# Order-3 Magic Tesseracts

This page starts with a brief history of magic tesseracts, along with a list of references.
Then I show the Order-3 Index #1 tesseract in diagram and text form.
This is followed by the other 57 order-3 magic tesseracts, listed in index order in text form.
Then is shown a catalog of all basic order-3 tesseracts using just 8 identifying numbers.
Finally I show a new type of classification based on placement of the even and odd numbers.

Introduction

It is difficult to say who constructed the first 4-dimensional magic hypercube. However, C. Planck, in his 1905 paper [1] mentions that he published a dimension 4 order-3 magic hypercube in The English Mechanic, March 16, 1888. In 1917 both Dr. Planck, and H.M. Kingery published octahedroids in Andrews Magic Squares and Cubes.[2]

John R. Hendricks (1929-2007) appears to be the first to construct and publish all order-3 magic tesseracts. By 1985 he had constructed 58 basic order-3 and proved that that was all there were. The first one was published in 1962 [3] (along with dimension 5 and 6 magic hypercubes)  when he introduced his new diagram for the tesseract. 30 of these were published in JRM between 1985 and 1990 [4] and all 58 in [5].

About this same time, Keh Ying Lin of Taiwan was also working on this same problem. He published Magic Cubes and Hypercubes of Order-3 in 1986. [6]
He also proved that there were 58 basic tesseracts of order-3. He showed method of construction, but did not actually list them. He also showed that there are 384 aspects of each , as a result of rotations/reflections.
He further stated that there are 2992 basic order-3 hypercubes of dimension 5 with 3840 aspects, and 543328 order-3, dimension 6 with 46080 aspects.

David Collison (1937-1991) verified by computer exhaustion 58 magic tesseracts of order-3, and sent Hendricks a computer printout of 7 and 8 dimension examples.[7]

References

Listed here are references I have used in the compiling of this page. I will cite a particular reference, if my information came from only that source.
However, most of these contain much the same information, so will not be cited separately.
Note that Hendricks published many more articles and books on the subject then those I have listed below.

[1] Planck, C. (M.A., M.R.C.S.) The Theory of Paths Nasik, Printed for private circulation by A. J. Lawrence, Printer, Rugby (England), 1905
[2]  Andrews, W. S., Magic Squares and Cubes, Dover Publ. 1960,. Pages 351-375. This book originally published in 1917 by Open Court Publishing.
[3] Hendricks, J. R., The Five and Six-Dimensional Magic Hypercubes of Order 3,
Canadian Math. Bulletin, 5:2:1962, pages171-190
[4] Hendricks, J. R., Ten Magic Tesseracts of Order 3, Journal of Recreational Mathematics, 18:2, 1986, pp 125-134 (T# 1 – 10)
The Third Order Magic Tesseract, Journal of Recreational Mathematics, 20:4, 1988, pp 251-256 (T# 21 - 24)
Another Magic Tesseract of order 3, Journal of Recreational Mathematics, 20:4, 1988, pp 275-276 (T# 11)
Creating More Magic Tesseracts of Order 3, Journal of Recreational Mathematics, 20:4, 1988, pp 279-283 (T# 12-16)
Groups of Magic Tesseracts, Journal of Recreational Mathematics, 21:1, 1989, pp 13-18 (T# 25-30)
More and More Magic Tesseracts, Journal of Recreational Mathematics, 21:1, 1989, pp 26-28 (T# 17-20)
[5] Hendricks, J. R., The Magic Square Course, 1991, 554 pages printed for a senior high school course he was teaching. (available at Strens Recreational Mathematics Collection,  University of Calgary (Canada)
[6] Keh Ying Lin, Magic Cubes and Hypercubes of Order-3, Discrete Mathematics 58:2, February 1986, pages 159-166 (this cited in [7]
[7] Hendricks, J. R., All Third-Order Magic Tesseracts, Self-published, 1999, 0-9684700-2-5  (Both Keh and Collison mentioned, all tesseracts diagrammed, species)
[8] Hendricks, J. R., Magic Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9
[9] Heinz & Hendricks, Magic Square Lexicon: Illustrated, HDH, 2000, 0-9687985-0-0

The 58 order-3 magic tesseracts

 Index # 1 This is the first of the 58 order-3 basic magic tesseracts. I show it here in both text form and as a diagram. The other 57 tesseracts will be listed below in text form only. ```48 62 13 70 24 29 5 37 81 7 42 74 50 55 18 66 26 31 68 19 36 3 44 76 52 60 11 4 39 80 47 61 15 72 23 28 65 25 33 9 41 73 49 57 17 54 59 10 67 21 35 2 43 78 71 22 30 6 38 79 46 63 14 51 56 16 64 27 32 8 40 75 1 45 77 53 58 12 69 20 34 ```

The layout for the text listings on this page are as per the system of coordinates shown on my introductory page.
Aale de Winkel also lists all 58 order-3 tesseracts in his encyclopedia, but in a different format. No claim is made here as to which is the correct one. They are simply two different methods (aspects) of presenting the numbers in the tesseract. Other ways of listing these tesseracts are also often seen, even on my pages!
It should be noted, however, that the diagram shown here is of the basic tesseract in the standard position. The lowest corner number is in the lower left front corner, with the four adjacent numbers in increasing order of x, y, z, and w.

