# Tesseracts - Update 2013

Introduction

This exercise was intended originally to be simply an addendum announcing that the Summary Table was now up-to-date and complete.
I decided to make it a bit more elaborate and so a separate page!

There is not much to report in Tesseract happenings in the last several years.
However, I missed some events that should have been reported in earlier years so will mention them on this page.
Two items I missed were Christian Boyer’s construction of 5 multimagic cubes in 2003 and some of Mitsutoshi
Nakamura’s contributions to the summary Table. After contributing 13 new classes of tesseracts in 2007, I forgot to mention the two
constructed in 2010.

Because relatively few persons have been involved with magic tesseract research, I am including a section called Contributors to Tesseract Knowledge.

I am also including a section on possible future investigations and suggesting a dimension-5 challenge!

While writing the section on Dimension-5, I realized that many readers would not be aware of, or have access to John Hendricks 1999 basic
program for generating any desired path for a dimension-5 nasik magic pentacube of order-32. [1]

[1] Hendricks, John R., Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published, 1999, 0-9684700-4-1

 Summary Table Contributers to Magic Tesseracts Suggestions for New Investigations

Summary Table

For convenience, this table has been reproduced from my Hypercube Math page.

More tables, and much information on the relationship between the dimensions of magic hypercubes is available on that page.

 Hypercubes –Minimum number of correct summations  - based on smallest order possible Magic Hypercube Lowest Order (m) r-agonals Min. Sums This order first Built by 1 2 3 4 Total Square Simple 3 2m 2 ---- ---- 2m + 2 8 ? Nasik  (Pandiagonal and Perfect) 4 2m 2m ---- ---- 4m 16 ? Cube Simple 3 3m2 ---- 4 ---- 3m2 + 4 31 Hugel  - 1876 Diagonal   (Boyer/Trump Perfect) 5 3m2 6m 4 ---- 3m2+6m+4 109 Trump/Boyer - 2003 Pantriagonal 4 3m2 ---- 4m2 ---- 7m2 112 Frost - 1878 PantriagDiag 8? 3m2 6m 4m2 ---- 7m2+6m 496 Nakamura - 2005 Pandiagonal 7 3m2 6m2 4 ---- 9m2 + 4 445 Frost – 1866 Nasik (Hendricks Perfect) 8 3m2 6m2 4m2 ---- 13m2 832 Barnard 1888 Tesseract Simple 3 4m3 ---- ---- 8 4m3 + 8 116 Planck – 1905 Triagonal 4 4m3 ---- 16m 8 4m3 + 16m + 8 328 Nakamura - 2007 Diagonal 4 4m3 12m2 ---- 8 4m3 + 12m2 + 8 456 Nakamura – 2007 Diagonal + Triagonal 8? 4m3 12m2 16m 8 4m3 + 12m2 + 16m + 8 2,952 Nakamura – 2007 Panquadragonal 4 4m3 ---- ---- 8m3 12m3 768 Hendricks – 1968 Triagonal + Pan4 4 4m3 ---- 16m 8m3 12m3 + 16m 832 Nakamura – 2007 Diagonal + Pan4 8? 4m3 12m2 ---- 8m3 12m3 + 12m2 6,912 Nakamura – 2007 Diagonal + Triagonal+Pan4 8? 4m3 12m2 16m 8m3 12m3 + 12m2 + 16m 7,040 Nakamura – 2007 Pandiagonal 9? 4m3 12m3 ---- 8 16m3 + 8 11,672 Nakamura – 2007 Triagonal + Pan2 16? 4m3 12m3 16m 8 16m3 + 16m + 8 65792 Nakamura – 2010 Pantriagonal 4 4m3 ---- 16m3 8 20m3 + 8 1,288 Nakamura – 2007 Diagonal + Pan3 8? 4m3 12m2 16m3 8 20m3 + 12m2 + 8 11016 Nakamura – 2010 Pan2 +Pan4 13? 4m3 12m3 ---- 8m3 24m3 52,728 Nakamura – 2007 Triagonal + Pan2 + Pan4 16? 4m3 12m3 16m 8m3 24m3 + 16m 98,560 Nakamura – 2007 Pan3 + Pan4 4 4m3 ---- 16m3 8m3 28m3 1,792 Nakamura – 2007 Diagonal + Pan3 + Pan4 8 4m3 12m2 16m3 8m3 28m3 + 12m2 15,104 Nakamura – 2007 Pan2 + Pan3 15? 4m3 12m3 16m3 8 32m3 + 8 108,008 Nakamura – 2007 Nasik  (Hendricks perfect) 16 4m3 12m3 16m3 8m3 40m3 163,840 Hendricks - 1998

There is still uncertainty for some classes as to the minimum possible order.

