Tesseracts - Update 2013
This exercise was intended originally to be simply an
addendum announcing that the Summary Table was now up-to-date and complete.
There is not much to report in Tesseract happenings in
the last several years.
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
additional classes he 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 investigationsand 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
 Hendricks, John R., Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published, 1999, 0-9684700-4-1
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.
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
Dr. C. Planck
published an Order-3 tesseract in 1888. It may be the one in Fig. 10 of his 1905
paper . He called it an octahedroid.
at the work these pioneers did! I wonder if he actually wrote out the other 255
planes? Following is a footnote from page 7
Stringham came up with the names tetrahedroid, hexahedroid,octahedroid, etc.
mentioned in Planck’s paper, but
Planck, The Theory of Path Nasiks, Printed for private circulation by A.
J. Lawrence, Printer, Rugby (England),1905 (Available from
Invented the Tesseract grid pattern in 1950 (published in 1962)  .
Published dimension 5 and 6 tesseracts in 1962. 
Found and published all 58 order-3 magic tesseracts 
Published a second method for construction of magic hypercubes which he later
called Digital Equations . These are
Conducted a math enrichment class on magic squares for several years at a
Winnipeg secondary school with the help of
Explored species in magic hypercubes 
Adapted the diagonal rule for constructing magic squares to all dimensions of magic hypercubes 
Explored grouping of Magic Tesseracts by various permutations 
Pioneered Inlaid Magic Cubes and constructed an Inlaid Magic tesseract. 
Introduced a classification system for magic hypercubes based on types of n-agonals 
Constructed and published an order-16 Nasik magic 0rder-16 tesseract. 
This was confirmed by Cliff Pickover of IBM
Constructed and published an order-32 Nasik magic dimension 5 hypercube. 
Constructed various spreadsheets to manipulate magic hypercubes . This
after he finally obtained a computer to use
Published 53 papers on magic hypercubes and ten (10) self-published books. He
also published 43 articles on other
 Hendricks recounts the sequence of events involved in this discovery.http://members.shaw.ca/johnhendricksmath/tesseracts.htm
 Hendricks, John R. The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, Vol. 5, No. 2, 1962, pp. 171-189.
 Hendricks, John R, Ten Magic Tesseracts of Order 3 (and other articles), Journal of Recreational Mathematics, 18:2:1985-86:125-134
 Hendricks, John R., Magic Tesseracts and n-Dimensional Magic Hypercubes, Journal of Recreational Mathematics, 6:3:1973: 193-201.
 Hendricks, John R., Species of Third-Order Magic Squares and Cubes, Journal of Recreational Mathematics, 6:3:1973:190-192.
 Hendricks, John R., The Diagonal Rule for Magic Cubes of Odd Order, Journal of Recreational Mathematics, 20:3:1988:192-195
 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.
 Self-published as an 8 page brochure. The Order-6 Inlaid Magic Tesseract may be seen at http://members.shaw.ca/johnhendricksmath/inlaid_t.htm
Also, Hendricks, John R., Inlaid Magic Squares and Cubes, Self-published, 1999, 0-9684700-1-7
 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.
 Perfect.Tess.xls, a pathfinder program especially for this nasik hypercube may be downloaded from http://magic-squares.net/downloads.htm#MS%20Excel
 Hendricks, John R., Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published, 1999, 0-9684700-4-1
 Seven other spreadsheets by John Hendricks may also be downloaded from the above site.
 The list of publications are at http://members.shaw.ca/johnhendricksmath/bibliography.htm
 Many of Hendricks out-of-print books are now available for free download in Portable Document Format (pdf) from my Downloads page.
A short 2001 note by Hendricks on the nasik tesseract is at the end of this web page
Found and constructed an example of the cube class that Hendricks missed 
Established the 18 classes of Tesseracts 
First to constructed examples of 15 of the 18 classes of tesseracts 
excellent site dealing with all dimensions of magic hypercubes. Complete with
elaborate sections on Theory, definitions,
still energetically working with tesseracts. On Jan. 20, 2013 he advised us
that he had constructed 1,2,3,4-agonal magic
. 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
. 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!
Christian Boyer constructed five multimagic tesseracts
in early 2003! They are two order 32 and one order 64 bimagic;
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.
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,
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)
Contributed to the development and popularization of The Unified Classification System for Magic Hypercubes.
Researched and published (via my web site ) the relationship between
all dimensions of magic hypercubes. Emphases on history and
Collaborated with John Hendricks on publishing Magic Square Lexicon:
Compiled and published an Omnibus DVD with all the information I have collected or found over the years
Constructed one 2-D diagram (of many possible versions) of the 4-D tesseract. 
(1) Heinz, H and Hendricks, R., A Unified
Classification System for Magic Hypercubes, Journal of Recreational
 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.
 Heinz, Harvey D., Magic Hypercube Omnibus DVD, 2011, 978-0-9687985-1-5 (free distrubion)
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.
About the same time David M. Collison, also from California constructed tesseracts up to dimension 8 using Hendricks’ methods.
Charlie Kelly has a magic Hypercube generator athttp://net.indra.com/~charliek/Instructions.htm http://mathworld.wolfram.com/MagicTesseract.html http://www.maa.org/mathland/mathtrek_10_18_99.html
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=
From the Math Book: From Pythagorus to the 57th Dimensionhttp://books.google.ca/books?id=JrslMKTgSZwC&pg=PA498&lpg=PA498&dq=magic+tesseracts&source=bl&ots=VscB2SMOKI&sig=
Suggestions for New Investigations
 Dwane Campbell has constructed tesseracts with special compact features on his site at http://magictesseract.com/
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
So I guess after square, cube, tesseract comes
penteract, hexeract, hepteract, octeract, etc.
 Aale De Winkel Mar. 28,2013 " but currently
too busy with a new spreadsheet generator (squares to penteract (n=5
Therefore you are indeed on firm footing with your "pente" root!"
 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".
 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.
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.  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 . 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.
 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.
 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".
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
John R. Hendricks August 28, 2001
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.
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.
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.
John R. Hendricks