Pan-Zigzag Magic Tori Magnify the "Dürer" Magic Square

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.

Pan-Zigzag Descendants of the Magic Torus that displays the "Dürer" Magic Square

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 zigzag₂. Harry White states that V zigzag₂ magic squares are also V zigzagA_2,3,4,5...n, and underlines the fact that V zigzagA₂ is the same as V zigzag₂. It should be noted that the V zigzagA₃ waves are knight moves, and in correspondence with Dwane Campbell (who has also written about zigzag patterns), Harry White points out that "every magic square that has the Zigzag₂ property in both rows and columns will also have the Pan Knight Move property." In addition to the pan-4-way V zigzag₂ characteristics already mentioned above, this magic torus is also U zigzag 2-way (up, down). For more details of these classifications please refer to Harry White's Zigag Magic Squares page.

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:

Latest Development

Peter D. Loly, a senior scholar of the Department of Physics and Astronomy of the University of Manitoba, has kindly analysed the « Michelet » pan zigzag square of order-8, using the critical tools developed since the publication of « Signatura of magic and Latin integer squares: isentropic clans and indexing », a paper which he co-authored with Ian Cameron and Adam Rogers in 2012.

« Signatura of magic and Latin integer squares: isentropic clans and indexing » is a paper based on a video presentation in celebration of George Styan’s 75th, (at a conference held on the 19th July, 2012 at Będlewo, Poland), and was published in “Discussiones Mathematicae Probability and Statistics”, Volume 33, No 1-2, pages 121-149, in 2013. The paper can be obtained via a direct download link at: https://www.discuss.wmie.uz.zgora.pl/ps/source/publish/view_pdf.php?doi=10.7151/dmps.1147.

Following studies of the Shannon entropies of order-9 Sudoku and Latin square matrices by Newton and DeSalvo (P.K. Newton and S.A. DeSalvo, “The Shannon entropy of Sudoku matrices”, Proc. R. Soc. A 466 (2010), 1957–1975.), Cameron, Rogers and Loly calculate for magic squares in their paper: “a percentage of compression factor for the reduction of the entropy from a reference maximum entropy which goes as the logarithm of the order of the squares, ln(n), in order to provide a comparison across different orders. NDS found average compressions in the range of 21 to 25%.”

The magic square of order-4 in Dürer's engraving "Melencolia I"

Illustrated above, the magic square of order-4, from Dürer’s famous 1514 engraving “Melencolia I”, has the Frénicle index 175. It is hosted in clan α of Cameron, Rogers and Loly’s Table 1, in which the authors give the following percentage of compression factor:
C= 37. 23% (the highest compression of Group G1-3(α) that hosts the ancient Jupiter magic square, Dürer’s famous 1514 square, as well as one of Franklin’s magic squares).

Based on the “Durer” semi-pandiagonal magic square of order-4, the “Michelet” pan zigzag magic square of order-8 is as below:

The "Michelet" order-8 pan-zigzag magic torus of order-8,
a direct descendant of the "Dürer" magic square of order-4

Peter Loly’s calculations show the following:

singular values: [260.0, 143.15, 35.777, 8.1950, 0, 0, 0, 0],
eigenvalues: 260, 0, -64, 64
Σ_{i}ⁿσ_{i} = 260 + 143.15 + 35.777 +8.1950 = 447.12
σ₁ = 260 / 447.12 = 0.58150
σ₂ = 143.15 / 447.12 = 0.32016
σ₃ = 35.777 / 447.12 = 8.0017 × 10⁻²
σ₄ = 8.1950 / 447.12 = 1.8328 × 10⁻²
H = -σ₁ln(σ₁) -σ₁₂ln(σ₂) -σ₃ln(σ₃) -σ₄ln(σ₄)
H = -0.58150ln(0.58150) -0.32016ln(0.32016) -8.0017×10⁻²ln(8.0017×10⁻²) -1.8328×10⁻²ln(1.8328×10⁻²) = (-1.0σ₁lnσ₁-1.0σ₃lnσ₃-1.0σ₄lnσ₄-1.0σ₁₂lnσ₂ = H) = 0.95528
C=(1-(H/(ln(n))))∗100%.

The resulting percentage of compression factor of the “Michelet” pan zigzag magic square of order-8 is therefore as follows:

C = (1-0.95528/ln(4)) ∗ 100 = 31.091%

This can now be compared with Peter Loly’s calculation of the percentage of compression factor of the “Dürer” semi-pandiagonal square:

singular values: [34.0,17. 889,4. 4721,6. 0749×10⁻³⁸],
eigenvalues: -8,0,8,34,
{{16, 3, 2, 13}, {5, 10, 11, 8}, {9, 6, 7, 12}, {4, 15, 14, 1}}
34.00 + 17.889 + 4.4721 = 56.361
σ₁ = 34 / 56.361 = 0.60325
σ₂ = 17.889 / 56.361 = 0.3174
σ3= 4.4721 / 56.361 = 7.9347 × 10⁻²
H = -0.60325ln( 0.60325) -0.3174ln(0.3174) -7.9347×10⁻²ln(7. 9347×10⁻²) = 0.8702
C = (1-0.8702/ln(4)) ∗ 100 = 37.228%

The 37.228% percentage of compression factor C of the “Dürer” semi-pandiagonal square can be seen to be totally consistent with the 37.23% highest percentage compression factor of Group G1-3(α), as previously announced in Table 1 of the paper co-authored by Ian Cameron, Adam Rogers and Peter Loly in 2012: « Signatura of magic and Latin integer squares: isentropic clans and indexing ».

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First-Order Panmagic Torus T1

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:

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 illustrated above, 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, previously 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 admit to having rather underestimated 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 proof of 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."

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Magic Diagonal Sequences of 4th-Order Magic Tori

The number of fourth-order magic series is 86. The complete list of these magic series has been published by Walter Trump, who gave me confirmation that all of these magic series were present on fourth-order magic squares, even if the geometrical arrangement of the series was often chaotic. He pointed out that for example the series 7+8+9+10 does not exist in any row, column, or main diagonal of a magic 4x4-square, even when considering all the 24 possible permutations of the four numbers. However the series 7+8+9+10 occurs in the 2x2 centre square or in the four corners of certain 4x4 magic squares. As defined in MathWorld , or in Wikipedia, a magic series is not a specific sequence of numbers, but a set of numbers that add up to the magic constant.

The magic constant (MCn) of an nth order magic square (or torus) can be calculated:

MCn = (n²)² + n²  x  1    =    n(n² + 1)

                 2             n               2

For a fourth-order magic square (or magic torus) the magic constant is therefore:

MC4   =  4(4² + 1)    =   34

                    2

Magic diagonal sequences on fourth-order magic tori

Magic diagonal sequences on the Magic Torus Type T4.01.1 (index n° T4.194)

On any magic diagonal of a fourth-order magic torus (or of an order 4 magic square) :

a + b + c + d = 34

Considering the continuous curved surface of a magic torus, each of the 86 4th-order magic series can be expressed in 3 essentially different sequences:

a, b, c, d  (sequence type A) - which is also the magic series

a, b, d, c  (sequence type B)

a, c, b, d  (sequence type C)

As they curve round the torus the sequences have no beginning or end. The magic torus concept is explained in an earlier article "From the Magic Square to the Magic Torus." The interrelationships of the 4th-order magic tori, first explored in the "Fourth-Order Magic Torus Chart," are now more extensively detailed in "Multiplicative Magic Tori."

Theoretically the total number of different magic diagonal sequences on fourth-order magic tori would be 86 magic series x 3 sequence types = 258. However, after checking the 255 different fourth-order magic tori I discovered that there are 48 exceptions, leaving only 210 essentially different diagonal magic sequences. My observations have enabled me to determine some of the rules that govern fourth-order magic diagonal sequences:

Rules for magic diagonal sequences on fourth-order magic tori

Rule 1: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if consecutive numbers with an even successor number occur, then the series cannot be magic along diagonals.

Rule 2: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a and b are odd numbers, then (c + d) - (a + b ) must be a multiple of 3.

Rule 3: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a and b are even numbers, then (c + d) - (a + b) must be a multiple of 7.

Rule 4: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a - 1 = 8 - b = c - 9 = 16 - d, then the series cannot be magic along diagonals.

Presentation of the list of magic sequences

In the list that follows the index numbers of the magic series are the same as those already published by Walter Trump. I have just added the suffixes A, B, and C to identify the 3 essentially different sequences that are derived from each series. I wish to emphasise that although I have always chosen the lowest number to be the first, none of the sequences have either a beginning or an end, as they each form a continuous loop round their torus.

It was difficult to decide how to present the results. I have chosen to list the different sequence types A, B, and C separately, and I have grouped the sequences by sets of number rhythms in order to facilitate comparisons and reveal patterns. I have also sorted between the complementary (a+d = b+c = n²+1) sequences and the non-complementary (a+d ≠ b+c) sequences.

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 "Magic Diagonal Sequences of Fourth-Order Magic Tori" will then be displayed, with options for zooming.

Observations

Please note that all of the torus diagonals that sum to the magic constant are taken into account - even those that never coincide with the centre of a traditional magic square - such as for example the third magic diagonal sequence (4, 6, 16, 8) of the Frénicle square n° 275 (magic torus type n°T4.04.2 - index n°T4.098).

The complementary (a + d = c + b = n² + 1) diagonal series seem quite symmetrical and regular when compared to their orthogonal and sub-magic 2x2 square cousins. I have therefore decided to use larger sets of complementary sequences in this present study.

However there are many other ways of ordering the sequences to accord with the specific number rhythms of the different magic torus types. This is why an ideal list, that will suit all the torus types, remains a difficult objective.

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