Thursday, 20 June 2019

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 traditional "Dürer" magic square of order-4 that is portrayed in "Melencolia I"
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 magic torus that magnifies the "Dürer" magic square of order-4 by using 2 x 2 squares
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 zigzagA2,3,4,5...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 16 subsquares of the "Dürer" magic torus of order-4 and of the descendant "Michelet" magic torus of order-8.
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:

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