440 Torus-Opposite Pairs of the 880 Frénicle Magic Squares of Order-4

Finding the torus-opposite pairs of magic squares in even-orders:

The following diagram, which illustrates the array of n² essentially-different square viewpoints of a basic magic torus of order-n, starting here with the 4x4 magic square that has the Frénicle index n° 1, shows how a torus-opposite magic square of order-4 can be identified:

Array of 16 essentially-different square viewpoints of a basic magic torus of order-4

This basic magic torus, now designated as type n° T4.05.1.01, can be seen to display a torus-opposite pair of Frénicle-indexed magic squares n° 1-458 (with Dudeney pattern VI), as well as 7 torus-opposite pairs of semi-magic squares that the discerning reader will easily be able to spot.

The relative positions of the numbers of torus-opposite squares can be expressed by a simple plus or minus vector. There are always two equal shortest paths towards the far side of the torus: These are, in toroidal directions, east or west along the latitudes, and in poloidal directions, north or south along the longitudes of the doubly-curved 2D surface. So, if the even-order is n, then:

v  =  ( ± n/2, ± n/2 )

"Tesseract Torus" by Tilman Piesk CC-BY-4.0
 https://commons.wikimedia.org/w/index.php?curid=101975795

Torus-opposite squares are not just limited to basic magic tori, as they also exist on pandiagonal, semi-pandiagonal, partially pandiagonal, and even semi-magic tori of even-orders. Therefore, for order-4, we can either say that there are 880 magic squares, or we can announce that there are 440 torus-opposite pairs of magic squares. Similarly, the 67,808 semi-magic squares of order-4 can be expressed as 33,904 torus-opposite pairs of semi-magic squares, etc.

Why torus-opposite pairs of magic squares cannot exist in odd-orders

A torus-opposite magic square always exists in even-orders because an even-order magic square has magic diagonals that produce a first magic intersection at a centre between numbers, and another magic intersection at a second centre between numbers on the far side of the torus (at the meeting point of the four corners of the first magic square viewpoint). However, in odd-orders, where a magic square has magic diagonals that produce a magic intersection over a number, then a sterile non-magic intersection always occurs between numbers on the far side of the torus (at the meeting point of the four corners of the initial magic square). This is why torus-opposite pairs of magic squares cannot exist in the odd-orders.

255 magic tori of order-4, listed by type, with details of the 440 torus-opposite pairs of the 880 Frénicle magic squares of order-4

The enumeration of the 255 magic tori of order-4 was first published in French on the 28th October 2011, before being translated into English on the 9th January 2012: "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus." In this previous article, the corresponding 880 fourth-order magic squares were listed by their Frénicle index numbers, but not always illustrated. The intention of the enclosed paper is therefore to facilitate the understanding of the magic tori of order-4, by portraying each case.

Here, Frénicle index numbers continue to be used, as they have the double advantage of being both well known and commonly accepted for cross-reference purposes. But please also note, that now, in order to simplify the visualisation of the magic tori, their displayed magic squares are not systematically presented in Frénicle standard form. 

In the illustrations of the 255 magic tori of order-4, listed by type with the presentation of their magic squares, the latter are labelled from left to right with their Frénicle index number, followed by, in brackets, the Frénicle index number of their torus-opposite magic square, and finally by the Roman numeral of their Dudeney complementary number pattern. Therefore, for the magic square of order-4 with Frénicle index n° 1 (that forms a torus-opposite pair of magic squares with Frénicle index n° 458; both squares having the same Dudeney pattern VI), its label is 1 (458) VI.

To find the "255 magic tori of order-4, listed by type, with details of the 440 torus-opposite pairs of the 880 Frénicle magic squares of order-4" please consult the following PDF:

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Third-Order Magic Torus T3

The first illustration shows a line path escaping from the boundaries of the traditional 3x3 magic square. The liberation of the line path is possible once we agree that magic squares are partial viewpoints of magic tori. This representation differs greatly from that of the traditional line path shown in the second illustration.

During the XVIIth century Claude-Gaspard Bachet de Méziriac (1581-1638) was the first to propose line paths that linked the numbers of magic squares in numerical order. The above illustration shows the line path of the traditional 3rd-order magic square "Saturn," previously associated with the astrological planet by Heinrich Cornelius Agrippa (1486-1535) in his "De occulta philosophia libri tres".

