History of the Magic Tesseract
         (indeed Magic Hypercubes, as well)

The analogy between squares and cubes is not perfect, for rows of numbers  can be arranged side-by-side to represent a visible square, squares can be piled one upon another to represent a visible cube, but cubes cannot be so combined in drawing as to picture to the eye their higher relations……….
[Magic squares and Cubes, by W.S. Andrews, Dover, 1917, Chapter 14.]

W.S. Andrews had said it. It could not be done. And, even if it could be done, it certainly could not be shown in one diagram on a piece of paper.

 Moreover, the Geometry experts had all agreed that the model of the tesseract, as shown in Figure 1, was the correct model of those proposed and because it had all the requirements. There were 16 corners, 32 edges and 8 cubes. It did not matter to them that the top & bottom cubes were mirror reflections and possibly reversing positive and negative zones in the z-direction. It did not matter that the figure was impossible to partition, which is one of the requirements for magic squares and magic cubes. For cubes, you could cut it into building blocks and assign numbers to each and then assemble them into a magic cube.

A magic square had to be written into a partitioned square. A magic cube must be done in building blocks and shown in layers. This is what the train of thought was around 1950.

When I was studying magic cubes, I liked to have all the numbers available  in one diagram so that I could see them all at once. That could not be done with building blocks. So, I devised the open lattice system. I also liked to think in terms of a coordinate system, so I devised one a little different from the one you are normally using. Brackets were not required and they were all whole numbers. (x, y, z) was simply written xyz unless the order exceeded nine which was a huge cube and would have to be done in layers anyway.

Used to project the lattice points revealed a slightly different structure to the tesseract capable of being partitioned.

Figure 3. Open lattice coordinate system for a magic tesseract of order 3.
The colored coordinates show one partition.

 It was in the spring of 1950. My friend Eric and I thought it was far too good a day to spend in lecture halls and classrooms and so we went to the beach. I had some paper and a pen and while sunning at the beach I sketched the first magic tesseract. It startled me that it came so easily because I knew there were 5.797126020747367985879734231578e+120 ways of placing numbers 1 to 81 into such an array. I guessed that 41 had to go dead centre. I assumed that symmetrically opposite (across the centre) numbers would sum 82. I was used to the balance between high numbers and low numbers of squares and cubes which helped. It took a couple of hours. Should I tell Eric, or not?

 I decided to tell Eric and discussed the situation. I had to be very careful in case someone else stole it. Eric told me about the “poor man’s copyright.”

When you make copies of your important creation, you send a copy to yourself by registered mail and never open it until it is published in your name. If it is published in someone else’s name, then take them to court and have the court open it and show the seal has never been broken and it predates anyone else. Good enough for me!
 

It was about five, or six years later and I was at Montreal.
An ex-meteorologist, now a professor of mathematics at Seattle, was home for Christmas and saw some of his friends, but heard of my hypercubes. He asked to see them, so I brought them to the office. His immediate reaction was that this stuff must be published. I told him that that is what I have been saying for the past 12 years. Then, he phoned a friend at McGill University and set up an appointment to see his friend. I left my article with him.

 A month must have passed and I got a letter from Winnipeg. They wanted to know how many reprints of the article I would I like. There was also a phone call from McGill with an invitation to speak to the Mathematics Club. I was told to prepare 45 minutes with 15 minutes for questions. An hour later they asked if I could continue for another hour. The article came out as The Five- and Six-Dimensional Hypercubes of Order Three in The Canadian Mathematical Bulletin May 1962. The Magic Tesseract was born.  

Andrews work turned out to be not in vain though, because his octahedroids turned out to be planar cross-sections of a tesseract.

 All 58 magic tesseracts of order three have been found now and are published in the Journal of Recreational Mathematics. Several people showed that there are only 58. It is a registered pattern now in Pickover’s book of patterns and mentioned by him in his book, The Zen of Magic Squares, Circles and Stars.
Magic Hypercubes have been constructed to the 8^th dimension by the late David M. Collison and up to Order 9 tesseracts by Meredith Houlton using my methods. In recent years I have constructed special magic tesseracts such as Inlaid and Perfect.

Three Early Articles on Tesseracts

The first article which extended magic squares and cubes more generally into n-dimensional space was number 1 below.

Then, number 2 below showed the first example of the equivalent to the pandiagonal magic square in 4-space.

These two articles set the stage for articles on how to handle the geometry. Some new concepts such as triagonal and quadragonal had to be coined.

1.  Hendricks, John R. The Five and Six Dimensional  Magic Hypercubes of Order 3, Canadian Mathematical Bulletin. Vol. 5, No. 2, 1962, pp. 171-189.

2. Hendricks, J.R., The Pan-4-agonal Magic Tesseract, The American Mathematical Monthly, Vol. 75, No. 4, April 1968, p.384.

3.  Hendricks, John Robert, Magic Tesseracts and n-Dimensional Magic Hypercubes, Journal of Recreational Mathematics, Vol. 6, No. 3, Summer, 1973, pp. 193-201.

4.  Hendricks, John Robert, Pan-n-agonals in Hypercubes, Journal of Recreational Mathematics, Vol. 7. No.  2. Spring, 1974, pp. 95-96.

See the complete bibliography of John R. Hendricks published articles and books.

Last updated Thursday March 22, 2007

Webmaster: Harvey Heinz   harveyheinz@shaw.ca