Magic cubes

9-fold Magic Square Pattern of Tao Te Ching Insights (Part #9)

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There are extensive resources on magic cubes and hypercubes [notably Harvey Heinz and Marián Trenkler] that may offer even more powerful ways of organizing the 81 insights. A magic cube is a three-dimensional version of the magic square in which the rows, columns, pillars (or "files"), and four space diagonals each sum to a single number known as the magic constant. If the cross section diagonals also sum to that constant, the magic cube is called a perfect magic cube; if they do not, the cube is called a semiperfect magic cube, or sometimes an Andrews cube (Gardner 1988). A pandiagonal cube is a perfect or semiperfect magic cube which is magic not only along the main space diagonals, but also on the broken space diagonals [more]. In a panmagic square, in addition to the main diagonals, the broken diagonals also sum to the magic constant.

Harvey Heinz (Magic Cubes - Introduction, 2003) has reviewed the variety of, often confusing, definitions and features of "magic cubes" (see also his Magic Cubes Definitions, which includes a discussion of cube features) and has allocated them to distinct classes according to the types of parts that must sum correctly for the more advanced cubes. His classes may be summarized here as:

• Simple: Containing no, or less then 3m, orthogonal magic squares. Only the rows, columns, pillars and triagonals are required to sum correctly for a simple magic cube.
• Pantriagonal: All pantriagonals must sum correctly. There may be some simple and/or pandiagonal magic squares, but not enough to satisfy any other classifications.
• Diagonal: All 3m planar arrays must be 'simple' magic squares.
• Pandiagonal: All 3m planar arrays must be 'pandiagonal' magic squares. The 6 oblique squares are always magic. One of them may be pandiagonal magic.
• Perfect: All 3m planar arrays must be 'pandiagonal' magic squares. In addition, all pantriagonals must sum correctly. These two conditions combine to provide another 6m pandiagonal magic squares.

Heinz notes that a magic cube is called normal if it consists of the numbers 1 to m³ (or 0 to m³ - 1). A magic cube is called associated if all pairs of two numbers diametrically equidistant from the center of the cube equal the sum of the first and last number in the series. If the associated cube (or other dimension of hypercube) is an odd order, then the center of the cube is a cell containing one half the sum of the first and last number in the series.

Heinz provides a generalized definition as follows: A hypercube of dimension n is perfect if all pan-n-agonals sum correctly, and all lower dimension hypercubes contained in it are perfect! He also provides spreadsheets for testing them. Heinz has collaborated with J. R. Hendricks to produce a A Unified Classification system for Magic Cubes (Journal of Recreational Mathematics, 2002).

The relationship of the 81 tetragrams of the Taoist classic Tai Hsuan Ching (or Tài Xuán Jïng) and the Tao Te Ching has most recently been explored in relationship to modern physics by Tony Smith (I Ching (Ho Tu and Lo Shu), Genetic Code, Tai Hsuan Ching, and the D4-D5-E6-E7-E8 VoDou Physics Model ). According to Smith:

To construct the Tai Hsuan Ching, consider the Magic Square sequence as a line 3 8 4 9 5 1 6 2 7 with central 5 and opposite pairs at equal distances. If you try to make that, or a multiple of it, into a 9x9 Magic Square whose central number is the central number 41 of 9x9 = 81 = 40+1+40, you will fail because 41 is not a multiple of 5.

However, since 365 = 5x73 is the central number of 729 = 364+1+364, you can make a 9x9x9 Magic Cube with 9x9x9 = 729 entries, each 9x9 square of which is a Magic Square. The Magic Cube of the Tai Hsaun Ching gives the same sum for all lines parallel to an edge, and for all diagonals containing the central entry. The central number of the Magic Cube, 365....

The total number for each line is 3,285 = 219 x 15. The total of all numbers is 266,085 = 5,913 x 45.

Since 729 is the smallest odd number greater than 1 that is both a cubic number and a square number, the 729 entries of the 9x9x9 Magic Cube with central entry 365 can be rearranged to form a 27x27 Magic Square with 729 entries and central entry 365. 27 = 3x3x3 = 13+1+13 is a cubic number with central number 14, and there is a 3x3x3 Magic Cube with central entry 14 (14 is the dimension of the exceptional Lie algebra G2) and sum 42...

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