In order to better visualise how the T3 torus works I have produced the diagrams below. The magic torus is a symmetrically stable number system. When the torus is twisted through 360° different number couples produce continuously balanced tensions. I have indicated some of the mathematical properties but you will notice others when you contemplate this beautiful counting machine:

There are 9 squares on the 3rd order magic torus. Beginning with the 3x3 magic square in Frénicle standard form (at the top left of the following illustration), and taking into account the scrolling effect we obtain another eight 3rd-order semi-magic squares:

After transformation of these squares into Frénicle standard form we obtain:

The 3rd-order magic torus displays just one magic square. Although there are two crossed magic diagonals on the torus, they only produce a single magic intersection over the number 5, whilst the second intersection occurs in the interval between the numbers 6, 4, 8 and 2. It is interesting to note that the mean number of 6+4+8+2 is 5, and that the mean numbers of the couples 6+4 and 8+2 are also 5.

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Table of Fourth-Order Magic Tori

In a previous article "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus," the 255 fourth-order magic tori have been identified and listed by type, following the numerical order of the Frénicle index numbers of the squares displayed on each torus. Although it was useful to begin this way, the classification by type is not practical when searching for a specific torus starting with any square.

I have therefore created an additional "Table of Fourth-Order Magic Tori" in which the 255 tori are also indexed and listed in numerical order for convenient reference. This table is, in a way, a homage to Bernard Frénicle de Bessy, as each torus is represented by a normalised square. The following illustration shows an example of the normalised square that represents the magic torus index n° T4.001 (type n° : T4.05.1.02), which displays not only 2 basic magic squares (Frénicle index numbers 2 and 448), but also 14 semi-magic squares:

The normalised squares are not necessarily traditional magic squares. Whether magic or semi-magic, they are just standard viewpoints of each magic torus. To see the other squares that are displayed by the torus, you need to displace the viewpoint, as explained in a previous article "From the Magic Square to the Magic Torus."

To view the complete "Table of Fourth-Order Magic Tori," please note that if you click on the button that appears at the top right hand side of the pdf viewer below, a new window will open and full size pages will then be displayed, with options for zooming.

Latest Developments

Further to studies published on the 28th March 2013, this table was revised on the 14th April 2013, to take into account the sub-magic 2x2 squares that are displayed on each torus, and then again on the 28th April 2013, to integrate new subdivisions that take into account different Dudeney types.

I wish to express my thanks to Aale de Winkel who pointed out an initial inversion of T4.003 and T4.004, which have since been interchanged to respect numerical order.

255 (the number of fourth-order magic tori) is now the fourth number of the sequence A270876 "Number of magic tori of order n composed of the numbers from 1 to n^2," published by the On-Line Encyclopedia of Integer Sequences (OEIS). 

Though the representation by normalised squares may not be ideal, it does give us a good insight of the subtle permutations that engender essentially different tori. The interrelationships of these tori are explored further and illustrated in an article "Multiplicative Magic Tori." published on the 21st January 2018. This post shows that the 255 magic tori of order-4 are different multiplied states of 82 Multiplicative Magic Tori (MMT) of order-4!

Since the 20th June 2019, twenty-seven of the 255 magic tori of order-4 are shown (using classic magic square geometry) to be Extra-Magic, having a parallel magic system with nodal intersections of 4 magic lines over numbers. These intersections do not yield traditional magic squares but are very significant for magic tori which have a limitless surface with no centre! Taking these findings into consideration, 136 of the 880 Frénicle-indexed magic squares are extra-magic! And when we take knight move magic diagonals into account, 6 intersecting magic lines can sometimes occur, and the total numbers of Extra-Magic Tori rise again! Since the 13th August 2019 a new article entitled "Extra-Magic Tori and Knight Move Magic Diagonals" confirms these findings and illustrates the different cases of extra-magic line intersections.

Since the 2nd September 2019, and the publication of a new article entitled "Even and Odd Number Patterns on Magic Tori of Orders 3 and 4", the Table of Fourth-Order Magic Tori is updated to include the details of the 4 essentially different even and odd number patterns.

Since the 24th April, 2024, a new article entitled "Plus or Minus Groups of Magic Tori of Order 4" now demonstrates that the 255 magic tori of order 4 (and the 880 magic squares that these tori display) originate from 137 plus or minus groups of order 4!
 

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From the Magic Square to the Magic Torus

This post is 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):

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:

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:

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

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):

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...

Inside-out torus by Surot [Public domain], via Wikimedia Commons

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