Fourth-Order Magic Torus T4.001

As shown in previous articles, the basic magic squares with Frénicle index n°002 and 448 are displayed on a same magic torus classified by type T4.05.1.02, and indexed (using normalised torus form) T4.001.

When studied on the surface of the magic torus, the line path between consecutive numbers is completely different to that of the magic square :

In order to better visualise how the T4.001 magic torus works I have produced the diagrams below. The magic torus is a symmetrically stable number system. 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:

The number couples of the magic torus produce different patterns on the magic squares that they display:

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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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Fourth-Order Panmagic Torus T4.194

The illustration shows the interchangeability of horizontal or vertical representations of a same panmagic torus. By twisting one side of the interlocked torus the other side turns, and vice versa. A simultaneous twisting and turning movement is also possible. The contact between the two sides of the interlocked torus takes place along perpendicularly connected circular tangents.

Interlocked Pandiagonal Torus - (excluded from CC licence)

 What happens at the intersection of these tangents? Here are two ways of seeing things:

The interlocked magic torus could symbolise the interaction between opposites such as behind and in front, outside and inside, etc. Alternatively, if one side of the torus represents the past and the other the future, the present could take place along the two perpendicular circular tangents.

Leaving philosophical considerations, and returning to mathematics, the above illustration portrays a pandiagonal or panmagic torus classified by type n°T4.01.1 - see the previous article "255 Fourth-Order Magic Tori, and 1 Third-Order Magic Torus" The torus displays 16 fourth-order pandiagonal squares (Frénicle index numbers 102, 104, 174, 201, 279, 281, 365, 393, 473, 530, 565, 623, 690, 748, 785, and 828). Since the publication of a new "Table of Fourth-Order Magic Tori" this torus is now also indexed and listed as the n° T4.194 in normalised square form:

Numbered following the Bernard Frénicle de Bessy index, the pandiagonal squares that are displayed on this pandiagonal torus are as follows:  

The illustration below shows a way to spot the similar number sequences in the different Frénicle squares showing that they come from the same magic torus:

In order to better visualise how the T4.194 torus works I have produced the diagrams below. The magic torus is a symmetrically stable number system. When the torus is twisted through 360° changing 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. Further below, the results of the study are illustrated by patterns on the panmagic square.

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