137 Plus-or-Minus Groups of Magic Tori of Order 4, Linked by Plus-and-Minus Operations

General information about the 255 magic tori of order 4, and about the 880 magic squares that those tori display, can be found in a previous article, published on the 15th June 2023, and entitled "440 Torus-Opposite Pairs of the 880 Frénicle Magic Squares of Order-4".

The 137 complete-torus, same-integer, plus-or-minus groups of magic tori of order 4 have been presented in a previous article, published on the 24th April 2024, and entitled "Plus or Minus Groups of Magic Tori of Order 4".

Further information about the magic tori can be found in the lists of references that are appended to the papers in each of the above-mentioned articles.

Searching for 2-or-more-integer plus-and-minus links between the 137 ± Groups of Magic Tori

As we are looking for connections between complete-torus, same-integer, plus-or-minus groups of magic tori, it seems logical to search for links with 2-or-more-integer plus-and-minus operations. The closest links between the tori will be those where only 4 out of the 16 numbers are changed, and these will be for example 3/4 & 1/4: (±0, ±1), or 3/4 & 1/4: (±0, ±2) etc., depending on the differences between the cell entries.

A variant of these links will be those where 12 out of the 16 numbers are changed, and these can be written, for example, as 3/4 & 1/4: (±1, ±0), or as 3/4 & 1/4: (±2, ±0) etc., depending on the differences between the cell entries. However, a change of only 1/4 of the cell entries seems less disruptive than a change of 3/4 of these, and we can therefore suppose that the previously-mentioned 3/4 & 1/4: (±0, ±1), or 3/4 & 1/4: (±0, ±2) are closer links.

Close links also include those where only 8 out of the 16 numbers are changed, and these will be, for example, 1/2 & 1/2: (±0, ±1), or 1/2 & 1/2: (±0, ±2) etc., depending on the differences between the torus cell entries.

In some cases 2-integer plus-and-minus operations cannot be found, but more-complex 3-integer links may exist, such as 1/2, 3/8 & 1/8: (±2, ±0, ±1) or 1/2, 1/4 & 1/4: (±4, ±0, ±8). 3-integer links like these are frequent between the paired complementary and self-complementary torus groups.

Sometimes even 3-integer plus-and-minus operations are unavailable, but there exist 4-integer links, such as 1/4, 1/4, 1/4, & 1/4: (±0, ±2, ±4, ±6), or 3/8, 1/4, 1/4 & 1/8: (±3, ±0, ±6, ±9), which, if not particularly close, show interesting patterns. Links like these are common within the paired complementary torus groups that are to be found in the pages that follow.

Some Precisions Concerning the Enclosed Presentation

Because of its A3 format, this enclosed PDF file can at first seem unwieldy, especially when it is displayed on small-screen mobile phones. However the large display size allows same-page illustrations of complete sets of ± groups, which can be comprised of up to 24 examples. Please bear in mind that, in order to reduce space requirements and facilitate reading, most of the 137 ± groups are represented by single magic square viewpoints of only one of their magic tori. But occasionally, a same group is represented more than once, so as to facilitate the comprehension of multiple links. To find our which other magic tori are within a ± group, please refer to the details given in the paper "Plus or Minus Groups of Magic Tori of Order 4", already mentioned above.

In order to simplify the annotation of the pages of the paper enclosed below, the links between the ± groups are written as ± operations. This is slightly ambiguous, as there must always be a same number of plusses as minuses of any particular integer. Here therefore, the ± sign in front of an integer will mean an equal number of plusses and minuses of that integer. This will become evident as soon as we compare any pair of magic square viewpoints which are connected by the links in question. For full details, please refer to the paper below.

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Plus or Minus Groups of Magic Tori of Order 4

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:

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:

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List of the 880 Frénicle Magic Squares of Order-4

Bernard Frénicle de Bessy was the first to determine that there were 880 essentially different magic squares of order-4, and his findings were published posthumously in the "Table Générale des Quarrez Magiques de Quatre," in 1693.

Title of Frénicle's study of magic squares published in 1693

The "Table Générale des Quarrez Magiques de Quatre" published in 1693

It was more than 300 years later, that it was discovered that these 880 magic squares of order-4 were displayed on 255 magic tori of order-4. At first listed by type in 2011, the 255 magic tori of order-4 were later given additional index numbers in a "Table of Fourth-Order Magic Tori" in 2012. In 2019 some of the magic tori of order-4 were found to be extra-magic, with nodal intersections of 4 or more magic lines and / or knight move magic diagonals. Other studies of the magic tori of order-4 have included sub-magic 2 x 2 squares (in 2013), magic torus complementary number patterns (in 2017), and even and odd number patterns (in 2019). Then after a gap of 5 years, the magic tori have been found to belong to 137 plus or minus groups (in 2024).

It has become increasingly important to provide an easily accessible document that recapitulates these different findings. I have therefore compiled the following "List of the 880 Frénicle Indexed Magic Squares of Order-4," with their corresponding Dudeney types and also full details of the magic tori:

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