Hypercube Aspects
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Hypercube Aspects
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Square aspectsIn a
magic square, there are 4 corners that can be placed in the upper left position.
Each of these corners has two rays and because 4 times 2 = 8, a magic square has
8 aspects.
Normalized position For enumerating and listing the magic squares of a given
order, one of the eight above positions must be designated the normal
position. This standard prevents confusion and permits easy comparison
when the list is sorted. Interestingly, the number of order-5 magic squares was not
established until December, 1973. At that time, Richard Schroeppel calculated
and proved that there were 275,305,224 basic order-5 magic squares.
[3] Number of basic magic squares for orders higher then five have only been estimated. [4] For higher dimension hypercubes, only order-3 numbers are known. There are 4 basic magic cubes, 58 basic magic tesseracts,. 2992 basic magic dimension 5, and 543,328 basic dimension 6 magic hypercubes. [5] In 1917, W. S. Andrews published all four order-3 cubes,
but presumably without realizing that that was all there were.
[6] Hendricks found all 58 basic order-3 magic tesseracts by 1998, and showed that there could be no more. This was confirmed independently by others, including M. Collison, A. W. Johnson, Jr., and Key Ying Lin. [8]
Normalized position – magic squares Before 1693, Frénicle established two simple rules to determine the standard position for order-4. These same rules are now used for normalizing all other orders of magic squares. 1. The smallest number in any corner of the magic square must be in the top left corner. If it is not, rotate the square until it is. 2. The second number in the top row must be of lower then the first number in the second row. These are the two numbers adjacent to the one in the top left corner.
Text list of the first 4 order-4 magic squares. Index 1st row 2nd row 3rd row 4th row 1 01 02 15 16 12 14 03 05 13 07 10 04 08 11 06 09 2 01 02 15 16 13 14 03 04 12 07 10 05 08 11 06 09 3 01 02 16 15 13 14 04 03 12 07 09 06 08 11 05 10 4 01 03 14 16 10 13 04 07 15 06 11 02 08 12 05 09
Normalized position – magic cubes The normalized position for magic cubes, tesseracts, etc., is different. Until John Hendricks started working with the higher dimension hypercubes in the second half of the twentieth century, very little consideration had been given to coordinates (or for that matter, lists of solutions). Because Hendricks work with higher dimensions was so dependent on coordinates, he decided to go with the conventional geometry. In magic hypercubes, the coordinates are always positive, so the lower, left front corner of the hypercube is the starting point (x, y, z, w,… = 1, 1, 1, 1, …). [8] The rules for normalizing a magic cube, tesseract, etc. are:
Here are the four basic order-3 magic cubes, normalized, and in increasing index order.
Order-3 magic tesseracts will be discussed on a
separate page. Unfortunately, only order-3 magic hypercubes can be reconstructed from a small group of numbers in a catalogue list. This is because order-3 has only 3 numbers per line, and the center number (which is known), is part of the majority of these lines.
Magic cube aspects We have seen above that there are 8
different variations (aspects) of the basic magic square. A cube has 8 corners, each of which may
be placed in the lower front left position. Each of these corners has 3 rays
which can be labeled in 6 different ways.
Magic tesseract aspects A magic tesseract has 384 aspects because there are 16 corners, each with 4 rays which can be arranged in 24 ways., and 16 x 24 = 384. No attempt will be made here to illustrate those 384 different aspects. For convenience, I will simply reproduce here an early cube already shown on another page, along with two variations.
[1] Bernard Frénicle de Bessy ,
Des Quarrez Magiques. Acad. R. des Sciences 1693 (this is a
posthumous paper, not a book)
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