Introduction
This exercise was intended originally to be simply an
addendum announcing that the Summary Table was now uptodate 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
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
investigations and suggesting a dimension5 challenge!
While writing the section on Dimension5, 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 dimension5 nasik magic
pentacube of order32. [1]
I have now made available for download from my
Downloads page, pdf
copies of 17 of Hendricks smaller books.
[1] Hendricks,
John R., Perfect nDimensional Magic Hypercubes of Order 2^{n},
Selfpublished, 1999, 0968470041
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) 
ragonals 

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 
3m^{2} 
 
4 
 
3m^{2} + 4 
31 
Hugel  1876 
Diagonal (Boyer/Trump Perfect) 
5 
3m^{2} 
6m 
4 
 
3m^{2}+6m+4 
109 
Trump/Boyer  2003 
Pantriagonal 
4 
3m^{2} 
 
4m^{2} 
 
7m^{2} 
112 
Frost  1878 
PantriagDiag 
8? 
3m^{2} 
6m 
4m^{2} 
 
7m^{2}+6m 
496 
Nakamura  2005 
Pandiagonal 
7 
3m^{2} 
6m^{2} 
4 
 
9m^{2} + 4 
445 
Frost – 1866 
Nasik (Hendricks Perfect) 
8 
3m^{2} 
6m^{2} 
4m^{2} 
 
13m^{2} 
832 
Barnard 1888 
Tesseract 








Simple 
3 
4m^{3} 
 
 
8 
4m^{3} + 8 
116 
Planck – 1905 
Triagonal 
4 
4m^{3} 
 
16m 
8 
4m^{3} + 16m + 8 
328 
Nakamura  2007 
Diagonal 
4 
4m^{3} 
12m^{2} 
 
8 
4m^{3 }+ 12m^{2}
+ 8 
456 
Nakamura – 2007 
Diagonal + Triagonal 
8? 
4m^{3} 
12m^{2} 
16m 
8 
4m^{3} + 12m^{2}
+ 16m + 8 
2,952 
Nakamura – 2007 
Panquadragonal 
4 
4m^{3} 
 
 
8m^{3} 
12m^{3} 
768 
Hendricks – 1968 
Triagonal + Pan4 
4 
4m^{3} 
 
16m 
8m^{3} 
12m^{3} + 16m 
832 
Nakamura – 2007 
Diagonal + Pan4 
8? 
4m^{3} 
12m^{2} 
 
8m^{3} 
12m^{3} + 12m^{2}

6,912 
Nakamura – 2007 
Diagonal + Triagonal+Pan4 
8? 
4m^{3} 
12m^{2} 
16m 
8m^{3} 
12m^{3} + 12m^{2}
+ 16m 
7,040 
Nakamura – 2007 
Pandiagonal 
9? 
4m^{3} 
12m^{3} 
 
8 
16m^{3} + 8 
11,672 
Nakamura – 2007 
Triagonal + Pan2 
16? 
4m^{3} 
12m^{3} 
16m 
8 
16m^{3} + 16m + 8 
65792 
Nakamura – 2010 
Pantriagonal 
4 
4m^{3} 
 
16m^{3} 
8 
20m^{3} + 8 
1,288 
Nakamura – 2007 
Diagonal + Pan3 
8? 
4m^{3} 
12m^{2} 
16m^{3} 
8 
20m^{3} + 12m^{2}
+ 8 
11016 
Nakamura – 2010 
Pan2 +Pan4 
13? 
4m^{3} 
12m^{3} 
 
8m^{3} 
24m^{3} 
52,728 
Nakamura – 2007 
Triagonal + Pan2 + Pan4 
16? 
4m^{3} 
12m^{3} 
16m 
8m^{3} 
24m^{3} + 16m 
98,560 
Nakamura – 2007 
Pan3 + Pan4 
4 
4m^{3} 
 
16m^{3} 
8m^{3} 
28m^{3} 
1,792 
Nakamura – 2007 
Diagonal + Pan3 + Pan4 
8 
4m^{3} 
12m^{2} 
16m^{3} 
8m^{3} 
28m^{3} + 12m^{2} 
15,104 
Nakamura – 2007 
Pan2 + Pan3 
15? 
4m^{3} 
12m^{3} 
16m^{3} 
8 
32m^{3} + 8 
108,008 
Nakamura – 2007 
Nasik (Hendricks perfect) 
16 
4m^{3} 
12m^{3} 
16m^{3} 
8m^{3} 
40m^{3} 
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 Order3 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. Republished 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 order3 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 nagonals
[9]

Constructed and published an order16 Nasik magic 0rder16 tesseract. [10]
This was confirmed by Cliff Pickover of IBM
on their computers