 Index # 2 66 26 31 | 52 60 11 | 05 37 81 07 42 74 | 68 19 36 | 48 62 13 50 55 18 | 03 44 76 | 70 24 29 04 39 80 | 65 25 33 | 54 59 10 47 61 15 | 09 41 73 | 67 21 35 72 23 28 | 49 57 17 |0 2 43 78 53 58 12 | 06 38 79 | 64 27 32 69 20 34 | 46 63 14 | 08 40 75 01 45 77 | 71 22 30 | 51 56 16 Index # 3 60 26 37 | 52 660 5 | 11 31 81 19 42 62 | 68 07 48 | 36 74 13 44 55 24 | 03 50 70 | 76 18 29 10 33 80 | 59 25 39 | 54 65 04 35 73 15 | 21 41 61 | 67 09 47 78 17 28 | 43 57 23 | 02 49 72 53 64 06 | 12 32 79 | 58 27 38 69 08 46 | 34 75 14 | 20 40 63 01 51 71 | 77 16 30 | 45 56 22 Index # 4 62 24 37 | 48 70 05 | 13 29 81 19 44 60 | 68 03 52 | 36 76 11 42 55 26 | 07 50 66 | 74 18 31 10 35 78 | 59 21 43 | 54 67 02 33 73 17 | 25 41 57 | 65 09 49 80 15 28 | 39 61 23 | 04 47 72 51 64 08 | 16 32 75 | 56 27 40 71 06 46 | 30 79 14 | 22 38 63 01 53 69 | 77 12 34 | 45 58 20 Index # 5 61 24 38 | 48 71 04 | 14 28 81 20 43 60 | 67 03 53 | 36 77 10 42 56 25 | 08 49 66 | 73 18 32 12 35 76 | 59 19 45 | 52 69 02 31 75 17 | 27 41 55 | 65 07 51 80 13 30 | 37 63 23 | 06 47 70 50 64 09 | 16 33 74 | 57 26 40 72 05 46 | 29 79 15 | 22 39 62 01 54 68 | 78 11 34 | 44 58 21

Here I have shown listings for index numbers 2 to 5. Complete listings for all 58 order-3 tesseracts are available for downloading in MS Word and Adobe PDF. My Hypercube Generator-3.xls which was used to generate these listings is available from the same downloads page.

Catalog of the 58 order-3 magic tesseracts

This is a sorted list of all the basic order-3 magic tesseracts. Shown is the index number. Then the lower left front corner number (c) followed by the 4 numbers adjacent to it (x, y, z, w). The numbers xy, xz, and yz are not required to identify the tesseract, but are required to reconstruct it.
T#
indicates the order that John Hendricks first constructed each tesseract. Species is a special classification. It will be discussed in the next section.

Species

 John Hendricks found another way to classify magic tesseract of order-3. He gave it the term species. [1]The 1 order-3 basic magic square has even numbers on all 4 corners. So there is only one species of order- 3 magic square. All 4 order-3 basic magic cubes have the same arrangement of even and odd numbers, as shown in the illustration to the right. So there is only 1 species of order-3 magic cubes. There are 58 basic order-3 magic tesseracts. In these, the even and odd numbers appear in 3 different arrangements. These are shown in the illustration below.

Species 1 is easy to spot. Find any corner with an odd number. Check the rays (row, column, pillar, file) passing through it. If the other 2 numbers in each is an even number , this is a species # 1 tesseract. There are only two basic magic tesseracts of this species.

For species # 2, again consider a corner with an odd number. Two of the rays will contain all odd numbers, and the other two rays will contain two even numbers. There are 24 tesseracts of this species.

The remaining 32 magic tesseracts are of species # 3. Through an odd numbered corner, either the remaining numbers are even in only one ray, or the remaining numbers are odd in only one ray.

[1] Hendricks, J. R., All Third-Order Magic Tesseracts, Self-published, 1999, 0-9684700-2-5  (Both Keh and Collison mentioned, all tesseracts diagrammed, species)

 This page was originally posted November 2007 It was last updated October 20, 2010 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz