PERFECT MAGIC HYPERCUBES
Once one comes up with something, someone tries to do better. For a magic square, this was achieved with the concept of a pandiagonal magic square. Not only do the rows, columns and two diagonals sum the magic sum, (which is the basic requirement,) but also so do all the broken diagonals, as well.
The red numbers in Figure 1 indicate a broken diagonal. If another copy is moved in beside it then the broken diagonal can become continuous. Computer people think in terms of the "wrap around." One way to show it effectively is to put the square on the surface of a donut. Others will fill the plane with identical squares. Some talk in terms of modular space, there are 4 rows, 4 columns, 4 diagonals one way and 4 diagonals the other way. That is 16 straight-line sums. For a square of order m, there would be 4m sums.
When it comes to the magic cube, there is not only a new z-direction put in, but there are two kinds of diagonals. The term triagonal was coined to differentiate the three-dimensional characteristic of the true diagonals of the cube from the regularly established planar diagonals of a magic square.
The perfect magic cube begins at order 8. All rows, columns and pillars must sum the magic constant and the four continuous triagonals to meet the requirements generally accepted by all. Instead of 4 ways to move suddenly there are 13 ways in a cube, unless you want to consider the reverse directions and make it 26. To be perfect means all diagonals and all triagonals, including the broken ones also do.
Many false claims of a perfect cube have been noticed with calling a pandiagonal cube perfect when the triagonals do not all sum the magic sum. The late David M. Collison showed a pandiagonal magic cube of order 7. But, it is not perfect.
The Perfect Magic Tesseract.
The lowest order where it occurs is at order 16. There are 65,536 numbers to put in. The magic sum will be S=524,296 by formula. There are 40 different directions through each cell which sum S. There are 32,768 “quadragonals” besides the 65,536 triagonals, and the 49,152 diagonals to sum S. Oh yes! There are 1536 pandiagonal magic squares included, as well as 64 perfect magic cubes embedded. With the tiniest type, you could show it on 1536 pieces of paper.
The alternative is to generate selected parts of it as needed by a program.
Figure. One page of the perfect magic tesseract of order 16.
A perfect magic tesseract means that all squares contained in it are pandiagonal magic squares. It also means that all magic cubes contained within it are perfect too. It must also be panquadragonal. It is totally and utterly wrapped around no matter how you look at it. Unfortunately, the smallest perfect magic tesseract is of order 16. The best one can do is to show it as a distorted projection on many pieces of paper taped together.
The program has been made so that all coordinates are between zero and 15 inclusive. If a negative route is chosen, then it shows the set of 16 numbers in reverse order, Only change the values in the green shaded zones and the rest is given. You are given a line at a time to do with what you want,
There are the numbers 1 to 65,536 used. The magic sum is 524,296. There are 32,768 four-dimensional diagonals (quadragonals); 65,536 triagonals; 49,152 diagonals to check out. Contained are 1536 pandiagonal magic squares of order 16 and 64 perfect magic cubes of order 16. And, oh yes, there are 16,384 rows, columns, pillars and files to check.
The easiest way is to use a coordinate system and show any row, column, or whatever you want. Pick any starting position (w, x, y, z) and any route from 1 to 40 and enter the values into the computer. Out come the numbers and the sum. If you want the reverse direction enter –5 instead of 5 and the numbers will all be reversed. For coordinates, 0 is the same as 16. For routes zero is not a route.
Some of the routes:
You figure out what the rest of them are, and I think they are all diagonals, triagonals, and quadragonals of one type, or another. .
If you would like a copy of the program, please request one by email.
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Last updated Thursday March 22, 2007
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