Constructed and published an order32 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) selfpublished 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. 171189.
[3] Hendricks, John R, Ten Magic Tesseracts of Order 3 (and other
articles), Journal of Recreational Mathematics, 18:2:198586:125134
[4] Hendricks, John R., Magic Tesseracts and nDimensional Magic Hypercubes,
Journal of Recreational Mathematics, 6:3:1973: 193201.
[5] Hendricks, John R., Species of ThirdOrder Magic Squares and Cubes,
Journal of Recreational Mathematics, 6:3:1973:190192.
[6] Hendricks, John R., The Diagonal Rule for Magic Cubes of Odd Order,
Journal of Recreational Mathematics, 20:3:1988:192195
[7] Hendricks, John R., Ten Magic Tesseracts of Order Three, Journal of
Recreational Mathematics, 18:2:198586:125134 (#110) and #21 24 JRM:20:4:
1988:251256, #11:pp 275276, #1216:pp 279283, and #2530 in
JRM:21:1:1989:1318 and #1720:2628 and finally # 3158 in JRM:22:1:1990;
1526.
Here they are numbered in the order found, not index order.
[8] Selfpublished as an 8 page brochure. The Order6 Inlaid
Magic Tesseract
may be seen at
http://members.shaw.ca/johnhendricksmath/inlaid_t.htm
Also,
Hendricks, John R.,
Inlaid Magic Squares and Cubes,
Selfpublished, 1999, 0968470017
[9] Heinz, H and Hendricks, R., A Unified Classification System for Magic
Hypercubes, Journal of Recreational Mathematics:32:1:200304:3036.
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
http://magicsquares.net/downloads.htm#MS%20Excel
[11] Hendricks, John R., Perfect nDimensional Magic Hypercubes of Order 2^{n},
Selfpublished, 1999, 0968470041
[12] Seven other spreadsheets by John Hendricks may also be downloaded from the
above site.
[13] The list of publications are at
http://members.shaw.ca/johnhendricksmath/bibliography.htm
[14] Many of Hendricks outofprint
books are
now
available for free download in Portable Document Format (pdf)
from my Downloads page.
http://magicsquares.net/downloads.htm#Books
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 1,2,3,4agonal magic
tesseracts of orders 8, 10, 12, 14, 15, and 18 and posted them on his site.
[1]
http://magicsquares.net/cthtm/c_update3.htm#A%20New%20Magic%20Cube%20Class
[2]. Heinz, Harvey and Nakamura,
Mitsutoshi, Magic Tesseract Classes, Journal of Recreational Mathematics,
JRM:35:1:2006:1114
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:35:1:2006:510.
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]
http://homepage2.nifty.com/googol/magcube/en/classes.htm#tesserallclasses.
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 xagonals 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 multimagic 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://magicsquares.net/cthtm/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, 189204, 2000 ACTA ARITHMETICA XCII.2 92000)
(aacub00.pdf) and
AN ALGORITHM FOR MAGIC TESSERACTS (06MagicTesseract.pdf) showing how to
construct a magic tesseract by formulae.
Scientific Bulletin of Chelm, Section of Mathematics and Computer Science
2/2006, 249251 ISBN 8392453646, ISSN 1896463X
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 2D diagram (of many possible
versions) of the 4D tesseract. [7]
(1) Heinz, H and Hendricks, R., A Unified
Classification System for Magic Hypercubes, Journal of Recreational
Mathematics:32:1:200304:3036.
In this first version we missed one cube classification.
(2). Heinz, Harvey, Hypercube Classes  An Update, Journal of Recreational
Mathematics: JRM:35:1:2006:510
(3). Heinz, Harvey and Nakamura, Mitsutoshi, Magic Tesseract Classes, Journal of
Recreational Mathematics, JRM:35:1:2006:1114
[4]
http://magicsquares.net/magic_tesseract_index.htm
[5] Heinz, H.D. and Hendricks, J. R., Magic Square Lexicon: Illustrated.
Selfpublished, 2000, 0968798500.
239 terms defined, about 200 illustrations and tables, 171 captioned.
[6] Heinz, Harvey D., Magic Hypercube Omnibus DVD, 2011, 9780968798515 (free
distrubion)
[7]
http://magicsquares.net/cthtm/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]
[1]
http://members.shaw.ca/johnhendricksmath/tesseracts.htm
Otherlinks
Charlie Kelly has a magic Hypercube generator at
http://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=9To1nPvY1AC&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
http://io9.com/5823271/themanydimensionsofthetesseract
Suggestions for New
Investigations

Investigate and construct Inlaid, Quadrant, AddMultiply , etc.
tesseracts
[1]

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 Dimension5 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 followup search revealed many sites that used the term penteract in referring
to a 5D 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/5cube".
[4] A site using the term penteract and also a
good source for penteract diagrams is
http://en.wikipedia.org/wiki/5cube
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, order3 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 nagonal (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 nDimensional Magic Hypercubes of Order 2n, Selfpublished, 1999,
0968470041. 36 pages plus covers, 8.5" x 11" flat stitched, some diagrams.
Theory with examples for a cube, tesseract, 5D
and 6D hypercubes. John Hendricks wrote a small basic program in 1999
(for his programmable calculator) that generates any desired path of an order 32
nasik hypercube5.
[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/5cube
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 fourdimensional
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
