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 flying 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.

Read more ...

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 Jellis, 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!

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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Thursday, 30 April 2020

John Horton Conway and LUX

On the 25th April 2020, I came across a newsletter of the French magazine "Tangente" announcing that, despite the ongoing COVID-19 difficulties, their next issue n° 194 would still be published at the end of May. They stated that this future magazine would contain a main section dedicated to the late John Horton Conway, famous for his "Game of Life" (a cellular automaton in which cells live, die and are born). Having been previously unaware of Conway's death, I immediately wished to inform the fellow members of a magic square circle of the sad news, and sent them the following e-mail: 

Dear all,
I have just learnt that John Horton Conway passed away on the 11th April in New Brunswick, New Jersey, at the age of 82, after complications from COVID-19. I did not know him personally, but I expect that some of you did. His well-known LUX method for magic squares is an algorithm for creating magic squares of order 4n+2, where n is a natural number, and his "lozenge" method is another method of construction of magic squares. So this is a considerable loss to the magic square community, and goes to show that the virus is indiscriminate and dangerous.

Here's hoping that a cure will be found soon.

Stay safe!

William 

I rapidly received replies from Peter Loly, Bob Ziff, Awani Kumar and Miguel Angel Amela:

Peter Loly stated that Conway's passing away reminded him of his colleague Richard Guy who had passed away aged 103 on the 30th September 2019, and also of the losses of John Hendricks and Harvey Heinz, both of whom he had met. He asked if I could send this information out on the blog.

Bob Ziff replied to say that Conway had had the bad luck of being in a nursing home for some other health problems, and concluded by writing "Let’s all hope that we stay healthy!"

Awani Kumar replied with a simple "So sad. RIP."

Miguel Angel Amela then replied with a photomontage of a very fitting "in memoriam" plaque: He proposed that John Horton Conway's name could be inscribed above a 6x6 magic square constructed with Conway's LUX method, and that the Spanish words "descansa en la LUX" could be inscribed below. The Spanish text "descansa en la LUX" means "rest in the LUX" or "rest in the LIGHT." Miguel stated that in Latin, the inscription would be R.I.L. or "REQUIESCAT IN LUX," and he suggested a poem written by Art Durkee, entitled "requiescat in lux." 

An "In Memoriam" plaque for John Horton Conway

Miguel Amela's idea of an "in memoriam" plaque in homage of John Horton Conway and his LUX method for constructing magic squares
Adaptation of Miguel Angel Amela's idea of an "in memoriam" plaque for John Horton Conway

Perhaps, one day, a plaque like this will be placed at Cambridge or Princeton, or at another of Conway's haunts.

John Horton Conway's LUX Method


Martin Erickson has already described Conway's LUX method in page 98 of "Aha Solutions" (published by the Mathematical Association of America in 2009), and he even used the same "in memoriam" square of order-6 as a base array to construct a LUX square of order-18! However, here it is useful to examine how the LUX method can be used to construct the "in memoriam" square itself, and the following pdf file explains the construction steps:



The "in memoriam" magic square can also be programmed, for example using languages such as Elixir, Haskell and Ruby, as shown in "Magic squares of singly even order," edited by the programming chrestomathy site "Rosetta Code."

Conclusions


This short article only intends to commemorate a small part of John Horton Conway's work on magic squares. Because I myself have never had the pleasure of meeting him, those who did are far better placed to write about his other accomplishments, and about his life in general. The obituary written by Catherine Zandonella for Princeton University gives us a good insight, not only of his many achievements, but also of his mathematical playfulness and of his kind attention to others that made him many friends. Another source of information with useful external links is John Horton Conway's biography written by John Joseph O'Connor and Edmund Frederick Robertson for the MacTutor History of Mathematics Archive at the University of St Andrews. 

Acknowledgements


I wish to thank Peter Loly, Bob Ziff, Awani Kumar and Miguel Angel Amela for replying to my e-mail. Their messages and contributions have inspired me to write this post.

With COVID-19 still active and dangerous at the time of writing, I urge you all to take heed of the social distancing guidelines, and keep safe!
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Sunday, 1 March 2020

A Tetrad Puzzle

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.

Definition of a Tetrad


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.

Tetrads Published in Martin Gardner's "Mathematical Games" Column


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.

Tetrad Puzzle Transformations


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:

1/ Assembly of Simple Tetrad Jigsaw Puzzle Pieces, with 3 Tabs and 3 Blanks

Simple congruent tetrad jigsaw puzzle pieces make a holeless 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.

Simple congruent tetrad jigsaw puzzle pieces make a tetrad with a hole.
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.

2/ Assembly of Tetrad Jigsaw Puzzle Pieces, with 10 Tabs and 8 Blanks


Congruent jigsaw puzzle pieces with 10 tabs and 8 blanks make this holeless tetrad.
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.

Congruent jigsaw puzzle pieces with 10 tabs and 8 blanks make this tetrad with a hole.
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.

3/ Assembly of Tetrad Jigsaw Puzzle Pieces, with 18 Alternate Tabs and Blanks


Congruent jigsaw puzzle pieces with 18 alternating tabs and blanks make this holeless tetrad.
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.

Congruent jigsaw puzzle pieces with 18 alternating tabs and blanks make this tetrad with a hole.
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.

Observation on the limitations of tetrad puzzle pieces with tabs and blanks


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:

4/ Assembly of Tetrad Jigsaw Puzzle Pieces, with 9 Positives and 9 Negatives

Congruent jigsaw puzzle pieces with 9 positives and 9 negatives make this holeless tetrad.
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.
Congruent jigsaw puzzle pieces with 9 positives and 9 negatives make this tetrad with a hole.
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.

5/ Assembly of Tetrad Jigsaw Puzzle Pieces, with 10 Positives and 8 Negatives

Congruent jigsaw puzzle pieces with 10 positives and 8 negatives make this holeless tetrad.
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.

Congruent jigsaw puzzle pieces with 10 positives and 8 negatives make this tetrad with a hole.
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.
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Saturday, 14 September 2019

List of the 880 Frénicle Magic Squares of Order-4

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 from "Divers ouvrages de mathematique et de physique / par Messieurs de l'Académie Royale des Sciences [M. de Frénicle ... et al.]."
Title of Frénicle's study of magic squares published in 1693

This is the first page of the Table General of Magic Squares of Order-4 by Bernard Frénicle de Bessy
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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Monday, 2 September 2019

Even and Odd Number Patterns on Magic Tori of Orders 3 and 4

Ancient references to the pattern of even and odd numbers of the 3 × 3 magic square appear in the "I Ching" or "Yih King" (Book of Changes). In the introduction to the Chou edition, the scroll of the river Loh or "Loh-Shu", which is represented by the magic square of order-3, is written with black dots (the yin symbol and emblem of earth) for the even numbers, and white dots (the yang symbol and emblem of heaven) for the odd numbers. The "Loh-Shu" is incorporated in the writings of Ts'ai Yuän-Ting who lived from 1135 to 1198:
The Scroll of Loh is an ancient drawing of the 3x3 magic square by Ts'ai Yuän-Ting.
The Scroll of Loh, according to Ts'ai Yüang-Ting

A normal magic torus (or magic square) of order-n uses consecutive numbers from 1 to n² and consequently, for the order-3, the magic constant (MC) = 15. The Loh-Shu Magic Torus of Order-3 is presented below, together with its corresponding even and odd number pattern P3:


The magic torus or magic square of order-3 and its even and odd number pattern P3.
The "Loh-Shu" Magic Torus of Order-3 and its Even and Odd Number (x mod 2) Pattern P3

The magic torus T3 of order-3 is shown here as seen from the Scroll of Loh magic square viewpoint.

The even and odd number pattern P3 has reflection symmetry (with 6 lines of symmetry); rotational symmetry (of order 4); and point symmetry (over number 5, between numbers 3 and 7, between numbers 1 and 9, and between numbers 4, 6, 2, and 8).

The Even and Odd Number Patterns of the Magic Tori of Order-4


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

The "Table of Fourth-Order Magic Tori" lists and describes each magic torus of order-4 using index numbers, while "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus" gives a group categorisation of the magic tori of order-4 using type numbers. Both of these classifications give the details of the Frénicle indexed magic squares that are displayed by each torus.

In the study that follows it can be seen that there are 4 essentially different even and odd number patterns on the magic tori of order-4:


The Order-4 Even and Odd Number Pattern P4.1
(represented by the pandiagonal torus T4.198)


A pandiagonal torus or pandiagonal square of order-4 and its even and odd number pattern P4.1.
The Pandiagonal Torus T4.198 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.1

The pandiagonal torus of order-4 (with index n° T4.198 and type n° T4.01.2) is shown here as seen from the Frénicle n° 107 magic square viewpoint.

The even and odd number pattern P4.1 has reflection symmetry (with 6 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The pattern P4.1 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.1 concerns 79 out of the 255 magic tori, and 404 out of the 880 magic squares of order-4. These include the 3 pandiagonal tori that display 48 pandiagonal squares; 30 semi-pandiagonal tori that display 240 semi-pandiagonal squares; 12 partially pandiagonal tori that display 48 partially pandiagonal squares; and also 34 basic magic tori that display 68 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.2
(represented by the semi-pandiagonal torus T4.059)


A semi-pandiagonal torus or semi-pandiagonal square of order-4 and its even and odd number pattern P4.2.
The Semi-Pandiagonal Torus T4.059 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.2

The semi-pandiagonal torus of order-4 (with index n° T4.059 and type n° T4.02.2.01) is shown here as seen from the Frénicle n° 21 magic square viewpoint.

The even and odd number pattern P4.2 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 4); and multiple point symmetry when reading as a torus. The pattern P4.2 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.2 concerns 72 out of the 255 magic tori, and 256 out of the 880 magic squares of order-4. These include 18 semi-pandiagonal tori that display 144 semi-pandiagonal squares; 6 partially pandiagonal tori that display 16 partially pandiagonal squares; and also 48 basic magic tori that display 96 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.3
(represented by the partially pandiagonal torus T4.186)


A partially pandiagonal torus or partially pandiagonal square of order-4 and its even and odd number pattern P4.3.
The Partially Pandiagonal Torus T4.186 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.3

The partially pandiagonal torus (with index n° T4.186 and type n° T4.03.1.3) is shown here as seen from the Frénicle n° 100 magic square viewpoint.

The even and odd number pattern P4.3 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 4); and multiple point symmetry when reading as a torus. The pattern P4.3 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.3 concerns 16 out of the 255 magic tori, and 44 out of the 880 magic squares of order-4. These include 6 partially pandiagonal tori that display 24 partially pandiagonal squares; and also 10 basic magic tori that display 20 basic magic squares.

The Order-4 Even and Odd Number Pattern P4.4
(represented by the basic magic torus T4.062)


A basic magic torus or basic magic square of order-4 and its even and odd number pattern P4.4.
The Basic Magic Torus T4.062 of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.4

The basic magic torus (with index n° T4.062 and type n° T4.05.1.12) is shown here as seen from the Frénicle n° 37 magic square viewpoint.

The even and odd number pattern P4.4 has reflection symmetry (with 8 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The pattern P4.4 also has negative symmetry (evens to odds and vice versa).

The even and odd number pattern P4.4 concerns 88 out of the 255 magic tori, and 176 out of the 880 magic squares of order-4. All of these 88 magic tori and the displayed 176 magic squares are basic magic.

Observations on the Even and Odd Number Patterns of the Magic Tori and Magic Squares of Order-4


The findings of the even and odd number patterns of the magic tori of order-4 are recapitulated and analysed in comparative tables, (together with the Dudeney complementary number patterns and the Magic Torus self-complementary and paired complementary number patterns), in the file below:



The Even and Odd Number Patterns of the Semi-Magic Tori of Order-4


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.

In "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus" the 4,038 semi-magic tori of order-4 have been sorted, and type numbers have been attributed taking into account the arrangements of their magic diagonals (if any). 

Each of the four even and odd number patterns already found above on the magic tori of order-4, also occur on semi-magic tori of order-4. For example, from left to right below, we find a semi-magic torus type T4.08.0W with pattern P4.1; a semi-magic torus type T4.08.0X with pattern P4.2; a semi-magic torus type T4.07.0Y with pattern P4.3; and a semi-magic torus type T4.09.00Z with pattern P4.4:

Semi-magic tori or semi-magic squares of order-4 that show the even and odd number patters P4.1 to P4.4.
Semi-Magic Tori of Order-4 with Even and Odd Number (x mod 2) Patterns P4.1, P4.2, P4.3 and P4.4

The even and odd number patterns P4.1 to P4.4 are not the only ones that can be found, but the patterns which are specific to the semi-magic tori of order-4 have neither been fully investigated nor precisely counted, and the examples that follow only represent some of the different varieties.

The Order-4 Even and Odd Number Pattern P4.5
(represented by the semi-magic torus type T4.06.0A)


A semi-magic torus or semi-magic square of order-4 and its even and odd number pattern P4.5.
The Semi-Magic Torus type T4.06.0A of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.5

This semi-magic torus type n° T4.06.0A was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.5 has diagonal reflection symmetry (with 4 lines of symmetry); and negative (evens to odds and vice versa) rotational symmetry (of order 2). The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.6
(represented by the semi-magic torus type T4.07.0A)


A semi-magic torus or semi-magic square of order-4 and its even and odd number pattern P4.6.
The Semi-Magic Torus type T4.07.0A of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.6

This semi-magic torus type n° T4.07.0A was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.6 has reflection symmetry (with 4 lines of symmetry); rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.7
(represented by the semi-magic torus type T4.09.00M)


A semi-magic torus or semi-magic square of order-4 and its even and odd number pattern P4.7.
The Semi-Magic Torus type T4.09.00M of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.7

This semi-magic torus type n° T4.09.00M was found by Walter Trump, and has been previously illustrated in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The even and odd number pattern P4.7 has both positive and negative reflection symmetry; negative rotational symmetry (of order 2); and multiple negative point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

The Order-4 Even and Odd Number Pattern P4.8
(represented by the semi-magic torus type T4.10.000X)


A semi-magic torus or semi-magic square of order-4 and its even and odd number pattern P4.8.
The Semi-Magic Torus type T4.10.000X of Order-4 and its Even and Odd Number (x mod 2) Pattern P4.8

3,726 semi-magic tori of the type 10 have been found by Walter Trump, and his findings are detailed in "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus".

The T4.10.000X's even and odd number pattern P4.8 has negative reflection symmetry (2 lines of symmetry); negative rotational symmetry (of order 2); and multiple point symmetry when reading as a torus. The blocks of even and odd numbers are translations of each other.

General Observations


The "Table of Fourth-Order Magic Tori" has been revised to include the even and odd number pattern references for each of the magic tori of order-4 (listed together with the Frénicle indexed magic squares that each magic torus displays).

Other interesting websites that examine, but do not enumerate, the different even and odd number patterns on magic squares of order-4, include "4x4 Magic Squares" by Dan Rhett and "Pattern in Magic Squares" by Vipul Chaskar.

A brief look at higher-orders suggests that the order-5 has many irregular patterns; the order-6 has many patterns with negative reflection symmetry (evens to odds and vice versa); the order-7 has many patterns with rotational symmetry; and the order-8 has patterns very like those of the order-4 examined above. All of these orders merit an in-depth analysis of their even and odd number patterns.

In conclusion, it is interesting to note that the Arab mathematicians of the tenth century constructed odd-order magic squares with striking even and odd number patterns (as for example the magic squares in figures b44 and b49 on pages 243 and 248 of Magic Squares in the Tenth Century: Two Arabic treatises by Anṭākī and Būzjānī, translated by Jacques Sesiano). And in the same context, Paul Michelet has recently brought to my attention a magnificent 15 x 15 magic square by Ali b. Ahmad al-Anṭākī (d. 987). Al-Anṭākī's bordered magic square is solid evidence of the Middle Eastern mathematicians' mastery of even and odd number patterns!
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