Contributors to Magic Tesseracts

Because a relatively few people have contributed to dimension 4 and higher magic hypercubes, it is possible to list them all here (to the
best of my knowledge and ability).

Dr. C. Planck

Dr. Planck published an Order-3 tesseract in 1888. It may be the one in Fig. 10 of his 1905 paper [1]. He called it an octahedroid.
Fig. 10 (above) appears on page 364 of Andrews as fig. 687. [2]
He published one plane of a nasik order 16 tesseract (fig. 16) in his Theory of Path Nasiks in 1905. [1]
I checked the square and it is pandiagonal magic so is nasik.

I marvel at the work these pioneers did! I wonder if he actually wrote out the other 255 planes? Following is a footnote from page 7
of Planck’s paper.

W. L. Stringham came up with the names tetrahedroid, hexahedroid,octahedroid, etc. mentioned in Planck’s paper, but
otherwise had nothing to do with magic hypercubes. [3]

[1] C. Planck, The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby (England),1905 (Available from
The University Library, Cambridge).
[2] W. S. Andrews, Magic Squares and Cubes. Open Court Publ.,1917. Pages 365,366, by Dr. C. Planck. Re-published by Dover Publ., 1960,
Pages 365,366 (no ISBN); Dover Publ., 2000, 0486206580; Cosimo Classics, Inc., 2004, 1596050373
[3] W. L. Stringham, American Journal of Mathematics Vol. III, 1880, pages 1 - 14

John Hendricks

• Invented the Tesseract grid pattern in 1950 (published in 1962) [1] [2].

• Published dimension 5 and 6 tesseracts in 1962. [2]

• Found and published all 58 order-3 magic tesseracts [3]

• Published a second method for construction of magic hypercubes which he later called Digital Equations [4]. These are
especially useful for tracing paths through the hypercube.

• Conducted a math enrichment class on magic squares for several years at a Winnipeg secondary school with the help of
a 554 page course book he wrote in 1991.

• Explored species in magic hypercubes [5]

• Adapted the diagonal rule for constructing magic squares to all dimensions of magic hypercubes [6]

• Explored grouping of Magic Tesseracts by various permutations [7]

• Pioneered Inlaid Magic Cubes and constructed an Inlaid Magic tesseract. [8]

• Introduced a classification system for magic hypercubes based on types of n-agonals [9]

• Constructed and published an order-16 Nasik magic 0rder-16 tesseract. [10] This was confirmed by Cliff Pickover of IBM
on their computers

• Constructed and published an order-32 Nasik magic dimension 5 hypercube. [11]

• Constructed various spreadsheets to manipulate magic hypercubes [12]. This after he finally obtained a computer to use
during the last 4 or 5 years of his life.

• Published 53 papers on magic hypercubes and ten (10) self-published books. He also published 43 articles on other
subjects. [13] [14]

