Sunday, 5 April 2020

From the Magic Square to the Magic Torus

This post remplaces an article that was previously published on Friday 9th March 2012, (a translation of the article "Passage du Carré Magique au Tore Magique," first published in French on the 11th November 2011).

To visualise a magic torus let's begin with the Frénicle index square n° 91 (Bernard Frénicle de Bessy's classification of 880 fourth order magic squares - published posthumously in 1693 in his book "Des Quarrez Magiques" - see the listing on Harvey Heinz's site - Frénicle n°s 1 - 200, Frénicle n°s 201 - 400, Frénicle n°s 401 - 600, Frénicle n°s 601 - 880):
order-4 Frénicle magic square index 91
Extending this magic square we can construct a matrix of possibilities sized (2N-1)² to discover the other squares that are displayed on the magic torus:

order 4 Frénicle magic square index 91 expanded

This way we can find another 15 squares. We can see that there are 4 partially pandiagonal squares and 12 semi-magic squares displayed on the magic torus:

order 4 Frénicle index 91 converting to magic torus step 1

Then we can convert the squares that we have found into Frénicle standard form:

order 4 Frénicle index 91 converting to magic torus step 2

We can see that the magic torus displays not only the partially pandiagonal square Frénicle n°91, but also 3 other partially pandiagonal squares (Frénicle n°150, 320, and 394), as well as 12 semi-magic squares.

We are looking at the partially pandiagonal torus type n° T4.03.1.2. (index n° T4.174):

order 4 perspective partially pandiagonal magic torus index T4.174 type T4.03.1.2

Ironically, although the procedure is invaluable in eliminating doubles, conversion into Frénicle standard form (by rotation, transposition, and / or reflection) has until today concealed the toroidal continuity of magic squares.

In the end, magic squares are to be observed in only two fundamental ways: either from outside the torus or from within, and it does not really matter which, as the torus can always be turned inside out...
animation inside-out magic torus
Inside-out torus by Surot [Public domain], via Wikimedia Commons

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