As the positive integer entries of normal magic tori, and of their displayed magic squares, are always arranged in same-sum orthogonal arrays, it would seem logical to compensate any additions or subtractions to those entries, by making corresponding subtractions or additions. How can we develop this simple observation?
Preliminary Definitions
A normal magic square is an N x N array of same-sum rows, columns and main diagonals. If the magic diagonal condition is not satisfied, the square is deemed to be semi-magic. The torus, upon which a magic or semi-magic square is displayed, can be visualised by joining the opposite edges of the magic or semi-magic square in question.
There are N x N square viewpoints on the surface of a torus of order N. A normal magic torus has N x N arrays of same-sum latitudes and longitudes, and at least one magic intersection of magic diagonals on its surface. In his paper "Conformal Tiling on a Torus" published by Bridges in 2011, John M. Sullivan shows that, on a square torus T1, 0, the diagonal grid lines “form (1, ±1) diagonals on the torus, each of which is a round (Villarceau) circle in space.”
Semi-magic tori can also have magic diagonals, but the latter can never produce the magic intersections required for magic squares, either because the magic diagonals are single or parallel, or because they do not intersect correctly. Additional information about magic and semi-magic tori, with further explanations of their diagonals, can be found in the references at the end of the enclosed paper.
In order 4, starting with any magic torus, and examining cases where half of the torus numbers are subjected to equal additions, and the other half are subjected to same-integer subtractions, we notice the following: From its initial state (which we can call 0), each magic square can be transformed by four plus or minus operations that will either produce alternative magic or semi-magic square viewpoints of the same torus, or square viewpoints of other essentially different magic or semi-magic tori. Here's a simple example of two transformations:
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| 3 Magic Tori of Order 4 linked by by Complete-Torus, Same-Integer, Plus or Minus Operations |
Complete-Torus, Same-Integer, Plus or Minus Groups in Order 4
Please note that, in order 4, the orthogonal totals of magic and semi-magic tori are always 34. Therefore, only the diagonal totals, which can vary, are announced here. These totals are interesting because of their symmetries, and in the example illustrated above, there are pairs of diagonals that always sum to 68 (twice the magic sum). Each of the 255 magic tori of order 4 is similarly examined, and the results are given in tables and observations at the end of the following PDF paper: "Groups of Magic Tori of Order 4 Assembled by Complete-Torus, Same-Integer, Plus or Minus Operations"
Complete-Torus, Same-Integer, Plus or Minus Groups in Orders N > 4
The same method cannot always be applied to higher even orders. A quick look at some examples from even orders N = 6, N = 8, and N = 10 shows that certain magic tori have either no solutions whatsoever, or only one that can be used in a complete-torus same-integer plus or minus matrix to produce another orthogonally magic torus. In singly-even orders, even divisors with odd quotients have to be ruled out. Each case will need to be tested, and systematic computer checks will be necessary for these higher even orders.
An adaptation is of course required for odd orders, as their odd square totals do not have even integer divisors. But partial-torus same-integer divisions produce plus or minus solutions, and there are a variety of approaches.
The following paper, entitled "Examples of Partial Groups of Magic Tori of Orders N > 4 Assembled by Complete-Torus (or Near-Complete in Odd Orders) Same-Integer, Plus or Minus Operations" shows how the method used for order 4 gives some good results in higher orders:
New Developments
Since the 9th October 2024, in order to facilitate cross-referencing, the "List of 880 Frénicle Magic Squares of Order 4" (initially published on the 14th September 2019), has been updated with the detail of the Plus or Minus Groups of Magic Tori.
Since the 14th October 2024, in order to facilitate cross-referencing, the paper entitled "137 Plus-or-Minus Groups of Magic Tori of Order 4 listed with the details of the four squares generated by each of the 880 Frénicle magic squares" is available below:
