Friday, 5 October 2012

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:
order-4 magic torus index n° T4.001 type N° T4.05.1.02 in normalised form
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 subsequent 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.

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