Thursday 15 June 2023

440 Torus-Opposite Pairs of the 880 Frénicle Magic Squares of Order-4

Finding the torus-opposite pairs of magic squares in even-orders:

The following diagram, which illustrates the array of 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:

This diagram shows how to identify a torus-opposite magic square in order-4. The method can be applied to all even-orders.
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, in toroidal directions, east or west along the latitudes, and in poloidal directions, 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.

Why torus-opposite pairs of magic squares cannot exist in odd-orders

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.

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

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 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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Tuesday 7 June 2022

Polyomino Area Magic Tori

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:

Colour diagram of an area magic torus of order-3, showing the 9 magic square viewpoints, created by William Walkington in 2017

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.

Intention and Definition

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.

Polyomino Area Magic Tori (PAMT) of Order-3


Colour diagram of the magic torus of order-3, displaying Agrippa's "Saturn" magic square, with graphics by William Walkington
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.

Diagram of irregular rectangular Polyomino Area Magic Torus of order-3 with tetromino tiles, by William Walkington in 2022
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.

Diagram of an irregular rectangular Polyomino Area Magic Torus of order-3 with tromino tiles, by William Walkington in 2022
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.

Colour diagram of an oblong Polyomino Area Magic Torus of order-3 with tromino tiles, created by William Walkington in 2022
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.

Colour diagram of a square Polyomino Area Magic Torus of order-3 with tromino tiles, V1 created by William Walkington in 2022
Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 36. Trominoes Version 1.
Square 18 ⋅ 18 = 324 units.

Colour diagram of a square Polyomino Area Magic Torus of order-3 with tromino tiles, V2 created by William Walkington in 2022
Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 36. Trominoes Version 2.
Square 18 ⋅ 18 = 324 units.

Colour diagram of a square Polyomino Area Magic Torus of order-3 with pentomino tiles, V1 created by William Walkington in 2022
Polyomino Area Magic Torus (PAMT) of Order-3. Magic sums = 60. Pentominoes Version 1.
Square 30 ⋅ 30 = 900 units.

Diagram of an irregular rectangular Polyomino Area Magic Torus of order-3 with monomino tiles, by William Walkington in 2022
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.

Colour diagram of a square Polyomino Area Magic Torus of order-3 with monomino tiles, V1 created by William Walkington in 2022
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.

Colour diagram of a square Polyomino Area Magic Torus of order-3 with monomino tiles, V2 created by William Walkington in 2022
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.

Polyomino Area Magic Tori (PAMT) of Order-4


Colour diagram of a pandiagonal Magic Torus of order-4, displaying the Frénicle 107 index number square, by William Walkington
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.

Colour diagram of a pandiagonal Polyomino Area Magic Torus of order-4, constructed with pentominoes by William Walkington
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.

Colour diagram of a square Polyomino Area Magic Torus of order-4 with domino tiles, V1 created by William Walkington in 2022
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.

Diagram of an irregularly shaped view of a Polyomino Area Magic Torus of order-4 with domino tiles, by William Walkington in 2022
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.

Polyomino Area Magic Tori (PAMT) of Order-5


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.

Colour diagram of a pandiagonal Polyomino Area Magic Torus of order-5, constructed with hexominoes by William Walkington
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.

Colour diagram of a square Polyomino Area Magic Torus of order-5 with monomino tiles, V1 created by William Walkington in 2022
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.

Colour diagram of a square Polyomino Area Magic Torus of order-5 with monomino tiles, V2 created by William Walkington in 2022
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.

Polyomino Area Magic Tori (PAMT) of Order-6


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").

Colour diagram of a partially pandiagonal Polyomino Area Magic Torus of order-6, made with heptominoes by William Walkington
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.

Colour diagram of a square Polyomino Area Magic Torus of order-6 with domino tiles, V1 created by William Walkington in 2022
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.

Observations

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:

Colour diagram of irregular rectangular Polyomino Area Magic Torus tiling of order-3 with monominoes, by William Walkington in 2022
Tiling with irregular rectangular shaped PAMT of order-3. Monominoes. S=24.

Diagram of irregular rectangular Polyomino Area Magic Torus tiling of order-3 with tetromino tiles, by William Walkington in 2022
Tiling with irregular rectangular shaped PAMT of order-3. Tetrominoes. S=15.

