Sunday, 27 March 2016

First-Order Panmagic Torus T1

The smallest normal magic square is of the first-order:

order-1 pandiagonal magic square or magic torus T1

The dotted red lines across the magic square represent the magic diagonals. For a basic magic square, each row, column, and main diagonal must sum to the magic constant. The magic constant of a magic square is equal to the division of the triangular number of its squared order by its order. The magic constant (mcN) of an Nth order magic square (or torus) can be calculated as follows:
mcN  =  (N²)² + N²  x  1    =    N(N² + 1)
                     2            N                2
For the first-order magic square (or magic torus) the magic constant should therefore be:
mc1   =  1(1² + 1)   =  1,  which is the case.

The blue border of the unique number cell is also the limit of the magic square itself. The video below shows how this blue border merges to form single latitude and longitude lines on the curved 2D surface of the first-order magic torus:

Gluing a Torus video by Geometric Animations - University of Hannover, hosted by YouTube

We can check some of the simple conditions that define a basic magic odd-order torus, deduced whilst observing the unique third-order T3 magic torus:

The number of N-order squares (both magic and semi-magic squares) that are displayed on each N-order magic torus = N². The first-order magic torus should therefore display 1² = 1 first-order square, which is the case.

Basic magic odd-N-order tori  display not only 1 basic N-order magic square, but also N²-1 semi-magic N-order squares. The first-order basic magic square should therefore display 1 basic order 1 magic square, and 1²-1 = 0 semi-magic order 1 squares, which is also the case.

What is more surprising, although quite logical once we consider the question, is that the order 1 magic torus even satisfies the basic conditions for pandiagonality!

If pandiagonal (magic along all of its diagonals), an N-order torus displays N² pandiagonal squares of order N. The first-order torus should therefore display 1² pandiagonal squares = 1 pandiagonal square, which once again is the case.

Taking into account the panmagic properties of this torus, the adjective "trivial," which is often used to describe the first-order square, now seems quite disrespectful. For mathematical contexts, the dictionary definitions of trivial include: "simple, transparent, or immediately evident." However, there is more to the order 1 torus than first meets the eye! Not only is it pandiagonal, but its number One also signifies mathematical creation and the very beginnings... The Pythagoreans referred to the number One as the "monad," which engendered the numbers, which engendered the point, which engendered all lines, etc. For Plotinus and other neoplatonists, the number One was the ultimate reality and the source of all existence. The first-order torus, and the number One that it displays, together represent the "Big Bang."

I must admit to having completely overlooked this first-order magic torus until Miguel Angel Amela kindly sent me a copy of one of his studies "Pandiagonal Latin Squares and Latin Schemes in the Torus Surface," on the 9th March 2016. By extrapolating his findings, I discovered for myself the pandiagonal characteristics of the first-order, and I wish to thank Miguel for this paper which has been an inspiration for me.

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