11-2017 By Douglas A. Engel, Littleton, Co

Space Numbers and How They Work Copyright (c) 1972-2017

(symmetry, math, games, magic squares, fractals…)

Jain magic square space number analysis. Ref. link: https://en.wikipedia.org/wiki/Most-perfect_magic_square .

The Parshvanath Jain temple in Khajuraho India has a most perfect magic square, meaning a magic square where the greatest number of possible magic sums appear. It was produced in about the year 1000. It is a 4x4 square with numbers 1 thru 16 placed in its 16 interior squares. The numbers are arranged so that all columns, main diagonals and 2x2 corner subsquares all add up to the magic constant 34. Many other symmetrical patterns also add up to 34. It is also known as a diabolical square since it has so many ways to derive the magic constant.

Since it is a 4x4 square and 2^4=16, it can be seen as four two dimensional power patterns, 2^0, 2^1, 2^2, 2^3 as shown above by the black and white 4x4 cells. Think of each power pattern as having zeroes in the white squares and the power number in the black squares. Thus the 2^3 pattern has 0’s and 8’s. The numbers of the magic square are produced by adding the power numbers in each position together. Then the lower right Jain square numbered cell would equal 0+0+2+1=3. Space numbers subtract 4 from the magic constant changing it from 34 to 30(numbers 0 thru 15) but it remains just as magic. Looking at the four power patterns you see immediately why it is so magical. For instance you can see that all the columns and rows have 2 black squares and two white squares and each 2x2 corner subsquare has two black squares and two white squares and similarly for the two main diagonals, and so on for the other magic patterns of the Jain magic square. In addition we can shuffle the power patterns(such as let the 2^0 and 2^2 power patterns exchange positions) however we like and the square stays most perfect, only the numbers change positions. We can also reverse the black and white of any power pattern and the magic is retained. Rotation of individual power patterns is not allowed as the two pairs of the power patterns are already ninety degree rotations of each other. Thus with shuffling and reversing we can have 24x16=384 different number arrangements of the Jain magic square. Of course many of these will be rotations or reflections of others others, but it shows the ease with which space numbers can be used to make more magic squares. The 4x4 magic square shown in Albrecht Durer’s 1514 engraving ‘Melancholia’ can be analysed the same way and produces a very similar space number with 384 number arrangements. Other magic squares could be produced using space numbers. For instance a 12x12 magic square would need to use power patterns in base 2 and base 3.

The Durer magic square as a space number

Here is an illustration of Albrecht Durer’s magic square from his 1514 engraving Melancholia. It is not quite as magic as the Jain magic square but has several interesting features detailed here https://en.wikipedia.org/wiki/Magic_square#Albrecht_D.C3.BCrer.27s_magic_square

The four power patterns can be exchanged, and black and white reversed for a total of 4!x(2^4)=192 new magic squares. There may be some duplicates since reversing all four is the same as a 180 degree rotation.

The figure below shows the Mars magic square as a base 5 space number. Since 5 is prime this is the only way it can be shown as a space number. It shows how much simpler an odd order magic square can be as opposed to an even order magic square.

Below we show the Sol magic square as a space number. Since it’s prime factors are 2 and 3 it is necessary to use both binary and trinary base numbers to break it into a space number. From this you see it has some complexity since it contains both even order and odd order properties. All even order magic squares are more complicated than odd order magic squares.