Diagram of irregular rectangular Polyomino Area Magic Torus tiling of order-3 with tromino tiles, by William Walkington in 2022
Tiling with irregular rectangular shaped PAMT of order-3. Trominoes. S=18.

Colour diagram of oblong Polyomino Area Magic Torus tiling of order-3 with tromino tiles, created by William Walkington in 2022
Tiling with oblong PAMT of order-3. Trominoes. S=24.

Colour diagram of oblong Polyomino Area Magic Torus tiling of order-4 with pentomino tiles, by William Walkington in 2022
Tiling with oblong pandiagonal PAMT of order-4. Pentominoes. S=34.

Colour diagram of oblong Polyomino Area Magic Torus tiling of order-5 with hexomino tiles, by William Walkington in 2022
Tiling with oblong pandiagonal PAMT of order-5. Hexominoes. S=65.

Colour diagram of square Polyomino Area Magic Torus tiling of order-5 with monomino tiles, created by William Walkington in 2022
Tiling with square pandiagonal PAMT of order-5. Monominoes. S=125.

Colour diagram of oblong Polyomino Area Magic Torus tiling of order-6 with heptomino tiles, by William Walkington in 2022
Tiling with oblong partially pandiagonal PAMT of order-6. Heptominoes. S=111.

Colour diagram of square Polyomino Area Magic Torus tiling of order-6 with domino tiles, by William Walkington in 2022
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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Monday 20 September 2021

How Dürer's "Melencolia I" is a painful but liberating metamorphosis!

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."
 

The Historical Context


The year 1514 CE came during a turbulent historical period, just three years before the Protestant Reformation, and the seemingly endless wars of religion which would follow. When Albrecht Dürer created "Melencolia I," he was expressing the philosophical, scientific, and humanist ideas of fifteenth-century Italy, and thus contributing to the beginning of a new phase of the Renaissance. Dürer was one of the first artists of Northern Europe to understand the importance of the Greek classics, and particularly the ideas of Plato and Socrates. The Renaissance idea was revolutionary, as it suggested that everyone was created "in imago Dei," in the image of God, and was capable of developing himself, or herself, to participate in the creation of the universe. This idea was now being gradually transmitted to all classes of society, thanks to the invention of the printing press; but it required a metaphorical language which could be deciphered by all, especially in a largely illiterate population. Although most of Dürer’s prints were intended for this wide public, his three master engravings (“Meisterstiche”), which include "Melencolia I," were aimed instead at a more discerning circle of fellow humanists and artists. The messages were more intellectual, using subtle symbols that would not be evident for common men, but could be decrypted by those initiated in the art.

The Metamorphosis of the Flying Creature

In the cartouche of Dürer's "Melencolia I", what at first looks like a flying bat is in fact a self-disembowelled flying rat!
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!""

How Melancholy leads to 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, is an illustration of the artist's melancholy, and is filled with symbols.
"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!

Acknowledgement

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.  

Latest Development

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.


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Sunday 2 August 2020

A Bimagic Queen's Tour

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.

Some Important Definitions


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.

The First Hypotheses


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.

The First Trials and Observations


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 some diagonal moves become inaccessible to the queen after "leaving" and "re-entering" the chess board
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.

When ± N/2, ± N/2 diagonal moves "leave" the chess board, they always "re-enter" it along a diagonal accessible to the queen.-
± 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:

A massive use of ±2, ±2 diagonal moves often produces a bimagic queen's tour on a torus board.
An open queen's tour on a torus board

The First Bimagic Queen's Tour


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:

Doubly-symmetric arrays of immutable bimagic series, specially created for a bimagic queen's tour
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:

A semi-magic torus that, after translation of the board viewpoint, reveals a bimagic queen's tour
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 and probably the first bimagic tour of any chess piece!
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 version of the bimagic queen's tour has an orthogonal bimagic constant of 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 path of the first bimagic queen's tour shows that only 4 orthogonal moves are used.
The Bimagic Queen's Tour Path

Conclusion


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.

* The answer to the open question has been given by Walter Trump on the 3rd August 2020. Please refer to the "Latest Developments" below!

Latest Developments!


The Answer to the Open Question!


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:



106 Bimagic Queen's Tours on an 8x8 Board!


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:




Other Publications about Bimagic Queen's Tours!


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.

Acknowledgements


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.

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