwane Campbell, who identified and listed these some years ago. 192 of these series have the advantage of having a regular distribution, with each of their eight numbers coming from different eighths of the sequence 1 to 64.

Thirdly, it seemed probable that by using symmetric arrays of bimagic series, this would be beneficial for overall balance, and increase the chances of success.

Although I was at first unable to find a complete bimagic queen's tour, testing the above hypotheses gradually produced encouraging results. On the 18th July I was able to write an e-mail to Joachim Brügge announcing that I had found several broken bimagic queen's tours, with 4, 3, and even 2 separate sequences.

In all of these broken tours, because of the characteristics of the bimagic series that were tested, the sequence breaks occurred precisely at certain junctions between the different quarters of N², and it was of utmost importance to find a regular sequence that could negotiate these obstacles with valid diagonal queen's moves.

Also during these early trials I noticed that when certain diagonal moves "exited" the board they "re-entered" it in cells that were no longer directly accessible to the queen. The following diagram shows how this occurs:

How certain diagonal moves break the queen's tour |

In order to improve the chances of success of a queen's tour, which was necessarily limited to the board, I realised it would be best to use ± N/2, ± N/2, i.e. ± 4, ± 4 diagonal queen's moves as illustrated below. The advantage of such moves was that when they "left" the board *they always "re-entered" it in cells that the queen could once again access*, even if along an alternative diagonal.

± N/2, ± N/2 diagonal moves never break the queen's tour |

In order to optimise the ``convergence" effect of these ± N/2, ± N/2 i.e. ± 4, ± 4 diagonals, they should be used once every two moves.

Despite the inconvenience of their spreading beyond the edges of the board, a large use of ± 2, ± 2 diagonal moves often produced complete, though irregular bimagic queen's tours, such as the one below, observed on the limitless surface of a semi-bimagic torus:

An open queen's tour on a torus board |

Finally, on the 23rd July 2020, after revising the selection of immutable bimagic series, the following method proved to be successful:

The tables below are *doubly-symmetric* arrays of immutable bimagic series, specially created for the x and y coordinates of a suitable semi-bimagic torus. Arranged in groups of four, the eight colours that represent the x (file) and y (rank) coordinates can be freely attributed the values of 0 to 7:

Tables of coordinates for the Bimagic Queen's Tour Torus |

However, to construct a bimagic tour we need a regular sequence that satisfies the ±1 / ±1 diagonal constraints of regular queen's moves, and in order to make the queen's tour "converge" on the board we need to use as many ± 4, ± 4 diagonals as we can; the optimum being once every two moves. Additionally, we need to check that the x and y coordinates selected for the first eight positions of the queen will also allow for regular diagonal transitions between each quarter of N² at the moves 16-17, 32-33 and 48-49... Once these verifications are complete we can then use the approved coordinates to plot the successive positions of the queen on the semi-bimagic torus shown below, starting with the first position 1 at coordinates (0x, 0y) in the lower left-hand corner:

The Semi-Bimagic Torus Before Translation |

When constructed, the semi-bimagic torus appears to only contain a broken tour; but after a translation of the 8 x 8 board viewpoint, as shown below, a complete open bimagic queen's tour is revealed:

The First Bimagic Queen's Tour |

Displaying beautiful symmetries, this is apparently not only the first bimagic queen's tour, but also the first-known bimagic tour of any chess piece!

In each rank and file, the orthogonal total of the successively numbered positions of the queen is the magic constant S1 = 260, and (as can be verified in the squared version below), the orthogonal total of the *squares* of the successively numbered positions of the queen is the *bimagic constant* S2 = 11180.

The Squared Bimagic Queen's Tour |

Observing the bimagic queen's tour path we can see the symmetries of the four orthogonal moves (8 - 9, 24 - 25, 40 - 41, and 56 - 57). *Thirty-two out of the sixty-three queen's moves are **±4, **±4 diagonal*.

The Bimagic Queen's Tour Path |

It is probable that many other examples exist, and that these will include closed bimagic queen's tours. However, it is an open question as to whether or not a *diagonally* bimagic queen's tour can be found on an 8 x 8 board! *

For those who are interested, a PDF file of "A Bimagic Queen's Tour" can be downloaded here.

For those who are interested, a PDF file of "A Bimagic Queen's Tour" can be downloaded here.

On the 3rd August 2020, Walter Trump was already able to answer the "open question" that I had formulated in my conclusion! He ran a computer check on the complete set of bimagic 8x8 squares that he had previously found with Francis Gaspalou. Walter found that *on an 8x8 board there are no diagonally bimagic queen's tours *(or diagonally bimagic knight's tours for that matter, although it was already known that none of the 140 magic knight's tours were diagonally magic)*.* The longest possible queen's tour on a bimagic square of order-8 consists of 21 moves, as illustrated in his PDF file below:

On the 8th August 2020, testing a program that he had devised on the first bimagic queen's tour, Walter Trump found a second example, which turned out to be a complementary bimagic queen's tour. On the 11th August 2020, continuing to search with his program, Walter Trump was able to find a total of 44 closed and 62 open bimagic queen's tours!

Walter Trump conjectures that, up to symmetry, there are no further bimagic queen's tours to be found on an 8x8 board. The program searched within the semi-bimagic 8x8 squares which were found by Walter Trump and Francis Gaspalou in 2014. Essentially different means up to symmetry and permutations of rows and columns. Unique means up to symmetry. Considering that there are more than 715 quadrillion unique semi-bimagic squares of order-8, the 106 unique queen's tours are quite rare!

These different tours are now listed and indexed in “106 Bimagic Queen’s Tours on an 8x8 Board;” a paper co-authored with Walter Trump which is available below:

On the 9th August 2020 Greg Ross published an article entitled "A Bimagic Queen's Tour" in his excellent Futility Closet - An Idler's Miscellany of Compendious Amusements. Many thanks Greg!

On the 24th August 2020, Walter Trump created an excellent web page entitled “Closed Bimagic Queen’s Tours on an 8x8 Board” which provides some interesting additional information!

On the 4th September 2020, Bogdan Golunski published 510 semi-bimagic squares of order-8 with open bimagic queen's tours which were calculated using a program that he had devised. His list includes rotations and reflections of the previously-known 62 open bimagic queen's tours, and although no new tours have been found, his program is shown to yield good results!

In a German book co-authored with Hans Gruber, entitled "Schach als Sujet in den Künsten und der Wissenschaft", and published on the 1st April 2022, Joachim Brügge has included a chapter "Die erste Darstellung einer semi-bimagischen Damenwandering von William Walkington (2020)." The chapter relates the discovery of the first bimagic queen's tour, and also that of the other bimagic queen's tours on an 8x8 board which were later found by Walter Trump.

I am indebted, not only to Awani Kumar, for initially bringing the subject to our attention and for his appeal for a solution, but also to Joachim Brügge, for having had the idea of a bimagic queen's tour in the first place, and for his kind encouragements during my research.

My thanks also go to Dwane Campbell, for publishing his findings on immutable bimagic series; series which proved so useful in the search for the bimagic queen's tour. Dwane has informed me that Aale de Winkel was the first to recognize that component binary squares could be bimagic, the basis of immutable series; so my thanks to Dwane go indirectly to Aale as well.

I am also most grateful to Francis Gaspalou, for editing and sending our circle of magic square enthusiasts an *"Analysis of the 240 Immutable Series of order 8"* in 2018, and for sending me the full list of all 38 069 bimagic series of order-8 when I asked him for information about these in July this year.

]]>After the first discoveries of area magic squares, I decided to do some more exploring and search for examples of perimeter magic squares. I found that the latter do indeed exist, and that although these appear to be similar to linear area magic squares, their construction is quite different for two reasons: Depending on the slopes (and lengths) of the slanting dissection lines of a perimeter magic square, the hinge points of these lines are offset when compared with those of an area magic square. The total perimeter of a perimeter magic square results from the slopes (and lengths) of the slanting dissection lines, and not from the addition of the individual cell perimeters.

Unless stated otherwise, the perimeter magic squares illustrated here have been created by the author of this post. Although perimeter magic squares are shown to be different from area magic squares, they were deduced using Walter Trump's area magic square construction techniques.

For want of a better word, please note that these perimeter magic squares are

On the 25th February 2017, the first Linear Perimeter Magic Square (L-PMS) of Order-3 was constructed using AutoCAD:

This L-PMS is drawn to a precision of 2 decimal places for each of the 9 cell perimeters, and the resulting total perimeter of the square is therefore approximately 47.372 (before computer validation and optimisation). The coordinates of this square are available upon request.

Maybe the square perimeters of such L-PMS of order-3 are proportional to the regular consecutive number cell perimeter sequences that they display, and those amongst you with programming skills will be able to generate different similarly-constructed examples, and thus identify the relationship?

The following Semi-Orthogonal Linear Perimeter Magic Squares (L-PMS) of order-4 *have cell perimeter entries rounded to 6 digits*. Because their central slanted lines are not derived from Pythagorean triangles, the lengths of the adjoining cell perimeters are not finite integers, but irrational "infinite" numbers. The precision of the cell perimeters can be further increased, but with each increase in precision the cell entries and the corresponding magic sums become higher, and the dimensions have to be modified accordingly.

Constructed on the 23rd February 2017, the following L-PMS of order-4 has a rounded magic sum of S = 1936532:

The upper slanted line is derived from a Pythagorean triangle with long leg 24, short leg 7, and hypotenuse 25. The central slanted line is derived from a triangle with long leg 120, short leg 8, and hypotenuse 120.26637102698... The lower slanted line is derived from a Pythagorean triangle with long leg 40, short leg 9, and hypotenuse 41. The total sum of the 16 cell perimeters is 1936532 x 4 = 7746128. The average cell perimeter is 7746128 / 16 = 484133. The perimeter of the magic square is 120000 x 4 x 4 = 1920000. The coordinates of this square are available upon request. This perimeter magic square's cell values are voluntarily rounded to 6 digits. Greater accuracy is possible, but this will inevitably lead to very high entries. If more or less digits are used for the cell perimeter values, then the slanted lines will need vertical translations to take into account the revised magic sums, and this will imply new dimensions. There are some interesting relationships between the broken diagonals of this linear perimeter magic square (L-PMS), as shown in the following diagram:

Constructed on the 24th February 2017, the following L-PMS of order-4 has a rounded magic sum of S = 1930156:

The upper slanted line is derived from a Pythagorean triangle with long leg 1200, short leg 49, and hypotenuse 1201. The central slanted line is derived from a triangle with long leg 120, short leg 26.9, and hypotenuse 122.978087479... The lower slanted line is derived from a Pythagorean triangle with long leg 60, short leg 11, and hypotenuse 61. The total sum of the 16 cell perimeters is 1930156 x 4 = 7720624. The average cell perimeter is 7720624 / 16 = 482539. The perimeter of the magic square is 120000 x 4 x 4 = 1920000. The coordinates of this square are available upon request. This perimeter magic square's cell values are voluntarily rounded to 6 digits. Greater accuracy is possible, but this will inevitably lead to very high entries. If more or less digits are used for the cell perimeter values, then the slanted lines will need vertical translations to take into account the revised magic sums, and this will imply new dimensions. There are some interesting relationships between the broken diagonals of this linear perimeter magic square (L-PMS), as shown in the following diagram:

Perhaps a reader with programming skills will be tempted to find the first semi-orthogonal linear perimeter magic square of order-4 with all slanted diagonals derived from Pythagorean triangles, and therefore endowed with finite integer values for its cell entries?

For comparison with the above examples of linear perimeter magic squares with rounded cell entries, the following linear perimeter *semi-magic* squares of order-4 are each constructed with all three slanting lines derived from Pythagorean triangles. Their individual cell perimeters can thus be defined as integers, and the resulting cell entries and magic sums of these perimeter *semi-magic* squares are therefore rational and finite.

Constructed on the 24th February 2017, the following linear perimeter semi-magic square (L-PSMS) has a magic sum of S = 15492:

The upper slanted line is derived from a Pythagorean triangle with long leg 24, short leg 7, and hypotenuse 25. The central slanted line is derived from a Pythagorean triangle with long leg 480, short leg 31, and hypotenuse 481. The lower slanted line is derived from a Pythagorean triangle with long leg 40, short leg 9, and hypotenuse 41. The total sum of the 16 cell perimeters is 15492 x 4 = 61968. The average cell perimeter is 61968 / 16 = 3873. The perimeter of the semi-magic square is 960 x 4 x 4 = 15360. The coordinates of this square are available upon request. There are some interesting relationships between the broken diagonals of this linear perimeter semi-magic square (L-PSMS), as shown in the following diagram:

Also constructed on the 24th February 2017, the following linear perimeter semi-magic square (L-PSMS) has a magic sum of S = 970:

The upper slanted line is derived from a Pythagorean triangle with long leg 40, short leg 9, and hypotenuse 41. The central slanted line is derived from a Pythagorean triangle with long leg 60, short leg 11, and hypotenuse 61. The lower slanted line is derived from a Pythagorean triangle with long leg 24, short leg 7, and hypotenuse 25. The total sum of the 16 cell perimeters is 970 x 4 = 3880. The average cell perimeter is 3880 / 16 = 242.5. The perimeter of the semi-magic square is 60 x 4 x 4 = 960. The coordinates of this square are available upon request. There are some interesting relationships between the broken diagonals of this linear perimeter semi-magic square (L-PSMS), as shown in the following diagram:

If anyone wishes to contribute linear or strict geometrical constructions of other perimeter magic squares, then please send me the x and y coordinates of the cell intersections that define the perimeters correctly up to two or more decimal places. No prizes can be given, but the authors of pertinent solutions will, if they wish, have their solutions published here.

A new post on "Polyomino Area Magic Tori" (PAMT) can be found in this blog since the 7th June 2022. Perhaps these PAMT will suggest new possibilities for the exploration of polyomino perimeter magic squares (PPMS) and polyomino perimeter magic tori (PPMT)...*
*

]]>
Further to the discoveries of the area magic squares of order-3, the first linear area magic square of order-4 was found by Walter Trump on the 14th January 2017.

On the 15th January 2017, Walter Trump's computer program was able to produce thousands of 4x4 linear area magic squares. Meanwhile, he was joined by fellow mathematician and programmer Hans-Bernhard Meyer, who on the 16th January 2017, found a different type of 4x4 linear area magic square (L-AMS). From then on Walter and Hans-Bernhard worked in collaboration, and they have kindly authorised me to illustrate samples of their findings below:

The linear area magic squares (L-AMS) of order-4 with vertical or horizontal lines have not only interesting geometric properties, but also surprising arithmetic features that are even more exciting. On the 8th February Hans-Bernhard Meyer informed me that the number and the structure of the fourth-order L-AMS with 3 horizontal or vertical lines had been totally clarified by parameterisations with 3 parameters, although this was not yet the case for L-AMS with only 2 horizontal or vertical lines. For full details of the mathematics behind these area magic squares please refer to the related links at the foot of this page.

On the 12th February 2017, in an e-mail correspondence, I asked Hans-Bernhard Meyer if it was possible to create a L-AMS of order-4 that had a perpendicular intersection of slanted lines. He responded on the 13th February 2017, sending the following L-AMS:

Congratulations to Hans-Bernhard for this interesting square!

Concerning the possible existence of L-AMS of order-4 without vertical or horizontal lines, on the 12th February 2017 Hans-Bernhard informed me that this question was still quite difficult to resolve.

Many other types of fourth-order AMS are also possible, and the following squares are area magic interpretations of some of the classical Frénicle squares of order-4:

On the 12th February 2017, in an e-mail correspondence, I asked Hans-Bernhard Meyer if it was possible to create a L-AMS of order-4 that had a perpendicular intersection of slanted lines. He responded on the 13th February 2017, sending the following L-AMS:

Congratulations to Hans-Bernhard for this interesting square!

Concerning the possible existence of L-AMS of order-4 without vertical or horizontal lines, on the 12th February 2017 Hans-Bernhard informed me that this question was still quite difficult to resolve.

Many other types of fourth-order AMS are also possible, and the following squares are area magic interpretations of some of the classical Frénicle squares of order-4:

Since the 3rd February 2017, full details of the findings of Walter Trump can be found in the chapter "Area Magic Squares" of his website: Notes on Magic Squares and Cubes.

Since the 25th January 2017, full details of the findings of Hans-Bernhard Meyer can be found in the article "Observations on 4x4 Area Magic Squares with Vertical Lines" of his website: Math'-pages.

Since the 25th January 2017, "Area Magic Squares of Order-6" relates the first findings of area magic squares of the sixth-order.

Since the 13th January 2017, "Area Magic Squares and Tori of Order-3" relates the first findings of area magic squares of the third-order.

In the N° 487 2018 May issue of "Pour La Science" (the French edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "Les Carrés Magiques d'Aires."

In the December 2018 issue of "Spektrum der Wissenschaft" (a Springer Nature journal, and the German edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "FLÄCHENMAGISCHE QUADRATE."

In the N° 487 2018 May issue of "Pour La Science" (the French edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "Les Carrés Magiques d'Aires."

In the December 2018 issue of "Spektrum der Wissenschaft" (a Springer Nature journal, and the German edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "FLÄCHENMAGISCHE QUADRATE."

On the 21st May 2021, Yoshiaki Araki (面積魔方陣がテセレーションみたいな件
@alytile) tweeted several order-4 and order-3 solutions in a polyomino area magic square thread. These included (amongst others) an order-4 example constructed with 16 assemblies of 5 to 20 dominoes. For more information on polyomino area magic squares please check the links at the end of the article "Area Magic Squares of Order-3."

Since the 22nd June 2021, Inder Taneja has published a paper entitled "Creative Magic Squares: Area Representations" in which he explores polyomino area magic using perfect square magic sums.

A new post on "Polyomino Area Magic Tori" can be found in these pages since the 7th June 2022.

After my previous post "Area Magic Squares and Tori of Order-3," Walter Trump continued his computer research of the linear area magic squares of order-4, now joined in this venture by a fellow mathematician and programmer Hans-Bernhard Meyer.

Having already some insight into the linear area magic squares of the odd-order-3 and of the doubly-even-order-4, it was tempting to search for a singly-even-order example. I therefore decided to do some exploring, and on the 22nd January 2017, I found this first linear area magic square of order-6:

For the area magic square shown above, the magic constant of each row, column and main diagonal is 1296 (6

From then on using a computer program to find many more examples, on the 26th January 2017, Hans-Bernhard Meyer drew to my attention a new linear area magic square with not only parallel vertical lines, but also a central horizontal line. He kindly authorised me to publish this example illustrated below:

The magic constant of this linear area magic square (L-AMS) is 474, and the total areas are 2,844. The last digits of the first and last numbers of the non-parallel rows are either 4 for the even number sequences, or 9 for the odd number sequences. I noticed that in addition to the central horizontal line, there was also a pair of broken magic diagonals which would enable an easy transformation, as illustrated below:

It can be seen that these are two essentially different L-AMS viewpoints of a same area magic cylinder. Please note that there are also another four *area semi-magic parallelograms* displayed on this cylinder (with cells 44, 124, 144, or 19 in their top left corners).

On the 29th January 2017, Hans-Bernhard Meyer informed me that the following can be proved:

Every 6x6 L-AMS with magic sum S and horizontal centre line has the property:

x04+x11+x18+x19+x26+x33 = x03+x08+x13+x24+x29+x34 = S

and conversely, every 6x6 L-AMS with magic sum S and

x04+x11+x18+x19+x26+x33 = x03+x08+x13+x24+x29+x34 = S

has a horizontal centre line.

He stated that there are many examples of such L-AMS and strongly supposed that the example illustrated above had the lowest possible magic sum. He was able to confirm that examples with 2 horizontal lines do not exist.

On the same day, Hans-Bernhard also informed me that in the order-6, the L-AMS with the lowest possible magic sum had a magic constant of 402. He had found that there were several of these L-AMS, and kindly sent me details of an example, which I have illustrated below:

On the 29th January 2017, Hans-Bernhard Meyer informed me that the following can be proved:

Every 6x6 L-AMS with magic sum S and horizontal centre line has the property:

x04+x11+x18+x19+x26+x33 = x03+x08+x13+x24+x29+x34 = S

and conversely, every 6x6 L-AMS with magic sum S and

x04+x11+x18+x19+x26+x33 = x03+x08+x13+x24+x29+x34 = S

has a horizontal centre line.

He stated that there are many examples of such L-AMS and strongly supposed that the example illustrated above had the lowest possible magic sum. He was able to confirm that examples with 2 horizontal lines do not exist.

On the same day, Hans-Bernhard also informed me that in the order-6, the L-AMS with the lowest possible magic sum had a magic constant of 402. He had found that there were several of these L-AMS, and kindly sent me details of an example, which I have illustrated below:

The total areas of this square are 2,412. The last digits of the first and last numbers of the non-parallel rows are either 2 for the even number sequences, or 7 for the odd number sequences. The even number sequence of the bottom row begins with 12, which Hans-Bernhard informs me is the lowest possible vertex area for L-AMS (linear area magic squares) of order-6.

These are only the first of an infinite number of examples that remain to be discovered, and I therefore expect to be regularly publishing updates to this post!

__Related links__

A former post Area Magic Squares and Tori of Order-3 can be found in these pages since the 13th January 2017.

On the 25th January 2017, Hans-Bernhard Meyer published an article entitled Observations on 4x4 Area Magic Squares with vertical lines in his website: Math'-pages.

On the 3rd February 2017, Walter Trump published a chapter entitled Area Magic Squares in his website: Notes on Magic Squares. This chapter includes many analyses and examples of area magic squares of the third and fourth-orders.

Since the 8th February 2017, "Area Magic Squares of Order-4" relates the first findings of area magic squares of the fourth-order.

In the N° 487 2018 May issue of "Pour La Science" (the French edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "Les Carrés Magiques d'Aires."

In the December 2018 issue of "Spektrum der Wissenschaft" (a Springer Nature journal, and the German edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "FLÄCHENMAGISCHE QUADRATE."

]]>A former post Area Magic Squares and Tori of Order-3 can be found in these pages since the 13th January 2017.

On the 25th January 2017, Hans-Bernhard Meyer published an article entitled Observations on 4x4 Area Magic Squares with vertical lines in his website: Math'-pages.

On the 3rd February 2017, Walter Trump published a chapter entitled Area Magic Squares in his website: Notes on Magic Squares. This chapter includes many analyses and examples of area magic squares of the third and fourth-orders.

Since the 8th February 2017, "Area Magic Squares of Order-4" relates the first findings of area magic squares of the fourth-order.

In the N° 487 2018 May issue of "Pour La Science" (the French edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "Les Carrés Magiques d'Aires."

In the December 2018 issue of "Spektrum der Wissenschaft" (a Springer Nature journal, and the German edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "FLÄCHENMAGISCHE QUADRATE."

On the 21st May 2021, Yoshiaki Araki (面積魔方陣がテセレーションみたいな件
@alytile) tweeted several order-4 and order-3 solutions in a polyomino area magic square thread.
These included (amongst others) an order-4 example constructed with 16
assemblies of 5 to 20 dominoes. For more information on polyomino area
magic squares please check the links at the end of the article "Area Magic Squares of Order-3."

Since the 22nd June 2021, Inder Taneja has published a paper entitled "Creative Magic Squares: Area Representations" in which he explores polyomino area magic using perfect square magic sums.

A new post on "Polyomino Area Magic Tori" can be found in these pages since the 7th June 2022.

In December 2016 I made a sketch design of an area magic square to illustrate a seasonal greetings card for 2017. Using the traditional order-3 magic square, I attempted to use continuous straight lines for the area cell borders. I quickly realised that such an assembly was impossible to achieve, but that with approximate areas I could nevertheless make a pleasing graphic pattern:

Sending this image with my best wishes to a circle of magic square enthusiasts on the 30th December 2016, I added the following postscript: "The areas are approximate, and I don't know if it is possible to obtain the correct areas with 2 vertically slanted straight lines through the square. Perhaps someone will be able to work this out in 2017?"

Lee Sallows was the first to respond on the 4th and 5th January 2017, proposing an approach based on his previous work on geomagic squares. Developing the ideas that can be found in the figure 4.1 on page 5 of his book *"Geometric Magic Squares: A Challenging New Twist Using Colored Shapes Instead of Numbers,"* his area square is illustrated below:

Lee Sallows points out that, when comparing the initial square of his book with this new puzzle square, "in many (but not all) cases adjacent piece outlines have been made to complement each other. For every gain in area at one position there is an identical loss of area at another. In this way the areas of all 9 pieces remain as they were, so that the square remains a magical dissection." Thank you for your work Lee! I particularly like your puzzle solution.

Meanwhile, I was wondering if a solution could be found for a third-order area magic torus, that is to say, one where the area cell intersections would meet correctly when wrapped. With this purpose in mind I set the following new rules for myself:

1/ Each cell will have an area that corresponds with its number from 1 to 9. The total areas will therefore be 45.

2/ The connections between the cells will remain unchanged, but as these cells will be of different sizes, the resulting latitudes, longitudes and diagonals will not necessarily be straight lines. The initially square cells will become regular or irregular quadrilaterals, (excluding complex quadrilaterals).

3/ The distances measured orthogonally between opposite sides of the resulting magic or semi-magic square viewpoints will always be √45, (the circumferences of the magic torus).

This third rule took into account the flattened square that we are accustomed to looking at, but it also implied that the torus would degenerate into a sphere... With these new constraints, on the 6th January 2017 I was able to come up with the area magic torus illustrated below:

These extended best wishes, expressed through the complete set of third-order magic and semi-magic squares, include messages that reflect the multiple nationalities of the members of the magic square circle. The magic square viewpoint of the torus is placed at the top left. The patterns still need adjusting in order to achieve the correct areas, but as there is more flexibility, I am fairly confident that at least one accurate solution can be found.

On the 6th January 2017 Walter Trump also sent us a new area magic design that used 4 continuous straight dissection lines. Although this schema could not be adapted to the number sequence 1 to 9, it was area magic:

Inder Jeet Taneja then suggested that the problem with the sequence 1 to 9 was that the total areas did not add up to a perfect square area. He proposed that instead of using the numbers 1 to 9, perhaps we should try using the numbers 5 + 6 + ... + 13 = 81 = 9² ?

On the same day, following Inder's suggestion, Walter Trump amazed us all with this first third-order linear area magic square! :

Bravo Walter for your achievement following Inder's suggestion! I hardly dared to believe that such a simple area pattern could produce a magic square!

Having other commitments to honour, Walter Trump then requested that somebody else searched for solutions using lower sequences. No other volunteers came forward, so I decided to look for myself, and came up with this second linear area magic square of the third-order using the sequence 3 to 11:

As I am not a programmer, I used Autocad to construct this linear area magic square manually. The areas of this provisional square are therefore only accurate up to two decimal places, before computer verification and area optimisation.

In the meantime Walter Trump continued writing a computer program to handle the equations and search iteratively for solutions. This was able to give orthogonal coordinates having at least 14 decimals after the commas. At first unable to solve the non-linear equations explicitly, Walter progressively improved his approach, and on the 8th January 2017 he sent us a message stating that for some sequences there were two solutions, and that the lowest number sequence ranged from 2 to 10!

On the 10th January 2017, Walter Trump created an algorithm for "Third-Order Linear Area Magic," which he has kindly authorised me to publish below. This explains the method of construction and gives the reason why two solutions exist for most, but not all of the sequences studied:

Using the coordinates generated by Walter Trump's computer program, both solutions of the 5 lowest sequences of the third-order linear area magic squares are illustrated in square form below. It will be seen that the 1 to 9 sequence fails, and the linear area magic squares are only possible from the sequence 2 to 10 upwards (when the continuous straight dissection lines produce no more than 4 intersections inside the squares):

During this time my first question concerning the sequence 1 to 9 remained unanswered. So, on the 9th January 2017, I created the first area magic square that used the numbers of the classical magic square of order-3, as illustrated below:

More recreational than mathematical, this "Mont Saint Michel" area magic square demonstrates that it is possible to use 2 continuous straight dissection lines - thus resolving the initial 2017 greetings card challenge. The design is an area magic square, because all of the cells are quadrilaterals (even if near triangular for the area 1), and all of their connections are preserved.

Inder Taneja has since followed up on his initial suggestions for perfect square sequences. On the 11th February 2017 he published some very interesting results in his new paper "Magic Squares with Perfect Square Number Sums."

This is an ongoing story, and there will probably be new developments in the near future. Apart from Inder Taneja, Lee Sallows and Walter Trump who I have already mentioned, I also wish to thank the other participants: These include Craig Knecht for his kind encouragements, Miguel Angel Amela for his encouragements and suggestions concerning accuracy, Dwane Campbell for his encouragements and suggestions, and last but not least, Francis Gaspalou for his encouragements and interventions: For the area magic squares of the third-order, Francis Gaspalou wrote the equations of the 4 straight lines using 4 parameters which were the 4 slopes of those lines. He found the exact conditions which had to be fulfilled by these 4 parameters in order to obtain a solution. Unfortunately it was not possible to solve the system and write explicit formulae for the solutions. Francis was nevertheless able to check that the two solutions per sequence found by Walter Trump's algorithm fulfilled the conditions. Thus by using this different method, Francis was able to confirm the validity of Walter's computer results. Francis Gaspalou has kindly authorised me to publish his equations here:

On the 13th January 2017 Bob Ziff intervened, stating that if we agree that the system of equations is soluble, then it is soluble to any number of digits, and proved his point by calculating the slopes with 1000 digits, using the professional software Mathematica and the four equations of Francis Gaspalou.

Also, on the 13th January 2017, Walter Trump created a third-order linear area nearly-magic square with integer coordinates. This square is semi-magic with a magic constant of 940800! Walter points out that when you divide all the areas by 140, then the coordinates are no longer integers, but irrational numbers that can be exactly described using fractions and square roots. Walter has kindly authorised me to publish his findings below:

On the 13th January 2017 Bob Ziff intervened, stating that if we agree that the system of equations is soluble, then it is soluble to any number of digits, and proved his point by calculating the slopes with 1000 digits, using the professional software Mathematica and the four equations of Francis Gaspalou.

Also, on the 13th January 2017, Walter Trump created a third-order linear area nearly-magic square with integer coordinates. This square is semi-magic with a magic constant of 940800! Walter points out that when you divide all the areas by 140, then the coordinates are no longer integers, but irrational numbers that can be exactly described using fractions and square roots. Walter has kindly authorised me to publish his findings below:

Francis Gaspalou has suggested that the next step might be to find a fourth-order area magic square with 6 continuous straight dissection lines, or even a concentric fifth-order area magic square. Why not imagine a contest, open to all, so as to find higher-order solutions of an equal mathematical importance to that of the third-order linear area magic squares?

If anyone wishes to contribute linear or other strict geometrical constructions of higher-order area magic squares, then please send me the x and y coordinates of the cell intersections that define the areas correctly up to two or more decimals after the commas. No prizes can be given, but the authors of pertinent solutions will, if they wish, have their solutions published here, and thus be able to bask in the reflected glory...

###
__Latest developments__

On the 19th January 2017 Greg Ross published an article on the subject of Area Magic Squares in his Futility Closet - An Idler's Miscellany of Compendious Amusements. This article has since been relayed by Reddit Mathpics, Simplementenumeros, Prime Puzzles, Thermally Stressed Dairy Cows, and by Daily Design Stream, amongst others.

On the 20th January 2017 I was aware that Walter Trump, (now joined by Hans-Bernhard Meyer), had already found that there are no cases of order-4 area magic squares with sequences of consecutive numbers. So I therefore began searching for non-consecutive number sequences that might also yield solutions in order-3, and produced the draft linear area magic square illustrated below:

I sent this square to Walter Trump and the other members of the magic square circle asking if it could be verified using a computer program. Walter was quick to react, sending me precise coordinates for two solutions. Drawn-up using his coordinates, the two versions of this square are illustrated below:

On the 3rd February 2017, responding to a Prime Puzzle challenge, Jan van Delden contributed the following palprime linear area magic square solution:

The palindromic primes used here are symmetrical numbers that remain the same when their digits are reversed. The magic sum of these 11-digit palindromic primes is 377,024,295,63. This area magic square is the conversion of a square that was originally sent by Carlos Rivera and Jaime Ayala to Harvey Heinz on the 22nd May 1999. Jan van Delden mentions that the direction coefficients of the lines are indicated in white, and that the quadrilaterals are numbered in the style of Francis Gaspalou. Full details of Jan van Delden's approach to the equations can be found in Prime Puzzles.

On the 19th January 2017 Greg Ross published an article on the subject of Area Magic Squares in his Futility Closet - An Idler's Miscellany of Compendious Amusements. This article has since been relayed by Reddit Mathpics, Simplementenumeros, Prime Puzzles, Thermally Stressed Dairy Cows, and by Daily Design Stream, amongst others.

On the 20th January 2017 I was aware that Walter Trump, (now joined by Hans-Bernhard Meyer), had already found that there are no cases of order-4 area magic squares with sequences of consecutive numbers. So I therefore began searching for non-consecutive number sequences that might also yield solutions in order-3, and produced the draft linear area magic square illustrated below:

I sent this square to Walter Trump and the other members of the magic square circle asking if it could be verified using a computer program. Walter was quick to react, sending me precise coordinates for two solutions. Drawn-up using his coordinates, the two versions of this square are illustrated below:

On the 3rd February 2017, responding to a Prime Puzzle challenge, Jan van Delden contributed the following palprime linear area magic square solution:

The palindromic primes used here are symmetrical numbers that remain the same when their digits are reversed. The magic sum of these 11-digit palindromic primes is 377,024,295,63. This area magic square is the conversion of a square that was originally sent by Carlos Rivera and Jaime Ayala to Harvey Heinz on the 22nd May 1999. Jan van Delden mentions that the direction coefficients of the lines are indicated in white, and that the quadrilaterals are numbered in the style of Francis Gaspalou. Full details of Jan van Delden's approach to the equations can be found in Prime Puzzles.

On the 4th February 2017 Jan van Delden contributed this second prime linear area magic square to Prime Puzzles:

Please note, that since the 5th March 2017, Jan has published a new paper entitled "Area Magic Squares of Order 3," in which he extends his findings.

Seemingly, the number of order-3 linear area squares is infinite!

A new post on Area Magic Squares of Order-6 can be found in these pages since the 25th January 2017.

On the same day Hans-Bernhard Meyer published an article entitled Observations on 4x4 Area Magic Squares with vertical lines in his website: Math'-pages.

Since the 3rd February 2017, Walter Trump has published a chapter entitled Area Magic Squares in his website: Notes on Magic Squares. This chapter includes many analyses and examples of area magic squares of the third and fourth-orders.

Since the 8th February 2017, "Area Magic Squares of Order-4" relates the first findings of area magic squares of the fourth-order.

Since the 11th February 2017, Inder Jeet Taneja, inspired by the research in area magic squares, has published a new paper entitled "Magic Squares with Perfect Square Number Sums."

Since the 5th March 2017, Jan van Delden has published a paper entitled "Area Magic Squares of Order 3" in which he presents an improved algorithm. His work also includes new findings in area

Since the 8th March 2017, following a tweet by Simon Gregg, Microsiervos published an article entitled "Cuadrados de áreas mágicas." This article has been relayed by Inoreader amongst others.

Since the 22nd March 2017, writing for EL PAÍS, Miguel Ángel Morales has included the subject of area magic squares in an article entitled "No solo de números consecutivos vive el cuadrado mágico."

Since the 19th April 2017, writing for "Geogebra," Georg Wengler has included an article entitled "Magisches Flächen-Quadrat."

In the N° 487 2018 May issue of "Pour La Science" (the French edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "Les Carrés Magiques d'Aires."

In the December 2018 issue of "Spektrum der Wissenschaft" (a Springer Nature journal, and the German edition of Scientific American), Professor Jean-Paul Delahaye has written an article entitled "FLÄCHENMAGISCHE QUADRATE."

Since the 25th March 2019, Akehiko Takahashi, the webmaster of "Math Dojo," includes Area Magic Squares in his article entitled "Magic Squares and Beyond."

On the 17th July 2019, a discussion in Japanese, about Area Magic Squares, began on the forum of "Japanese Traditional Mathematical Calculator."

Circa 2019, on the French recreational mathematics site "Diophante," Michel Lafond published the first linear area magic square in a problem n° B138, entitled "Carré magique géométrique," and asked mathematicians to prove its existence. Several solutions are proposed.

On the 1st January 2020, (exactly three years after the original area magic square greetings card!), an article about area magic patchwork entitled "Flächenmagische Quadrate - aus Stoff," was published by the German blogger "Siebensachen-zum-Selbermachen."

On the 12th January 2020, the recreational mathematician Ed Pegg Jr., published a "Magic Square with areas" on his site "Mathpuzzle."

On the 12th January 2020, the recreational mathematician Ed Pegg Jr., published a "Magic Square with areas" on his site "Mathpuzzle."

On the 20th May 2021, Morita Mizusumashi (盛田みずすまし
@nosiika) tweeted a nice polyomino area magic square of order-3 constructed with 9 assemblies of *5* to *13 monominoes*.

On the 21st May 2021, Yoshiaki Araki (面積魔方陣がテセレーションみたいな件
@alytile) tweeted several order-4 and order-3 solutions in a polyomino area magic square thread. These included (amongst others) an order-4 example constructed with 16 assemblies of 5 to 20 dominoes.

On the 24th May 2021, Yoshiaki Araki then tweeted an order-3 polyomino area magic square constructed using assemblies of 1 to 9 same-shaped pentominoes! Edo Timmermans, the author of this beautiful square, had apparently been inspired by Yoshiaki Araki's previous posts!

Since the 22nd June 2021, Inder Taneja has published a paper entitled "Creative Magic Squares: Area Representations" in which he explores polyomino area magic using perfect square magic sums.

A new post on "Polyomino Area Magic Tori" can be found in these pages since the 7th June 2022.

]]>Published by Carlos Rivera on his website "Prime Puzzles" at the end of May 2019, the Puzzle 391 "9 dots and 8 lines graph" has stimulated a lot of conversation between magic square enthusiasts, and has inspired several ongoing experiments. It has also prompted me to look into the magic line graphs of magic tori, and to search for cases having a maximum number of magic line intersections.

The extra-magic tori of orders 3 and 4 are here presented in detail, together with some further observations on the extra-magic tori of orders 5 to 8, the eighth-order being particularly significant for those who are interested in chess moves...

Extra-Magic can be defined as the presence of any magic line intersections on Magic Tori that are not usually taken into account in the study of Magic Squares.

For example, the central intersection of the magic diagonals of a magic square takes place *between the number cells of even-order* magic squares, and *over a number cell of odd-order* magic squares. When we begin to study the wrap-around characteristics of the magic tori that display the magic squares, we notice that a same pair of magic diagonals produces two intersections on a magic torus. In even-orders the two intersections of a same pair of magic diagonals always produce two distinct magic squares on the magic torus. In odd-orders the two intersections of a same pair of magic diagonals only produce a single magic square, because the second intersection (at the opposite side of the torus), occurs between number cells, and cannot be at the centre of a second magic square.

Therefore, in even-orders, the cases where the intersections of magic diagonals occur over number nodes (2 magic orthogonals + 2 magic diagonals), are often ignored, because these never produce magic squares. However, when looking at the limitless surface of a magic torus, *which has no centre*, a concentration of magic lines *that intersect over a number in even-orders* becomes a very interesting feature.

A second example of extra-magic is the presence of knight move magic diagonals that occur on some magic tori (and on the magic squares that the magic tori display). The present study takes into account the eight traditional chess knight moves of (2, 1), (1, 2), (-1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -2) and (2, -1). Other fairy chess "leaper" move magic diagonals also exist, such as the Camel (3, 1) and Zebra (3, 2). However, in order to simplify the graphics of the magic line graphs, only traditional knight moves are considered here. It should be noted that knight move diagonals can never intersect at the centres of even-order magic squares, but they contribute to the weave of magic lines and are always an interesting feature when we observe the limitless surface of magic tori.

Because knight move magic diagonals connect *n* numbers (where *n* is the order of the magic torus), and one of the knight's steps has a length of *2*, the diagonal has to make *2* loops round the magic torus. In lower-*n*-orders, the reduced number *n* of the necessary knight moves results in a proximity of the *2* loops, which in turn allows the knight to use *2 magic diagonal paths to connect a same series of numbers! *Please refer to "Knight Move Magic Diagonals on Magic Tori" to see why knight move magic diagonals can sometimes have 2 paths for a same series of numbers.

with 4 Knight Move Magic Diagonals

It came as quite a surprise to find that there are 4 knight move magic diagonals on the traditional Lo Shu magic square!

Lo-Shu Extra-Magic Torus T3 of Order-3 |

The extra**-**magic of the knight move diagonals produces multiple intersections of magic lines over the number 5. After the addition of the knight move magic lines that sum up to the magic constant *MC = 15*, the maximum number of magic lines at a nodal intersection is 8, (at number node 5). The total number of magic lines on the "Lo Shu" Magic Torus T3 is therefore 6 orthogonals (in red) + 2 classic diagonals (in red) + 4 knight move diagonals (in blue) = 12 magic lines.

To the best of my knowledge, the knight move magic diagonals of the Lo Shu have not been mentioned in magic square literature before, but should I be mistaken, can you please advise me, and I will cite any previous authors accordingly. It should be noted that on page 37 of his 1877 paper "On the General Properties of Nasik Squares", Andrew Hollingworth Frost describes (1, 2) "pathlets" of construction for a 3 x 3 square, *but his diagram E is not the traditional Lo Shu. *

Legend of the Extra-Magic Line Graphs of Magic and Semi-Magic Tori |

More details of the Magic Tori of Order-4 *(and of the Frénicle indexed squares that these tori display)* can be found in "The Table of Fourth-Order Magic Tori" *(revised on the 20th June 2019 to include the details of the Extra-Magic Tori)*, and in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The total numbers of Extra-Magic Tori (with extra-magic intersections and / or knight move magic diagonals), that were previously given at the end of the "The Table of Fourth-Order Magic Tori", remain unchanged. Those concerning uniquely the knight move magic diagonals are now listed for more convenient reference in "Knight Move Magic Diagonals in Magic Tori and Magic Squares of Order-4". Miguel Angel Amela has since confirmed my findings although we have a slightly different approach to the count of the double-slanting same-series knight move magic diagonals. For the sake of symmetry, and for comparison with higher-orders, I have chosen to count both of the same-series slants of knight move magic diagonals for the orders*-n* where *n < 6*.

Only consecutively numbered magic tori (*with numbers from 1 to n²*) are studied here, and consequently, for *order-4*, the magic constant *MC = 34*.

with 16 Extra-Magic Intersections

Extra-Magic Pandiagonal Torus T4.198 of Order-4 |

Please note that here the torus viewpoint is deliberately centred on the number node 6. This is not a standard magic square presentation, but it means to show that on the limitless surface of the magic torus there is no fixed centre. The maximum number of magic lines at intersections (at each number node) is 4. The total number of magic lines on the order-4 pandiagonal torus index n° T4.198 is 8 orthogonals (in red) + 8 classic diagonals (in red) = 16 magic lines.

There are a total of 3 Pandiagonal Tori type T4.01 that each display 16 pandiagonal squares. The corresponding 48 pandiagonal squares are Dudeney type I.

with 8 Knight Move Magic Diagonals

Extra-Magic Semi-Pandiagonal Torus T4.080 of Order-4 |

The magic line graph of the semi-pandiagonal torus index n° T4.080 (type n° T4.02.3.02) is here presented as seen from the magic square viewpoint Frénicle n° 33. After the addition of knight move magic diagonals (1, 10, 16, 7) + (1, 7, 16, 10) + (11, 4, 6, 13) + (13, 11, 4, 6) + (8, 15, 9, 2) + (8, 2, 9, 15) + (14, 5, 3, 12) + (14, 12, 3, 5) the maximum number of magic lines at a nodal intersection is 5 (at each number node). The total number of magic lines on the magic torus is 8 orthogonals (in red) + 4 classic diagonals (in red) + 8 knight move diagonals (in blue) = 20 magic lines.

All of the 12 Semi-Pandiagonal Tori Type T4.02.3 have 8 knight move magic diagonals, and each torus displays 8 semi-pandiagonal squares. The corresponding 96 semi-pandiagonal squares are Dudeney type V.

with 4 Extra-Magic Intersections and 4 Knight Move Magic Diagonals

Extra-Magic Partially Pandiagonal Torus T4.186 of Order-4 |

The magic line graph of the partially pandiagonal torus index n° T4.186 (type n° T4.03.1.3) is here presented as seen from the magic square viewpoint Frénicle n° 793. After the addition of the knight move magic diagonals (3, 12, 13, 6) + (3, 6, 13, 12) + (4, 5, 14, 11) + (4, 11, 14, 5), the maximum number of magic lines at a nodal intersection is 6 (at extra-magic number nodes 4, 5, 13, 12). The total number of magic lines on the magic torus is 8 orthogonals (in red) + 4 classic diagonals (in red) + 4 knight move diagonals (in blue) = 16 magic lines.

2 out of the 6 Partially Pandiagonal Tori Type T4.03.1 have similar knight move magic diagonals (index n° T4.186 and T4.246), and both tori display 4 partially pandiagonal squares. The corresponding 8 partially pandiagonal squares are Dudeney Type VI.

with 4 Extra-Magic Intersections and 4 Knight Move Magic Diagonals

Extra-Magic Partially Pandiagonal Torus T4.060 of Order-4 |

The magic line graph of the partially pandiagonal torus index n° T4.060 (type n° T4.03.2.07) is here presented as seen from the magic square viewpoint Frénicle n° 472. After the addition of the knight move magic diagonals (1, 8, 14, 11) + (1, 11, 14, 8) + (5, 4, 10, 15) + (5, 15, 10, 4), the maximum number of magic lines at a nodal intersection is 6 (at extra-magic number nodes 4, 5, 11, 14). The total number of magic lines on the magic torus is 8 orthogonals (in red) + 4 classic diagonals (in red) + 4 knight move diagonals (in blue) = 16 magic lines.

All of the 12 Partially Pandiagonal Tori Type T4.03.2 have similar knight move magic diagonals, and each torus displays 4 partially pandiagonal squares. The corresponding 48 partially pandiagonal squares, which are evenly mixed on each torus, are either Dudeney type VII (24 squares), or Dudeney type IX (24 squares).

with 4 Extra-Magic Intersections and 4 Knight Move Magic Diagonals

Extra-Magic Partially Pandiagonal Torus T4.187 of Order-4 |

The magic line graph of the partially pandiagonal torus index n° T4.187 (type n° T4.03.3.2) is here presented as seen from the magic square viewpoint Frénicle n° 202. After the addition of the knight move magic diagonals (3, 4, 13, 14) + (3, 14, 13, 4) + (5, 6, 11, 12) + (5, 12, 11, 6) the maximum number of magic lines at a nodal intersection is 6 (at extra-magic number nodes 4, 5, 12, 13). The total number of magic lines on the magic torus is 8 orthogonals (in red) + 4 classic diagonals (in red) + 4 knight move diagonals (in blue) = 16 magic lines.

Both of the Partially Pandiagonal Tori Type T4.03.3 have similar knight move magic diagonals, and each torus displays 4 partially pandiagonal squares. The corresponding 8 partially pandiagonal squares are Dudeney type XI.

with 2 Extra-Magic Intersections

Extra-Magic Partially Pandiagonal Torus T4.096 of Order-4 |

The magic line graph of the partially pandiagonal torus index n° T4.096 (type n° T4.04.1) is here presented as seen from the magic square viewpoint Frénicle n° 40. The maximum number of lines at intersections (over the extra-magic number nodes 1 and 13) is 4. The total number of magic lines on the magic torus is 8 orthogonals (in red) + 3 classic diagonals (in red) = 11 magic lines.

The 4 Partially Pandiagonal Tori Type T4.04 have 2 extra-magic intersections, and each of these tori displays 2 partially pandiagonal squares. The corresponding 8 partially pandiagonal squares, which are evenly mixed on each torus, are either Dudeney type VIII (4 squares) or Dudeney type X (4 squares). The tori type T4.04 were initially found thanks to Walter Trump's computing skills.

with 4 Knight Move Magic Diagonals

Extra-Magic Basic Magic Torus T4.028 of Order-4 |

The magic line graph of the magic torus index n° T4.028 (type N° T4.05.1.78) is here presented as seen from the magic square viewpoint Frénicle n° 502. After the addition of knight move magic diagonals (1, 5, 13, 15) + (1, 15, 13, 5) + (2, 4, 12, 16) + (2, 16, 12, 4) the maximum number of magic lines at a nodal intersection is 5 (at number nodes 4, 5, 12, 13). The total number of magic lines on the magic torus is 8 orthogonals (in red) + 2 classic diagonals (in red) + 4 knight move diagonals (in blue) = 14 magic lines.

4 out of the 92 Basic Magic Tori Type T4.05.1 have 4 knight move magic diagonals (index n° T4.028, T4.102, T4.153 and T4.203), and each of these tori displays 2 basic magic squares. The corresponding 8 basic magic squares are Dudeney type VI.

with 8 Knight Move Magic Diagonals

Extra-Magic Basic Magic Torus T4.071 of Order-4 |

The magic line graph of the magic torus index n° T4.071 (type n° T4.05.1.35) is here presented as seen from the magic square viewpoint Frénicle n° 134. After the addition of knight move magic diagonals (1, 6, 15, 12) + (1, 12, 15, 6) + (2, 5, 16, 11) + (2, 11, 16, 5)+ (3, 8, 13, 10) + (3, 10, 13, 8) + (4, 7,14, 9) + (4, 9, 14, 7) the maximum number of magic lines at a nodal intersection is 5 (at number nodes 1, 4, 6, 8, 9, 11, 13, 16). The total number of magic lines on the magic torus is 8 orthogonals (in red) + 2 classic diagonals (in red) + 8 knight move diagonals (in blue) = 18 magic lines.

6 out of the 92 Basic Magic Tori Type T4.05.1 have 8 knight move magic diagonals (index n° T4.071, T4.076, T4.134, T4.140, T4.227 and T4.232), and each of these tori displays 2 basic magic squares. The corresponding 12 basic magic squares are Dudeney type VI.

with 4 Knight Move Magic Diagonals

Extra-Magic Basic Magic Torus T4.110 of Order-4 |

The magic line graph of the magic torus index n° T4.110 (type n° T5.05.2.29) is here presented as seen from the magic square viewpoint Frénicle n° 568. After the addition of knight move magic diagonals (1, 9, 11, 13) + (1, 13, 11, 9) + (4, 6, 8, 16) + (4, 16, 8, 6) the maximum number of magic lines at a nodal intersection is 5 (at number nodes 6, 8, 11, 13). The total number of magic lines on the magic torus is 8 orthogonals (in red) + 2 classic diagonals (in red) + 4 knight move diagonals (in blue) = 14 magic lines.

2 out of the 32 Basic Magic Tori Type T4.05.2 have 4 knight move magic diagonals (index n° T4.110 and T4.116), and both of these tori display 2 basic magic squares. The corresponding 4 basic magic squares, which are evenly mixed on each torus, are either Dudeney type VII (2 squares) or Dudeney type IX (2 squares).

More details of the Semi-Magic Tori of Order-4 can be found in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

Only consecutively numbered semi-magic tori (*with numbers from 1 to n²*) are studied here, and consequently, for the *order-4*, the magic constant *MC = 34*.

*Extra-magic semi-magic tori of order-4 have neither been fully investigated nor precisely counted, and the examples shown below are a small selection of some of the different varieties.*

with 8 Knight Move Magic Diagonals

Extra-Magic Semi-Magic Torus Type n° T4.06.0A of Order-4 |

The magic line graph of the semi-magic torus type n° T4.06.0A is based on a semi-magic torus that was first identified by Walter Trump. After the addition of knight move magic diagonals (1, 6, 16, 11) + (1, 11, 16, 6) + (2, 3, 15, 14) + (2, 14, 15, 3) + (4, 7, 13, 10) + (4, 10, 13, 7) + (5, 8, 12, 9) + (5, 9, 12, 8) the maximum number of magic lines at a nodal intersection is 5 (at all number nodes). The total number of magic lines on the semi-magic torus is 8 orthogonals (in red) + 4 classic diagonals (in red) + 8 knight move diagonals (in blue) = 20 magic lines.

with 4 Knight Move Magic Diagonals

Extra-Magic Semi-Magic Torus Type n° T4.07.0A of Order-4 |

The magic line graph of the semi-magic torus type n° T4.07.0A is based on a semi-magic torus that was first identified by Walter Trump. After the addition of knight move magic diagonals (1, 4, 13, 16) + (1, 16, 13, 4) + (2, 5, 12, 15) + (2, 15, 12, 5) the maximum number of magic lines at a nodal intersection is 5 (at number nodes 4, 5, 12, 13). The total number of magic lines on the semi-magic torus is 8 orthogonals (in red) + 2 classic diagonals (in red) + 4 knight move diagonals (in blue) = 14 magic lines.

In the Order-5, studies of knight move magic diagonals have previously been made by Miguel Angel Amela in his paper "Magic and Semi-Magic Squares of Order 5 - Rectangular Pandiagonality" dated Spring 2011: Here he has identified 144 magic squares and 3456 semi-magic squares with "rectangular pandiagonality", and he has kindly authorised me to use one of these for an illustration of this extra-magic in a basic magic torus of the order-5.

Only consecutively numbered magic tori (*with numbers from 1 to n²*) are studied here, and consequently for the *order-5*, the magic constant *MC = 65*.

*Extra-magic tori of order-5 have neither been fully investigated nor precisely counted, and the examples shown below are a small selection of some of the different varieties.*

with 6 Knight Move Magic Diagonals

Extra-Magic Pandiagonal Torus Type T5.01 of Order-5 |

This magic line graph, based on a pandiagonal torus type T5.01 of order-5, attributed to Philippe de la Hire (c. 1705) by Percy Alexander MacMahon in his 1902 paper "Magic Squares and Other Problems upon a Chess Board", shows the presence of extra-magic with 6 knight move magic diagonals. After taking these into account, the maximum number of magic lines at a nodal intersection is 8 (at number nodes 12 and 13). The total number of magic lines on the pandiagonal torus is 10 orthogonals (in red) + 10 classic diagonals (in red) + 6 knight move diagonals (in blue) = 26 magic lines.

with 2 Knight Move Magic Diagonals

Extra-Magic Partially Pandiagonal Torus Type T5.03 of Order-5 |

This magic line graph, based on a partially pandiagonal torus type n° T5.03.00A of order-5 found by Walter Trump in 2012, shows the presence of extra-magic with 2 knight move magic diagonals. After the addition of the knight move diagonals (5, 9, 17, 21, 13) and (5, 17, 13, 9, 21), the maximum number of magic lines at a nodal intersection is 6 (at number nodes 5 and 21). The total number of magic lines on the partially pandiagonal torus is 10 orthogonals (in red) + 7 classic diagonals (in red) + 2 knight move diagonals (in blue) = 19 magic lines.

with 2 Knight Move Magic Diagonals

Extra-Magic Partially Pandiagonal Torus Type T5.04 of Order-5 |

This magic line graph, based on a partially pandiagonal torus type n° T5.04.000A of order-5 found by Walter Trump in 2012, shows the presence of extra-magic with 2 knight move magic diagonals. After the addition of the knight move diagonals (6, 9, 15, 11, 24) and (6, 11, 9, 24, 15), the maximum number of magic lines at a nodal intersection is 6 (at number nodes 11 and 15). The total number of magic lines on the partially pandiagonal torus is 10 orthogonals (in red) + 6 classic diagonals (in red) + 2 knight move diagonals (in blue) = 18 magic lines.

with 12 Knight Move Magic Diagonals

Extra-Magic "Knight's Tour" Partially Pandiagonal Torus Type T5.06 of Order-5 |

This magic line graph, based on a "knight's tour" partially pandiagonal torus type T5.06 of order-5, published by William Symes Andrews in his book "Magic Squares and Cubes" in 1908 (figure 20), shows the presence of extra-magic with 12 knight move magic diagonals. After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 8 (at number node 13). The total number of magic lines on the partially pandiagonal torus is 10 orthogonals (in red) + 6 classic diagonals (in red) + 12 knight move diagonals (in blue) = 28 magic lines.

with 20 Knight Move Magic Diagonals

Extra-Magic Basic Magic Torus Type T5.10 of Order-5 |

This magic line graph, based on a basic magic torus type T5.10 of order-5 published by Miguel Angel Amela in his paper "Magic and Semi-Magic Squares of Order 5 - Rectangular Pandiagonality" in spring 2011, shows the presence of extra-magic with knight move pandiagonality. After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 8 (at number node 13). The total number of magic lines on the magic torus is 10 orthogonals (in red) + 2 classic diagonals (in red) + 20 knight move diagonals (in blue) = 32 magic lines.

*In order-5, Miguel Angel Amela has found a total of 144 Magic Tori with "Rectangular Pandiagonality". Each of these displays 1 magic square and 24 semi-magic squares, and the corresponding grand totals are therefore 144 magic squares and 3456 semi-magic squares with knight move pandiagonality!*

with 4 Knight Move Magic Diagonals

Extra-Magic Semi-Magic Torus Type T5.14 of Order-5 |

This magic line graph of a semi-magic torus type n° T5.14.000000000X of order-5, shows the presence of extra-magic with 4 knight move magic diagonals. After the addition of the knight move magic diagonals (2, 12, 19, 22, 10), (2, 22, 12, 10, 19), (5, 8, 12, 24, 16) and (5, 24, 8, 16, 12), the maximum number of magic lines at a nodal intersection is 7 (at number node 12). The total number of magic lines on the semi-magic torus is 10 orthogonals (in red) + 1 classic diagonal (in red) + 4 knight move diagonals (in blue) = 15 magic lines.

In the Order-6, studies of knight move magic diagonals have previously been made by Miguel Angel Amela in his paper "Compact Magic and Semi-Magic Squares of Order n = 4k+2 - Rectangular Pandiagonality" dated Spring 2011, and Miguel has kindly authorised me to use one of his squares for an illustration of this extra-magic in a compact semi-magic torus of order-6.

Only consecutively numbered magic tori (*with numbers from 1 to n²*) are studied here, and consequently, for the *order-6*, the magic constant *MC = 111*.

*Extra-magic tori of order-6 have neither been fully investigated nor precisely counted, and the examples shown below are selected samples of some of the different varieties.*

with 6 Knight Move Magic Diagonals

Extra-Magic Partially Pandiagonal Torus of Order 6 with 6 KM Magic Diagonals |

This magic line graph, based on a partially pandiagonal torus of order-6 which Harry White has kindly authorised me to use, shows the presence of extra-magic with 6 knight move magic diagonals. After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 5 (at number nodes 19, 20, 22). The total number of magic lines on the partially pandiagonal torus is 12 orthogonals (in red) + 3 classic diagonals (in red) + 6 knight move diagonals (in blue) = 21 magic lines.

with 16 Extra-Magic Intersections and 4 Knight Move Magic Diagonals

Extra-Magic Partially Pandiagonal Torus of Order-6 with 4 KM Magic Diagonals |

This magic line graph, based on a partially pandiagonal torus of order-6 which Francis Gaspalou has kindly authorised me to use, shows the presence of both extra-magic intersections and knight move magic diagonals. After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 6 (at number nodes 7, 11, 26, 30). The total number of magic lines on the partially pandiagonal torus is 12 orthogonals (in red) + 8 classic diagonals (in red) + 4 knight move diagonals (in blue) = 24 magic lines.

with 2 Knight Move Magic Diagonals

Extra-Magic Basic Magic Torus of Order-6 with 2 KM Magic Diagonals |

This magic line graph, based on a basic magic torus of order-6 by William Symes Andrews in figure 311 of the 1917 re-edition of his book "Magic Squares and Cubes", shows the presence of extra-magic with 2 knight move magic diagonals. After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 4 (at number nodes 5, 14, 23, 32). The total number of magic lines on the basic magic torus is 12 orthogonals (in red) + 2 classic diagonals (in red) + 2 knight move diagonals (in blue) = 16 magic lines.

*Extra-magic semi-magic tori of order-6 have neither been fully investigated nor precisely counted, and the examples shown below are selected samples of some of the different varieties.*

with 2 Extra-Magic Intersections and 1 Knight Move Magic Diagonal

Extra-Magic Semi-Magic Torus of Order-6 with 1 KM Magic Diagonal |

This magic line graph, based on a semi-magic torus of order-6 previously found by Walter Trump and already illustrated in "Sixth and Higher-Order Magic Tori", shows not only the presence of 2 extra-magic intersections, but also a single knight move magic diagonal. (The latter was not noticed at the time of the first illustration). Please note that here the torus viewpoint is deliberately centred on the number node 27. This is not a standard magic square presentation, but it means to show that on the limitless surface of the magic torus there is no fixed centre. After the addition of the knight move magic diagonal, the maximum number of magic lines at intersections (number nodes 3, 27, 28) is 4. The total number of magic lines on the semi-magic torus is 12 orthogonals (in red) + 2 classic diagonals (in red) + 1 knight move diagonal (in blue) = 15 magic lines.

with 24 Knight Move Magic Diagonals

Extra-Magic Compact Semi-Magic Torus of Order-6 with 24 KM Magic Diagonals |

This magic line graph is based on a compact semi-magic torus of order-6 with knight move pandiagonality, found by Miguel Angel Amela and published in his 2011 paper "Compact Magic and Semi-Magic Squares of Order n = 4k+2 - Rectangular Pandiagonality". After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 6 (at all number nodes). The total number of magic lines on the semi-magic magic torus is 12 orthogonals (in red) + 0 classic diagonals (in red) + 24 knight move diagonals (in blue) = 36 magic lines.

In a compact magic square, the sum of each *2 x 2* subsquare is equal to *4/n* of the magic constant *(MC)*, and for the order-*6* the sum of each of the *2 x 2* subsquares is therefore equal to *4/6 x 111 = 74*. Miguel Amela proves that, regardless of whether they are magic or semi-magic, all compact squares of orders *n = 4k+2* are also knight move pandiagonal. *Taking into account squares with consecutive numbers from 1 to n², he has calculated that there are 53136 compact (and therefore knight move pandiagonal) semi-magic squares of order-6. This implies that there are 1476 compact semi-magic tori with these characteristics in order-6.*

In the Order-7, studies of knight move magic diagonals have previously been made by Miguel Angel Amela and he has kindly authorised me to use one of his basic magic squares for an illustration of knight move pandiagonality.

Only consecutively numbered magic tori (*with numbers from 1 to n²*) are studied here, and consequently, for the *order-7*, the magic constant *MC = 175*.

*Extra-magic tori of order-7 have neither been fully investigated nor precisely counted, and the examples shown below are selected samples of some of the different varieties.*

with 16 Knight Move Magic Diagonals

Extra-Magic "Knight Tour" Pandiagonal Torus of Order-7 with 16 KM Magic Diagonals |

This magic line graph is based on a "knight's tour" pandiagonal Torus of Order-7 published by Emory McClintock in his 1896 paper "On the Most Perfect Forms of Magic Squares, with Methods for their Production" (figure A): Although McClintock mentions "leaper" moves such as the Zebra (3, -2) in the description of his construction method by "step summation", he does not point out the traditional knight move magic diagonal characteristics of his pandiagonal square. After the addition of the 16 knight move magic diagonals that we find on this torus, the maximum number of magic lines at a nodal intersection is 8 (at number node 25). The total number of magic lines on the pandiagonal torus is 14 orthogonals (in red) + 14 classic diagonals (in red) + 16 knight move diagonals (in blue) = 44 magic lines.

with 22 Knight Move Magic Diagonals

Extra-Magic "Knight Tour" Partially Pandiagonal Torus of Order-7 with 22 KM Magic Diagonals |

This magic line graph is based on a "knight's tour" partially pandiagonal torus of order-7 that George Jelliss has kindly authorised me to use. After the addition of the 22 knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 8 (at number node 25). George Jelliss points out that the *"geometry of 'straight lines' is distinctly non-euclidean."* For example, numbers such as (22, 23, 24, 25, 26, 27, 28) are connected in at least 3 different ways along knight (1, 2), camel (3, -1), and zebra (2, -3) move magic diagonals, and many more possibilities exist if we stretch out the knight move steps even further... But here we only take into account the traditional knight move magic diagonals, and the total number of magic lines on the partially pandiagonal torus is therefore 14 orthogonals (in red) + 8 classic diagonals (in red) + 22 knight move diagonals (in blue) = 44 magic lines.

with 28 Knight Move Magic Diagonals

Extra-Magic Basic Magic Torus of Order-7 with 28 KM Magic Diagonals |

This magic line graph is based on a *knight move pandiagonal* basic magic torus of order-7 that Miguel Angel Amela has kindly authorised me to use. This torus was probably created when he wrote his paper "Magic and Semi-Magic Squares of Order 5 - Rectangular Pandiagonality" in spring 2011. After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 8 (at number node 25). The total number of magic lines on the basic magic torus is 14 orthogonals (in red) + 2 classic diagonals (in red) + 28 knight move diagonals (in blue) = 44 magic lines.

*Extra-magic semi-magic tori of order-7 have neither been fully investigated nor precisely counted, and the example shown below is randomly chosen.*

with 6 Knight Move Magic Diagonals

Extra-Magic Semi-Magic Torus of Order-7 with 6 KM Magic Diagonals |

This magic line graph of a semi-magic torus of order-7 shows the presence of 6 knight move magic diagonals. The maximum number of magic lines at a nodal intersection is 5 (at number nodes 12, 28, 35 and 39). The total number of magic lines on the semi-magic torus is 14 orthogonals (in red) + 3 classic diagonals (in red) + 6 knight move diagonals (in blue) = 23 magic lines.

Of the same format as traditional chessboards, magic squares of order-8 have not only been the focus of much previous research into "knights' tours", *but have also been (since 1877, if not earlier) the subject of studies of knight move magic diagonals.* Further historical details are given in the observations made after the analysis of a "Caïssan Beauty", studied below.

Only consecutively numbered magic tori (*with numbers from 1 to n²*) are studied here, and consequently, for the *order-8*, the magic constant *MC = 260*.

*Extra-magic tori of order-8 have neither been fully investigated nor precisely counted, and the examples shown below are selected samples of some of the different varieties.*

with 64 Extra-Magic Intersections and 32 Knight Move Magic Diagonals

Extra-Magic "Caïssan Beauty" Pandiagonal Torus of Order-8 with 32 KM Magic Diagonals |

This magic line graph, based on a "Caïssan Beauty" pandiagonal torus of order-8 created by Nārāyana Pandita (Sanskrit: नारायण पण्डित) in 1356, shows the presence of 64 extra-magic intersections as well as knight move pandiagonality. The maximum number of magic lines at nodal intersections is 8 (at all number nodes). The total number of magic lines on the "Caïssan Beauty" pandiagonal torus is 16 orthogonals (in red) + 16 classic diagonals (in red) + 32 knight move diagonals (in blue) = 64 magic lines.

First published by Nārāyana Pandita in his 1356 "Gaṇita Kaumudī" (Part II), an original version of the torus can be seen online, on page 396 of a 1942 reprint. Only those versed in Sanskrit will be able to read whether Nārāyana Pandita mentioned the knight move pandiagonal characteristics.

What we know for sure is that in 1877, Andrew Hollingworth Frost wrote an article "On the General Properties of Nasik Squares" in "The Quarterly Journal of Pure and Applied Mathematics", Volume 15, in which he presented *the same Nārāyana Pandita torus seen from a different magic square viewpoint*, referring to a diagram *S* in his description: *"By the method adopted in No. XXV. 1865, of this Journal, squares of the form 4n had four summations only, in the lines of the row [sic], columns, and diagonals. This square has ten summations through each number, as shewn in the diagram S, the hollow dots giving the pathlets of contsruction [sic]."* After verification of the diagram *S*, we see that it shows 2 orthogonal pathlets, 2 diagonal (1, 1) pathlets, 2 diagonal (1, 2) pathlets, 2 diagonal (2, 1) pathlets, 2 diagonal (1, 3) pathlets = 12 pathlets! Although this is a clearly documented example of the early awareness of knight move magic diagonals, Frost may have been preceded by others...

In 1881, "Ursus" (Henry James Kesson?) wrote an article entitled "Caïssan Magic Squares" in "The Queen, The Lady's Newspaper & Court Chronicle", presenting *the same Nārāyana Pandita torus*,seen from yet another magic square viewpoint, in his Figure D. "Ursus" was clearly aware of all the chess move magic diagonals, and he defined a magic square to be Caïssan whenever it was pandiagonal and knight-Nasik. "Ursus" seems to have been the first person to explicitly connect "Caïssa" (named as the *patron goddess* of chess players by Sir William Jones in his 1763 "Caïssa, or The Game at chess, a poem") with magic squares.

In 1900, Charles Planck wrote an article entitled "The *n* queens problem" in "The British Chess Magazine", and presented *the same Nārāyana Pandita torus*, seen from a once again different magic square viewpoint in his Figure II, stating that he believed it to be the *"first complete Caïssan square of 64 cells which has been constructed"*!

Although the different magic square viewpoints of Nārāyana Pandita's original that were presented by these distinguished authors were certainly coincidental, it is a relief to find at least one writer of the same period who produced two totally original "Caissan Beauties": In 1896, Emory McClintock presented a paper to the American Mathematical Society entitled "On the Most Perfect Forms of Magic Squares, with Methods for their Production" in which his squares D and E were refreshingly new examples, both coming from completely different "Caïssan Beauty" pandiagonal tori.

George P. H. Styan has written an excellent paper entitled "An illustrated introduction to Caïssan squares: the magic of chess". Published in 2012, in Volume 16, Number 1, of "Acta et Commentationes Universitatis Tartuensis de Mathematica" it retraces the history of Caïssan squares in great detail with numerous references, and I fully recommend it to readers who wish to learn more.

with 32 Extra-Magic Intersections and 4 Knight Move Magic Diagonals

Extra-Magic Partially Pandiagonal Torus of Order-8 with 4 KM Magic Diagonals |

This magic line graph, based on a partially pandiagonal torus of order-8 which Walter Trump has kindly authorised me to use, shows not only the presence of 32 extra-magic intersections but also 4 knight move magic diagonals. (The torus is also bimagic, but the squared version has no such diagonals). After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 5 (at 16 number nodes 6, 7, 14, 15, 22, 23, 30, 31, 34, 35, 42, 43, 50, 51, 58, 59). The total number of magic lines on the partially pandiagonal torus is 16 orthogonals (in red) + 12 classic diagonals (in red) + 4 knight move diagonals (in blue) = 32 magic lines.

*Extra-magic semi-magic tori of order-8 have neither been fully investigated nor precisely counted, and although only one of many, the example below is specifically chosen for its historical importance.*

with 2 Knight Move Magic Diagonals

Extra-Magic Semi-Magic "Knight Tour" Torus of Order-8 with 2 KM Magic Diagonals |

This magic line graph is based on the first semi-magic knight's tour torus that was created by William Roxby Beverley and published as a letter dated 5th June 1847 "On the Magic Square of the Knight's March" in the August 1848 edition of "The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science". After the addition of the knight move magic diagonals, the maximum number of magic lines at a nodal intersection is 3 (at 16 number nodes 7, 8, 9, 10, 33, 34, 35, 36, 37, 38, 43, 44, 45, 46, 46, and 48). The total number of magic lines on the semi-magic torus is 16 orthogonals (in red) + 0 classic diagonals (in red) + 2 knight move diagonals (in blue) = 18 magic lines.

More information about William Beverley is given in the "History of Knight Tours" and "Magic Knight Tours" by George Jelliss. At this stage it is useful to point out that Beverley's first semi-magic knight's tour square of order-8 has sometimes been mistakenly attributed to Leonhard Euler by otherwise reputable sources, and although Euler did indeed present knights' tours in his "Solution d'une question curieuse qui ne paroit soumise a aucune analyse" which was published in 1759, none of his solutions were either semi-magic or magic!

Once we take knight move magic diagonals into account, this can modify our perception of the "most magic" solutions. In this study we have only used traditional knight move steps and there are already many examples with a dense weave of magic lines. What a magic line graph looks like after the addition of non-traditional knight move lines is difficult to say, but we can anticipate that in several cases there will be very little blank space, or even none whatsoever...

In order-4, we immediately notice that the total number of magic lines of a pandiagonal torus type T4.01 (which has 12 magic lines) is inferior to the total number of lines of the extra-magic semi-pandiagonal torus type n° T4.02.3 (which has 20 magic lines). Also, 2 out of 6 partially pandiagonal tori type T4.03.1, and each of the partially pandiagonal tori types T4.03.2 and T4.03.3 have 16 magic lines, which is more than the number of magic lines of a pandiagonal torus type T4.01. Even the basic magic tori with knight move magic diagonals have a greater number of magic lines than the pandiagonal torus: 4 of the basic magic tori type T4.05.1 have 14 magic lines and 6 have 18 magic lines, whist 2 of the basic magic tori type T4.05.2 again have 14 magic lines. What is perhaps more surprising is that semi-magic tori such as the torus types T4.06.0A and T4.07.0A have respectively 20 and 14 magic lines; both examples exceeding the pandiagonal torus T4.01 in their magic line totals. The semi-magic torus type T4.06.0A even equals the highest number of magic lines of the "most magic" semi-pandiagonal torus type T4.02.3! As the semi-magic tori of order-4 have not yet been explored in detail, there may be other "more magic" cases that remain to be discovered...

In order-5, there are also surprises, and without going into all the detail it can be seen that a basic magic torus type T5.10 with knight move pandiagonality has a total of 32 magic lines, which is many more than that of a pandiagonal torus type T5.01, which only has 6 knight move diagonals and a total of 26 magic lines. The order-5 has not been completely explored, so there may be other cases that will modify our appreciation of which tori are the "most magic".

In order-6, there are no pandiagonal tori when using consecutive numbers from *1 to n²*, as Charles Planck proved in his 1919 Monist article "Pandiagonal Magics of Orders 6 and 10 with Minimal Numbers". In the magic line graphs that are illustrated here, a partially pandiagonal example with 4 knight move magic diagonals has 24 magic lines; a basic magic example with 2 knight move magic diagonals has 16 magic lines; and a compact *semi-magic* example with knight move pandiagonality has 36 magic lines! Another semi-magic example worthy of interest is the torus with 1 knight move magic diagonal, and 2 extra-magic intersections of 6 magic lines, giving a total of 15 magic lines: This also exceeds the total number of magic lines of a "regular" basic magic torus of order-6, which, with 12 orthogonals + 2 diagonals, has only 14 magic lines.

In order-7, a "knight's tour" pandiagonal torus with 16 knight move magic diagonals, a "knight's tour" partially pandiagonal torus with 22 knight move magic diagonals, and a basic magic torus that is knight move pandiagonal, all have the same total of 44 magic lines.

In order-8, the "Caïssan Beauty" with 64 = *n²* magic lines is a pure marvel for both magic square and chess enthusiasts!

It can be seen that semi-magic tori (and the semi-magic squares that they display) often rival, and sometimes even better their magic tori cousins when we include the knight move magic diagonals in the counts of the magic lines. A reappraisal of the definitions seems to be called for. Both magic and semi-magic tori have almost similar magic weaves, so logically no hard distinction should be made between them.

In the squared version of the extra-magic partially pandiagonal torus of order-8 (with 32 extra-magic intersections and 4 knight move magic diagonals) tested above, it was disappointing to find no knight move bimagic diagonals. When testing another likely candidate , the order-32 square constructed by Su Maoting in February 2006 (which is the first known normal pandiagonal bimagic square using consecutive integers, and downloadable from Christian Boyer's Multimagic Squares site), it turns out to have 64 knight move magic diagonals, but once again, none of these are bimagic.

It is therefore still an open question as to whether knight move bimagic diagonals exist on bimagic tori, and if so, whether there are cases with knight move bimagic pandiagonality.

It is interesting to note that when 3 traditional knight move magic lines form a triangle, then this triangle will always be Pythagorean, with a 3 : 4 : 5 proportion. This and other observations on knight move geometry are discussed further on the web page "Geometry of Knight's Moves" by George Jelliss. The geometry of other non-traditional "leapers" can be found on two more of his pages: "Theory of Moves" and "Theory of Journeys".

My special thanks go to Miguel Angel Amela, for his invaluable assistance in the search for previous documentation on knight move magic diagonals, and for taking the time to check and confirm my findings concerning the knight move magic diagonals of the magic tori of order-4. Miguel has also kindly authorised me to append his papers "Magic and Semi-Magic Squares of Order 5 - Rectangular Pandiagonality" and "Compact Magic and Semi-Magic Squares of Order n = 4k+2 - Rectangular Pandiagonality" to this post, and has given me his permission to use his knight move pandiagonal squares of orders 5, 6 and 7 for magic line graphs.

Although, to the best of my knowledge, all of the magic line graphs illustrated here are original, many of them are based on magic squares or magic tori that have been constructed by others:

In the public domain we find the Lo Shu magic square of order-3 by Anonymus, and the magic squares of order-4, once again by Anonymus (but first listed by Bernard Frénicle de Bessy). Also coming from the public domain, because they were published before 1923, other magic squares used here include a pandiagonal square of order-5 attributed to Philippe de la Hire by Percy Alexander MacMahon; a "knight's tour" partially pandiagonal square of order-5 together with a basic magic square of order-6 by William Symes Andrews; a "knight's tour" pandiagonal square of order-7 by Emory McClintock; a "Caïssan Beauty" pandiagonal square of order-8 created by Nārāyana Pandita; and a "knight's tour" semi-magic square by William Roxby Beverley. Each of these authors deserves to be remembered.

Contributions of other magic tori or magic squares, by the magic square specialists of today, include (in the order of the authors' appearance) the semi-magic tori T4.06.0A and T4.07.0A of order-4, the partially pandiagonal tori T5.03.00A T5.04.000A of order-5, a semi-magic torus of order-6, and a partially pandiagonal bimagic torus of order-8 by Walter Trump; a basic magic torus T5.10 of order-5, a compact semi-magic torus of order-6, and a basic magic torus of order-7 (all knight move pandiagonal) by Miguel Angel Amela; a partially pandiagonal torus of order-6 by Harry White; a partially pandiagonal torus of order-6 by Francis Gaspalou; and a "knight's tour" partially pandiagonal torus of order-7 by George Jelliss. My hearty thanks go to each of these authors for allowing me to use their work. Without their contributions, certain magic line graphs of this paper would not have been possible.

The sites that host the hyperlinked documentation include (in the order of their "appearance") Prime Puzzles, Wikipedia, Google Books, Nature, Internet Archive, JStor, Mayhematics, Wikisource, The University of Tartu, McGill University, Multimagic Squares, and Bibnum Education.

On the 20th August 2019 Greg Ross published an article entitled "Magicker" on the subject of knight move magic diagonals in his Futility Closet - An Idler's Miscellany of Compendious Amusements.

]]>Magic squares have fascinated mathematicians for centuries and they continue to do so today. However, many questions remain unanswered, and this study proposes a different perspective in order to shed new light on what magic squares are and how they work. Considering magic squares to be flattened partial viewpoints of convex or concave magic tori, the implied Gaussian surfaces require a modular arithmetic approach that is tested and analysed here.

Until today, most magic square construction methods use base squares, whilst *"Magic Torus Coordinate and Vector Symmetries"* proposes modular coordinate equations that not only define specific magic tori, but also generate their magic torus descendants.

The construction of Agrippa's traditional magic squares is analysed in detail for each of the seven planetary magic tori, and modular coordinate equations are defined that generate descendant tori throughout the respective higher-orders, whether they be odd, doubly-even, or singly-even.

The unique third-order magic torus and also a fifth-order pandiagonal torus of Latin construction are examined in detail. The modular coordinate equations that are defined generate an interesting variety of higher-odd-order descendant tori.

The study also explores the construction of the 12 different types of fourth-order magic tori, as well as the generation of their doubly-even magic torus descendants. In addition, a fourth-order Candy-style perfect square pandiagonal torus, and a fourth-order unidirectionally pandiagonal semi-magic torus, are each analysed in detail, and their torus descendants generated, throughout their respective higher-orders.

At the end of the study observations are made, and some conclusions are drawn as to the signification of the findings, and the potential for future research.

Some sample illustrations of a fourth-order partially pandiagonal torus and its partially pandiagonal torus descendants are shown below:

T4.195 4th-order partially pandiagonal torus |

8th-order partially pandiagonal torus, direct descendant of T4.195 |

12th-order partially pandiagonal torus, direct descendant of T4.195 |

16th-order partially pandiagonal torus, direct descendant of T4.195 |

To download *"Magic Torus Coordinate and Vector Symmetries"* from Google Drive, please use the following link: MTCVS 161019. The 134 Mo pdf file exceeds the maximum size (25 Mo) that Google Drive can scan, but as __the file is virus free__ you can download it safely. Depending on the speed of your computer, the download can take up to two or three minutes to complete.

The subject of pan-zigzag magic squares was first brought to my attention by Harry White, who forwarded an e-mail with a pan-zigzag example inspired by the "Dürer" Magic Square, that Paul Michelet had sent to him on the 15th May 2019.

Paul Michelet, a chess endgame and problem composer, had constructed a magic torus of order-8 that was pan-zigzag, by replacing the entries of the "Dürer" Magic Square with consecutively numbered 2 x 2 squares.

The objective of the present study is to find higher-orders of pan-zigzag magic tori that are direct descendants of both the "Dürer" magic torus of order-4, and the "Michelet" pan-zigzag magic torus of order-8.

In Paul Michelet's solution that follows, the 2 x 2 squares "magnify" the T4.077 magic torus of order-4 that displays the "Dürer" Magic Square:

The order-4 magic torus T4.077 that displays the "Dürer" magic square. The red dots show the 4 magic diagonals. The magic constant (MC) is 34. |

The "Michelet" order-8 pan-zigzag magic torus, which is a magnified descendant of the "Dürer" magic torus T4.077.The red dots show the 4 magic diagonals. The magic constant (MC) is 260. |

In the order-8 pan-zigzag example illustrated above, each of the entries of the "Dürer" magic square are replaced by consecutively numbered 2 x 2 squares. The resulting magic torus has four magic diagonals that produce 8 magic intersections *and therefore eight essentially different magic squares*. All of the vertical zigzags, such as 61, 62, 20, 19, 33, 34, 16, 15, or 64, 63, 17, 18, 36, 35, 13, 14, add up to the magic constant of 260, and all of the horizontal zigzags, such as 64, 63, 9, 10, 5, 6, 52, 51, or 62, 61, 11, 12, 7, 8, 50, 49, also add up to 260. This magic torus is therefore *pan-4-way V zigzag _{2}*. Harry White states that

The sums of each of the N/4 x N/4 subsquares of the "Michelet" magic torus of order-8 are related to those of the original "Dürer" magic torus of order-4:

The totals of the 16 subsquares of the T4.077 "Dürer" magic torus of order-4 compared with those of the 16 subsquares of the "Michelet" magnified magic torus descendant of order-8. |

For more examples of pan-zigzag magic tori that continue to magnify the "Durer" magic square *in higher-orders*, the complete paper is below. Please note that although the preview page does not display the hyperlinks, these are accessible when you download Pan-Zigzag Torus Descendants that Magnify the "Durer" Magic Square:

The present study is inspired by Miguel Angel Amela's paper dated September 2016, entitled "Powers of Associative Magic Squares," and by Francis Gaspalou's paper dated 7th October 2016, entitled "Note on the features of the associative magic squares which give magic solutions when raised at an even power."

Both of these studies refer to an earlier paper by Charles K. Cook, Michael R. Bacon, and Rebecca A. Hillman; “The Magicness of Powers of Some Magic Squares,” published in The Fibonacci Quarterly, Volume 48, Number 4; 2010.

Both of these studies refer to an earlier paper by Charles K. Cook, Michael R. Bacon, and Rebecca A. Hillman; “The Magicness of Powers of Some Magic Squares,” published in The Fibonacci Quarterly, Volume 48, Number 4; 2010.

In the present study, which approaches the subject from a magic torus viewpoint, the use of formulae for generation enables an analysis of comparable magic tori throughout the different doubly-even and odd-orders. *Please note that there are no associative magic squares of singly-even-orders, as demonstrated by C. Planck in his "Pandiagonal magics of orders 6 and 10 with minimal numbers," published in "The Monist," Volume 29, N° 2 (April 1919), pages 307-316.*

The pages that follow show that both associative and non-associative matrices can yield magic solutions when squared. Although the study was initially intended for associative magic squares only, other types of magic squares, displayed on related magic tori, are now also included for the sake of comparison.

The research also reveals that recurring pandiagonal patterns appear on the squared matrices, throughout the higher-orders of magic tori generated by a same formula.

1/4 pandiagonal squared matrix of an associative magic square viewpoint of an order-12 semi-pandiagonal magic torus |

To view the paper "Pandiagonality of the Squared Matrices of Associative and Related Magic Squares," please note that if you click on the button that appears at the top right hand side of the pdf viewer below, a new window will open and full size pages will then be displayed, with options for zooming.

In his paper "Self-complementary magic squares of 4x4," Mutsumi Suzuki has shown that there are 352 self-complementary magic squares of order-4. Although Suzuki does not use Frénicle standard form, his examples are essentially different, and the total of these 352 self-complementary squares can therefore be compared with the grand total of the 880 essentially different magic squares of order-4 previously identified by Bernard Frénicle de Bessy.

On his page "Self-Similar Magic Squares," Harvey Heinz has studied Mutsumi Suzuki’s work, and points out that the 352 self-complementary magic squares include 48 Dudeney Type III associated semi-pandiagonal, 96 Dudeney Type VI semi-pandiagonal, and 208 Dudeney Type VI “simple” magic squares. The Dudeney Types referred to by Harvey Heinz come from Henry Dudeney’s classification of the 880 Frénicle magic squares into 12 types, each of which corresponds to a different complementary number pattern.

Today we know that the 880 Frénicle indexed magic squares are partial viewpoints of 255 essentially different magic tori of Order-4 (OEIS sequence A270876). A previous post "Complementary Number Patterns on Fourth-Order Magic Tori" has shown that some of the 12 Dudeney complementary number patterns become redundant on fourth-order magic tori, and this has resulted in the creation of a new reduced set of complementary number pattern types which are better adapted:

Therefore, with less complementary number patterns, what are the implications for the totals of self-complementary and paired complementary magic tori of order-4? In the study that follows, all of the 255 magic tori of order-4 are listed, and the details of their complements are given. In the final observations, the different cases of self-complementarity or paired complementarity are analysed, together with the respective complementary number patterns.

Therefore, with less complementary number patterns, what are the implications for the totals of self-complementary and paired complementary magic tori of order-4? In the study that follows, all of the 255 magic tori of order-4 are listed, and the details of their complements are given. In the final observations, the different cases of self-complementarity or paired complementarity are analysed, together with the respective complementary number patterns.

Please note that if you click on the button that appears at the top right hand side of the pdf viewer below, a new window will open and full size pages of the paper "Self-Complementary and Paired Complementary Magic Tori of Order-4" will then be displayed, with options for zooming.

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It is well known that sequenced concentric rings are present on classic bordered magic squares or tori. The subject of bordered magic squares is already well documented, and links for further reading are suggested at the end of this post.

However, this study is about *sequenced concentric rings in general*, and not about bordered magic squares in particular. Sequenced concentric rings exist on many different types of magic squares or magic tori, and various cases are explored up to order-10 together with one order-18 example.

It will be seen that the classic linear arithmetic sequences of bordered magic squares can also be found on non-bordered magic squares or tori. Also, some alternative arithmetic sequences, *and a geometric sequence*, are brought to light using simple algebra equations, and examples of each are given in the following document:

For those amongst you who may wish to know more about the particular cases of bordered magic squares, the sites of Harry White and Inder Taneja contain further information and examples.

Also, one of the earliest books on the subject of bordered magic squares is Bernard Violle's *"Traité complet des carrés magiques pairs et impairs, simples et composés, à bordures, compartiments, croix, chassis, équerres, bandes détachées etc, suivi d'un traité des cubes magiques, et d'un essai sur les cercles magiques : avec atlas de 51 grandes feuilles, comprenant 400 figures"*, which was first published in 1837. From page 44 to 213, its step-by-step diagrams, that explain how to construct bordered magic squares, can be easily understood by non-French readers.

]]>My recent interest in tetrads has been sparked off by an article entitled "Combines pour Tétrades," that was written by Professor Jean-Paul Delahaye, and published in the January 2020 edition of the magazine Pour La Science (N° 507).

An interesting conversation ensued with my friend Walter Trump, who, having already done pioneer work on the subject since 1970, then decided in February 2020 to create his own specialised website "Further Questions About Tetrads." Here he gives a step-by-step analysis and relates his own tetrad story. For readers who wish to learn more about tetrads and their history, I thoroughly recommend this site, which not only gives clear explanations, but also includes an exhaustive list of references.

According to Martin Gardner, in page 121 of his 1989 book "Penrose Tiles to Trapdoor Ciphers ...and the Return of Dr. Matrix," it was Michael R.W. Buckley, who, in the Journal of Recreational Mathematics (1975, Vol. 8), was the first to propose the name tetrad, for four simply connected planar regions, each pair of which sharing a finite portion of a common boundary.

Here, we will only be considering tetrads with congruent regions, each of which being transformable into each other by a combination of rigid motions that include translation, rotation, and reflection.

The following definition therefore applies: Tetrads are constituted of four congruent regions, when each of which shares a finite portion of their boundary with each of the others.

Here, we will only be considering tetrads with congruent regions, each of which being transformable into each other by a combination of rigid motions that include translation, rotation, and reflection.

The following definition therefore applies: Tetrads are constituted of four congruent regions, when each of which shares a finite portion of their boundary with each of the others.

In his "Mathematical Games" column for Scientific American, Martin Gardner wrote a Problem 7.8 entitled "Exploring Tetrads," and illustrated several tetrad examples in Figure 7.11. The part D of this Figure 7.11 showed a 22-sided polygonal tetrad solution with congruent pieces that had bilateral symmetry and a bilaterally symmetric border. Martin Gardner asked if an alternative solution might exist, with polygons of fewer sides.

In the Answer 7.8 that appeared in the "Mathematical Games" column of April 1977, Martin Gardner stated that Robert Ammann, Greg Frederickson and Jean L. Loyer had each found an 18-sided polygonal tetrad with bilateral symmetry, thus improving on the 22-sided solution that he had previously published. The Ammann-Frederickson-Loyer solution, illustrated in Figure 7.33, is the smallest holeless tetrad that can made from a polyiamond with mirror symmetry. We can see that each of the four pieces of this tetrad share a common boundary with each of the others. This assembly can be achieved by the translation or the rotation of copies of a starting piece. Although reflection is also possible, no reflection of the pieces is necessary for the construction of the tetrad.

While examining the different tetrads that have already been found, I noticed that some of these puzzles have completely interlocking pieces, whilst others do not. This gave me the idea of searching for jigsaw piece assemblies.

The following 5 propositions are transformations of the bilaterally symmetric tetrad found by Robert Ammann, Greg Frederickson and Jean L. Loyer. The addition of tabs, and the subtraction of blanks, can cause the jigsaw puzzle pieces to become* asymmetric*. Also, during the design process, we can decide whether or not certain pieces have to be flipped over, (reflected), in order to construct the tetrad:

Tetrad Jigsaw Puzzle n°1 making a holeless tetrad: Solution n°1a. |

In this first solution, which is a holeless tetrad, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, *the pink and blue pieces are also reflected.*

Tetrad Jigsaw Puzzle n°1 making a tetrad with a hole: Solution n°1b. |

In this second solution, which is a tetrad with a hole, the pieces are always translations and/or rotations of each other.

Tetrad Jigsaw Puzzle n°2 making a holeless tetrad: Solution n°2a. |

In this first solution, which is a holeless tetrad, the pieces are always translations and/or rotations of each other*.*

Tetrad Jigsaw Puzzle n°2 making a tetrad with a hole: Solution n°2b. |

In this second solution, which is a tetrad with a hole, the pieces are always translations and/or rotations of each other.

Tetrad Jigsaw Puzzle n°3 making a holeless tetrad: Solution n°3a. |

In this first solution, which is a holeless tetrad, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, *the green piece is also reflected.*

Tetrad Jigsaw Puzzle n°3 making a tetrad with a hole: Solution n°3b. |

In this second solution, which is a tetrad with a hole, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, *the green piece is also reflected.*

When using jigsaw puzzle pieces, *unless the sizes of the tabs and blanks are considerably reduced*, conflicting graphics can sometimes occur in acute angles. However, it is possible to get round this difficulty by using plus (+) signs instead of tabs, and minus (-) signs instead of blanks, as shown in the examples that follow:

Tetrad Jigsaw Puzzle n°4 making a holeless tetrad: Solution n°4a. |

In this first solution, which is a holeless tetrad, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, *the blue and pink pieces are also reflected.*

Tetrad Jigsaw Puzzle n°4 making a tetrad with a hole: Solution n°4b. |

In this second solution, which is a tetrad with a hole, the pieces are always translations and/or rotations of each other*.*

Tetrad Jigsaw Puzzle n°5 making a holeless tetrad: Solution n°5a. |

In this first solution, which is a holeless tetrad, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, *the blue, green and pink pieces are also reflected.*

Tetrad Jigsaw Puzzle n°5 making a tetrad with a hole: Solution n°5b. |

In this second solution, which is a tetrad with a hole, the pieces are not just translations and/or rotations of each other: Supposing that the starting piece is yellow, *the green piece is also reflected.*

## Observations

The necessary congruence of tetrad pieces implies that each can be transformed into each other by a combination of rigid motions which include translation, rotation, and reflection. The tetrad jigsaw puzzle exercises presented above show that, depending on the arrangement of the tabs and blanks of the pieces, reflection is sometimes, but not always needed, in order to achieve the correct assembly.

There could be a case for a new sub-classification of tetrads in general, depending on whether or not a reflection of their components is required.

In the examples given above we notice that there are jigsaw puzzle piece solutions with 9 tabs (+) and 9 blanks (-), with 10 tabs (+) and 8 blanks (-), or with 8 tabs (+) and 10 blanks (-). Why do these differences occur? The following document is an analysis of the different combinations of the positives (+) and negatives (-) of the tetrad puzzle pieces, depending the choices made during the design process:

We can also construct other shapes that are not necessarily tetrads. Readers who are interested, are invited to explore the many possibilities, and create their own patterns.

## Acknowledgements

I wish to thank Craig Knecht and Walter Trump for their encouragements and useful comments during the preparatory phase of this post.

]]>The necessary congruence of tetrad pieces implies that each can be transformed into each other by a combination of rigid motions which include translation, rotation, and reflection. The tetrad jigsaw puzzle exercises presented above show that, depending on the arrangement of the tabs and blanks of the pieces, reflection is sometimes, but not always needed, in order to achieve the correct assembly.

There could be a case for a new sub-classification of tetrads in general, depending on whether or not a reflection of their components is required.

In the examples given above we notice that there are jigsaw puzzle piece solutions with 9 tabs (+) and 9 blanks (-), with 10 tabs (+) and 8 blanks (-), or with 8 tabs (+) and 10 blanks (-). Why do these differences occur? The following document is an analysis of the different combinations of the positives (+) and negatives (-) of the tetrad puzzle pieces, depending the choices made during the design process:

We can also construct other shapes that are not necessarily tetrads. Readers who are interested, are invited to explore the many possibilities, and create their own patterns.

I wish to thank Craig Knecht and Walter Trump for their encouragements and useful comments during the preparatory phase of this post.

As is clearly stated in the OEIS sequence A006052* "Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections,"* the first and smallest magic square is of order-1.

The dotted red lines across this magic square represent its magic diagonals. For a basic magic square, each row, column, and main diagonal must sum to the magic constant. The magic constant of a magic square is equal to the division of the triangular number of its squared order by its order. The magic constant (mcN) of an Nth order magic square (or torus) can thus be calculated as follows:

The dotted red lines across this magic square represent its magic diagonals. For a basic magic square, each row, column, and main diagonal must sum to the magic constant. The magic constant of a magic square is equal to the division of the triangular number of its squared order by its order. The magic constant (mcN) of an Nth order magic square (or torus) can thus be calculated as follows:

mcN = __(N²)² + N²__ x __ 1 __ = __N(N² + 1)__

2 N 2

For the first-order magic square (or magic torus) the magic constant should therefore be:

mc1 = __1(1² + 1)__ = 1, which is the case.

2

The blue border of the unique number cell is also the limit of the magic square itself. The video below shows how this blue border merges to form single latitude and longitude lines on the curved 2D surface of the first-order magic torus:

Gluing a Torus video by Geometric Animations - University of Hannover, hosted by YouTube

We can check some of the simple conditions that define a basic magic odd-order torus, deduced whilst observing the unique third-order T3 magic torus:

The number of N-order squares (both magic and semi-magic squares) that are displayed on each N-order magic torus = N². The first-order magic torus should therefore display 1² = 1 first-order square, which is the case.

Basic magic odd-N-order tori display not only 1 basic N-order magic square, but also N²-1 semi-magic N-order squares. The first-order basic magic square should therefore display 1 basic order-1 magic square, and 1²-1 = 0 semi-magic order-1 squares, which is also the case.

What is more surprising, although quite logical once we consider the question, is that the order-1 magic torus even satisfies the basic conditions for pandiagonality!

If pandiagonal (magic along all of its diagonals), an N-order torus displays N² pandiagonal squares of order N. The first-order torus should therefore display 1² pandiagonal squares = 1 pandiagonal square, which once again is the case.

Taking into account the panmagic properties of this torus, the adjective *"trivial,"* which is often used to describe the first-order square, now seems almost depreciatory, (notwithstanding the fact that for mathematical contexts, the dictionary definitions of trivial include: *"simple, transparent, or immediately evident"*). There is more to the order-1 torus than first meets the eye! Not only is it pandiagonal, but its number One also signifies mathematical creation and the very beginnings... The Pythagoreans referred to the number One as the "monad," which engendered the numbers, which engendered the point, which engendered all lines, etc. For Plotinus and other neoplatonists, the number One was the ultimate reality and the source of all existence. The first-order torus, together with the number One that it displays, both symbolise the "Big Bang."

I must admit to having completely overlooked this first-order magic torus until Miguel Angel Amela kindly sent me a copy of one of his studies *"Pandiagonal Latin Squares and Latin Schemes in the Torus Surface,"* on the 9th March 2016. By extrapolating his findings, I discovered for myself the pandiagonal characteristics of the first-order, and I wish to thank Miguel for this paper which has been an inspiration to me.

The pandiagonal torus T1 of order-1 comes first again in the new OEIS sequence A270876*"Number of magic tori of order n composed of the numbers from 1 to n^2."*

]]>The pandiagonal torus T1 of order-1 comes first again in the new OEIS sequence A270876

The 880 fourth-order magic squares were first identified and listed by Bernard Frénicle de Bessy in his « Table générale des quarrez magiques de quatre, » which was published posthumously in 1693, in his book « Des quarrez magiques. » The census of these 880 4x4 squares is given in the appendix to the book « New Recreations With Magic Squares » by William H. Benson and Oswald Jacoby (Edition 1976), and also on the website of the late Harvey D. Heinz: Frénicle n° 1 à 200, Frénicle n° 201 à 400, Frénicle n° 401 à 600, Frénicle n° 601 à 880.

Further to Bernard Frénicle de Bessy's initial findings, Henry Ernest Dudeney then classified the 880 fourth-order magic squares under 12 pattern types, in "The Queen" in 1910, (later republished in his book "Amusements in Mathematics" in 1917). These patterns were determined through the observation of the relative positions of complementary pairs of numbers (which add up to N²+1 when N is the order of the square).

Although the different Dudeney pattern types can indeed be observed on fourth-order magic squares, they cannot be used for the classification of the fourth-order magic tori that display these squares. A Dudeney pattern type can be misleading because it depends on a bordered magic square viewpoint, whilst the magic torus that displays the magic square has a limitless surface...

In the study that follows the complementary number patterns are therefore tested, by comparing a developed torus surface and two traditional Frénicle magic square viewpoints for each of the 12 different types of fourth-order magic tori.

In addition to the Frénicle index numbers and the Dudeney types that have already been mentioned above, the present study also uses the index and type numbers of the fourth-order magic tori. For readers who may not be acquainted with the subject, essential information can be found in the following pages:

In a previous article "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus," the fourth-order magic tori have been classed in 12 types, and each magic torus has received a specific type number.

In a previous article "Table of Fourth-Order Magic Tori," so as to facilitate the identification of a magic torus beginning with any magic square, a standard torus form has also been defined, and each of the 255 fourth-order magic tori has received a specific index number.

The 3 pandiagonal tori type T4.01 are here represented by the pandiagonal torus with index n° T4.01.1 (torus type n° T4.01.1). This pandiagonal torus displays 16 pandiagonal squares with Frénicle index n°s 102, 104, 174, 201, 279, 281, 365, 393, 473, 530, 565, 623, 690, 748, 785 and 828. All of these squares, (including the Frénicle index n°s 102 and 174 illustrated here), show complementary number patterns that are Dudeney type I.

The 24 semi-pandiagonal tori type T4.02.1 are here represented by the semi-pandiagonal torus with index n° T4.115 (torus type n° T4.02.1.09). This semi-pandiagonal torus displays 8 semi-pandiagonal squares with Frénicle index n°s 48, 192, 255, 400, 570, 734, 763 and 824. Half of these squares, (including the Frénicle index n° 192 illustrated here), show complementary number patterns that are Dudeney type IV. The other half, (including the Frénicle index n° 48 illustrated here), show complementary number patterns that are Dudeney type VI.

The 12 semi-pandiagonal tori type T4.02.2 are here represented by the semi-pandiagonal torus with index n° T4.059 (torus type n° T4.02.2.01). This semi-pandiagonal torus displays 8 semi-pandiagonal squares with Frénicle index n°s 21, 176, 213, 361, 445, 591, 808 and 860. Half of these squares, (including the Frénicle index n° 21 illustrated here), show complementary number patterns that are Dudeney type II. The other half, (including the Frénicle index n° 176 illustrated here), show complementary number patterns that are Dudeney type III.

The 12 semi-pandiagonal tori type T4.02.3 are here represented by the semi-pandiagonal torus with index n° T4.085 (torus type n° T4.02.3.01). This semi-pandiagonal torus displays 8 semi-pandiagonal squares with Frénicle index n°s 32, 173, 228, 362, 425, 577, 798 and 853. All of these squares, (including the Frénicle index n°s 173 and 228 illustrated here), show complementary number patterns that are Dudeney type V.

The 6 partially pandiagonal tori type T4.03.1 are here represented by the partially pandiagonal torus with index n° T4.108 (torus type n° T4.03.1.1). This partially pandiagonal torus displays 4 partially pandiagonal squares with Frénicle index n°s 46, 50, 337 and 545. All of these squares, (including the Frénicle index n°s 46 and 545 illustrated here), show complementary number patterns that are Dudeney type VI.

As already mentioned in "A New Census of Fourth-Order Magic Squares," Dudeney overlooked the partially pandiagonal characteristics, and mistakenly classified these squares as "simple."

The 12 partially pandiagonal tori type T4.03.2 are here represented by the partially pandiagonal torus with index n° T4.195 (torus type n° T4.03.2.06). This partially pandiagonal torus displays 4 partially pandiagonal squares with Frénicle index n°s 208, 525, 526 and 693. Half of these squares, (including the Frénicle index n° 693 illustrated here), show complementary number patterns that are Dudeney type VII. The other half, (including the Frénicle index n° 208 illustrated here), show complementary number patterns that are Dudeney type IX.

###
__Complementary number patterns of the partially pandiagonal tori type T4.03.3__

###
__Complementary number patterns of the partially pandiagonal tori type T4.04__

As already mentioned in "A New Census of Fourth-Order Magic Squares," Dudeney overlooked the partially pandiagonal characteristics, and mistakenly classified these squares as "simple."

###
__Complementary number patterns of the basic magic tori type T4.05.1__

###
__Complementary number patterns of the basic magic tori type T4.05.2__

###
__Complementary number patterns of the basic magic tori type T4.05.3__

###
__Complementary number patterns of the basic magic tori type T4.05.4__

##
__Conclusions__

##
__Further Developments__

]]>
As already mentioned in "A New Census of Fourth-Order Magic Squares," Dudeney overlooked the partially pandiagonal characteristics, and mistakenly classified these squares as "simple."

The 2 partially pandiagonal tori type T4.03.3 are here represented by the partially pandiagonal torus with index n° T4.217 (torus type n° T4.03.3.1). This partially pandiagonal torus displays 4 partially pandiagonal squares with Frénicle index n°s 181, 374, 484 and 643. All of these squares, (including the Frénicle index n°s 484 and 643 illustrated here), show complementary number patterns that are Dudeney type XI.

As already mentioned in "A New Census of Fourth-Order Magic Squares," Dudeney overlooked the partially pandiagonal characteristics, and mistakenly classified these squares as "simple."

The 4 partially pandiagonal tori type T4.04 are here represented by the partially pandiagonal torus with index n° T4.096 (torus type n° T4.04.1). This partially pandiagonal torus displays 2 partially pandiagonal squares with Frénicle index n°s 40 and 522. The square with Frénicle index n° 40 (illustrated here) shows a complementary number pattern that is Dudeney type VIII, whilst the other square with Frénicle index n° 522 (illustrated here) shows a complementary number pattern that is Dudeney type X.

The 92 basic magic tori type T4.05.1 are here represented by the basic magic torus with index n° T4.002 (torus type n° T4.05.1.01). This basic magic torus displays 2 basic magic squares with Frénicle index n°s 1 and 458. Both of these squares, illustrated here, show complementary number patterns that are Dudeney type VI.

The 32 basic magic tori type T4.05.2 are here represented by the basic magic torus with index n° T4.049 (torus type n° T4.05.2.01). This basic magic torus displays 2 basic magic squares with Frénicle index n°s 23 and 767. The square with Frénicle index n° 767 (illustrated here) shows a complementary number pattern that is Dudeney type VII, whilst the square with Frénicle index n° 23 (illustrated here) shows a complementary number pattern that is Dudeney type IX.

The 4 basic magic tori type T4.05.3 are here represented by the basic magic torus with index n° T4.005 (torus type n° T4.05.3.1). This basic magic torus displays 2 basic magic squares with Frénicle index n°s 3 and 613. Both of these squares, illustrated here, show complementary number patterns that are Dudeney type XII.

The 52 basic magic tori type T4.05.4 are here represented by the basic magic torus with index n° T4.019 (torus type n° T4.05.4.1). This basic magic torus displays 2 basic magic squares with Frénicle index n°s 8 and 343. The square with Frénicle index n° 8 (illustrated here) shows a complementary number pattern that is Dudeney type VIII, whilst the square with Frénicle index n° 343 (illustrated here) shows a complementary number pattern that is Dudeney type X.

We can see that the Dudeney pattern types I, V, XI, and XII are not only valid for magic squares, but also for the magic tori that display these squares. On the other hand, the Dudeney pattern types II and III are shown to be partial viewpoints of a single pattern on the fourth-order magic tori. Again, the Dudeney pattern types IV and VI, the Dudeney pattern types VII and IX, and also the Dudeney pattern types VIII and X, are shown to be partial viewpoints of single patterns on the fourth-order magic tori.

We also notice that some of the detected complementary number patterns have contrasting hemitorus arrangements.

It is necessary to adapt the Dudeney pattern types, and add new symbols that not only reveal the real nature of the complementary number relationships, but also facilitate their comprehension. With this in mind, the 8 figures that follow have been selected to illustrate the essentially different complementary number pattern types that occur on fourth-order magic tori:

It is necessary to adapt the Dudeney pattern types, and add new symbols that not only reveal the real nature of the complementary number relationships, but also facilitate their comprehension. With this in mind, the 8 figures that follow have been selected to illustrate the essentially different complementary number pattern types that occur on fourth-order magic tori:

To be read with a previous table that compared the 12 Dudeney pattern types with the 12 Magic torus types, published in "A New Census of Fourth-Order Magic Squares," the new table that follows, recapitulates the latest findings, and shows the repartition of the 8 complementary number patterns on the magic squares of the fourth-order magic tori:

New articles entitled "Self-Complementary and Paired Complementary Magic Tori of Order-4" and "Multiplicative Magic Tori" now extend the above findings.