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 ».