[1] Hendricks recounts the sequence of events involved in this discovery. http://members.shaw.ca/johnhendricksmath/tesseracts.htm
[2] Hendricks, John R. The Five and Six Dimensional  Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, Vol. 5, No. 2, 1962, pp. 171-189.
[3] Hendricks, John R, Ten Magic Tesseracts of Order 3 (and other articles), Journal of Recreational Mathematics, 18:2:1985-86:125-134
[4] Hendricks, John R., Magic Tesseracts and n-Dimensional Magic Hypercubes, Journal of Recreational Mathematics, 6:3:1973: 193-201.
[5] Hendricks, John R., Species of Third-Order Magic Squares and Cubes, Journal of Recreational Mathematics, 6:3:1973:190-192.
[6] Hendricks, John R., The Diagonal Rule for Magic Cubes of Odd Order, Journal of Recreational Mathematics, 20:3:1988:192-195
[7] Hendricks, John R., Ten Magic Tesseracts of Order Three, Journal of Recreational Mathematics, 18:2:1985-86:125-134 (#1-10) and #21 -24 JRM:20:4:
1988:251-256, #11:pp 275-276, #12-16:pp 279-283, and #25-30 in JRM:21:1:1989:13-18 and #17-20:26-28 and finally # 31-58 in JRM:22:1:1990; 15-26.
Here they are numbered in the order found, not index order.
[8] Self-published as an 8 page brochure. The Order-6 Inlaid
Magic Tesseract may be seen at
Also, Hendricks, John R., Inlaid Magic Squares and Cubes, Self-published, 1999, 0-9684700-1-7
[9] Heinz, H and Hendricks, R., A Unified Classification System for Magic Hypercubes, Journal of Recreational Mathematics:32:1:2003-04:30-36.
In this first version we missed one cube classification.
[10] Perfect
.Tess.xls, a pathfinder program especially for this nasik hypercube may be downloaded from
[11] Hendricks, John R., Perfect n-Dimensional Magic Hypercubes of Order 2
n, Self-published, 1999, 0-9684700-4-1
[13] The list of publications are at
http://members.shaw.ca/johnhendricksmath/bibliography.htm
[14] Many of Hendricks out-of-print books are

A short 2001 note by Hendricks on the nasik tesseract is at the end of this web page

Mitsutoshi Nakamura

• Found and constructed an example of the cube class that Hendricks missed [1]

• Established the 18 classes of Tesseracts [2]

• First to constructed examples of 15 of the 18 classes of tesseracts [3]

• Has an excellent site dealing with all dimensions of magic hypercubes. Complete with elaborate sections on Theory, definitions,
classes and multiple algorithms and sample hypecubes [4]

• He is still energetically working with tesseracts. On Jan. 20, 2013 he advised us that he had constructed 1,2,3,4-agonal magic
tesseracts of orders 8, 10, 12, 14, 15, and 18 and posted them on his site.

[1] http://magic-squares.net/c-t-htm/c_update-3.htm#A%20New%20Magic%20Cube%20Class
[2]. Heinz, Harvey and Nakamura, Mitsutoshi, Magic Tesseract Classes, Journal of Recreational Mathematics, JRM:35:1:2006:11-14
An example of each of the 18 classes of tesseracts may be downloaded from Mitsutoshi’s site at
http://homepage2.nifty.com/googol/magcube/en/classes.htm#tesserallclasses.
[3]. Heinz, Harvey, Hypercube Classes - An Update, Journal of Recreational Mathematics: JRM:35:1:2006:5-10.
When I first posted the table
Hypercubes –Minimum
Number of Correct Summations there were 15 (out of 18) blanks in the column First to construct a
tesseract of this class
. Mitsutoshi Nakamura quickly filled in 13 of these. (I was recently made aware that he had constructed the
Triagonal + Pan2 and the Diagonal + Pan3 tesseracts in 2010. Either of these (or for that matter some others) may not be the
minimum possible. I have added these two to the table and it is now complete!

[4]

Dwane Campbell

• Dwane Campbell has recently posted a new Web site on Tesseracts; http://magictesseract.com/

• He includes material on magic squares and cubes and much material on Magic Tesseracts. His favorite method of construction
uses base lines and he has a hypercube generator available for download that uses that system to construct hypercubes of any
order or dimension.

• He helped expand the knowledge of the features compact and complete and mentions them in tesseracts he has constructed.

Other Contributors

Christian Boyer

Christian Boyer constructed five multimagic tesseracts in early 2003! They are two order 32 and one order 64 bimagic;
An order 243 trimagic and an order 256 trimagic and perfect bimagic. All were independently verified by two other qualified persons

Here are some characteristics (from the above page) of the trimagic hypercube of order 256. It uses numbers from 0 to 4,294,967,295.
Its 16,777,216 rows, 16,777,216 columns, 16,777,216 pillars and 16,777,216 files are trimagic. Its 8 quadragonals are trimagic.
Its 4,096 triagonals and its 786,432 diagonals are bimagic. This hypercube is also perfect bimagic, since the x-agonals of all the
dimensions are bimagic (and its quadragonals
are even trimagic). Its magic sums are:

• S1 = 549755813760

• S2 = 1574122160406792590720

• S3 = 5070602398551734364826868121600

His large website appears in three languages with the main focus on multi-magic squares and cubes. He includes methods of construction,
tables of records and offers prizes for some enigmas.

See his page with details at http://www.multimagie.com/English/Hypercubes.htm
I have some details of these large tesseracts at http://magic-squares.net/c-t-htm/c_monster.htm

Marian Trenkler

Marian Trenkler has written several papers on magic tesseracts.

Trenkler, M. Magic -Dimensional Cubes of Order (mod 4). Acta Arith. 92, 189-204, 2000 ACTA ARITHMETICA XCII.2 92000)
(aa-cub-00.pdf) and
AN ALGORITHM FOR MAGIC TESSERACTS (06-MagicTesseract.pdf) showing how to construct a magic tesseract by formulae.
Scientific Bulletin of Chelm, Section of Mathematics and Computer Science 2/2006, 249-251 ISBN 83-924536-4-6, ISSN 1896-463X

Harvey Heinz

• Contributed to the development and popularization of The Unified Classification System for Magic Hypercubes. [1]. [2]. [3].

• Researched and published (via my web site [4]) the relationship between all dimensions of magic hypercubes. Emphases on history and
statistics of Cubes and Tesseracts

• Collaborated with John Hendricks on publishing Magic Square Lexicon: Illustrated to help clarify terminology used with magic hypercubes. [5]

• Compiled and published an Omnibus DVD with all the information I have collected or found over the years (to 2011). [6]

• Constructed one 2-D diagram (of many possible versions) of the 4-D tesseract. [7]

(1) Heinz, H and Hendricks, R., A Unified Classification System for Magic Hypercubes, Journal of Recreational Mathematics:32:1:2003-04:30-36.
In this first version we missed one cube classification.
(2). Heinz, Harvey, Hypercube Classes - An Update, Journal of Recreational Mathematics: JRM:35:1:2006:5-10
(3). Heinz, Harvey and Nakamura, Mitsutoshi, Magic Tesseract Classes, Journal of Recreational Mathematics, JRM:35:1:2006:11-14
[4]
http://magic-squares.net/magic_tesseract_index.htm
[5]  Heinz, H.D. and Hendricks, J. R., Magic Square Lexicon: Illustrated. Self-published, 2000, 0-9687985-0-0.
239 terms defined, about 200 illustrations and tables, 171 captioned.
[6] Heinz, Harvey D., Magic Hypercube Omnibus DVD, 2011, 978-0-9687985-1-5 (free distrubion)
[7] http://magic-squares.net/c-t-htm/t_unfolded.htm

Merideth Houlton and David M. Collison

Meredith Houlton, a secondary school student came to Victoria in her summer holidays and studied tesseracts under John Hendricks.
She then returned home and constructed dimensions 5, 7 and 9 tesseracts for her science fair project. [1]

About the same time David M. Collison, also from California constructed tesseracts up to dimension 8 using Hendricks’ methods.[1]

Charlie Kelly has a magic Hypercube generator at http://net.indra.com/~charliek/Instructions.htm

Cliff Pickover account in The Zen of Magic Squares about the testing of the nasik order 16 tesseract. http://books.google.ca/books?id=9To1nPvY-1AC&pg=PA119&lpg=PA119&dq=magic+tesseracts&source=bl&ots=O3bTFSl3jz&sig=Xm0pJ9NgraPMAycsj2aLstfPUbQ&hl=en&sa=X&ei=
jK8wUZ2fLMmkiQKN0IDABg&ved=0CD8Q6AEwAzgK#v=onepage&q=magic%20tesseracts&f=false

From the Math Book: From Pythagorus to the 57th Dimension http://books.google.ca/books?id=JrslMKTgSZwC&pg=PA498&lpg=PA498&dq=magic+tesseracts&source=bl&ots=VscB2SMOKI&sig=
A6lkZyW13glteSw9GCKU90UWFEU&hl=en&sa=X&ei=jK8wUZ2fLMmkiQKN0IDABg&ved=0CEUQ6AEwBTgK#v=onepage&q=
magic%20tesseracts&f=false

Suggestions for New Investigations

• Develop more user friendly versatile tools for investigating features of constructed tesseracts.

[1] Dwane Campbell has constructed tesseracts with special compact  features on his site at http://magictesseract.com/

Dimension 5

The Penteract

New names are required for Dimension-5 and higher hypercubes!

Shortly after posting this page, I received an email from Aale de Winkel (on another subject) that casually mentioned the term penteract. [1]
Then an email from Paul Pasles confirming this is a foundation for the ‘pente’ root as a follow up to ‘tesse’. [2]
and finally Miguel Amela sent two links that used the term penteract! [3]
A follow-up search revealed many sites that used the term penteract in referring to a 5-D hypercube. [4]

So I guess after square, cube, tesseract comes penteract, hexeract, hepteract, octeract, etc.
The only possible stumbling block is that these prefixes usually end with an ‘a’ (instead of an ‘e’)

[1] Aale De Winkel Mar. 28,2013 " but currently too busy with a new spreadsheet generator (squares to penteract (n=5 hypercube))."
[2]Paul Pasles, Mar. 29,2013, "Just a quick word from (not so ancient) Greek mathematician. The tesseract comes from "tessera," which is "four" in Greek.
Therefore you are indeed on firm footing with your "pente" root!
"
[3] Miguel Amela, Mar. 30, 2013,
"A new mathematical word ? See "penteracto" in Spanish:
http://es.wikipedia.org/wiki/Penteracto and the "penteract" word in English: http://en.wikipedia.org/wiki/5-cube".
[4]
A site using the term penteract and also a good source for penteract diagrams is
http://en.wikipedia.org/wiki/5-cube

The investigation of magic tesseracts has proved not too popular. Probably because of the large number range and large number of paths through each cell.
Much faster computers and inexpensive memory available today lessens the significance of these obstacles.

Dimension 5 magic hypercubes will be even less popular because both of the above conditions are much greater yet! Most of those constructed to date were order 3. This, of course is the smallest possible order so consists of the smallest range of numbers and smallest constant. Also, all order 3 hypercubes are associated which makes them much easier to construct. However, order-3 has limited potential for additional features

The first example of a nasik order 32 penteract was constructed and published by John Hendricks in 1999. [2] Admittedly this large magic object is not shown as a complete unit. But by allowing any desired path to be generated on demand is maybe the only practical way to show these large objects!

Currently there is work going on with penteract investigations.  Aale de Winkel reported he had created about 37 penteracts using a method developed by George Chen [2]. Dwane Campbell has a penteract generator available for download, and among other penteracts he has constructed an order 32 that has many compact features. Others too, are working on penteract and higher dimension hypercubes so hopefully, in coming days, we will see a great expansion in the knowledge of higher dimension magic hypercubes.

The Challenge:

• Find and list 54 (?) classes for dimension 5 hypercubes

• Construct example magic penteracts for some (or all) of the classes. This will involve coining names for the classes by utilizing the name of the featured n-agonal (s) for that class. Or maybe a new method of naming can be devised.

• Design a simplified diagram of a penteract (probably showing just the corners). [3]

[1] Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published, 1999, 0-9684700-4-1. 36 pages plus covers, 8.5" x 11" flat stitched, some diagrams.
Theory with examples for a cube, tesseract, 5-D and 6-D hypercubes. John Hendricks wrote a small basic program in 1999 (for his programmable calculator) that generates any desired path of an order 32 nasik hypercube-5.
[2] Aale de Winkel, Apr. 1/13, "I already created about 37 order 3 penteracts using your cube to penteract method. Looked into tesseract to penteract for a bit but hadn’t found the true connection. Will study your just obtained. Any interest in the order 3 penteracts? Not yet created a summarizing spreadsheet".
[3]Some ideas for a simplified diagram of a penteract may be found at
http://en.wikipedia.org/wiki/5-cube

A short note on the order 16 Perfect magic tesseract

NOTES ON A

PERFECT MAGIC

TESSERACT OF

ORDER 16

John R. Hendricks August 28, 2001

Closely examining the magic cube of order three above, you will notice that the diagonals of the plane faces do not sum 42, which is the magic sum.
Only the central squares are magic squares. So, although this is a magic cube by definition, it is not perfect. Perfect for a cube means that all magic squares are pandiagonal and the cube itself is pantriagonal. The smallest perfect magic cube is of order 8,

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 one 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 is so made that coordinates are reduced to between zero and 15 inclusive.

If a negative route is chosen, then it shows the set of 15 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 65536 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.

John R. Hendricks

 This page was originally posted March 22, 2013 It was last updated April 03, 2013 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz