
Introducing a new reference book on magic
squares, cubes, tesseracts, magic stars, etc.
Front cover of the book 
Magic Square Lexicon:
Illustrated
By Harvey D. Heinz and John R. Hendricks
ISBN 0968798500, 228 pages 5 ˝ x 8 ˝,
perfect bound, laminated cover.
171 captioned illustrations and tables, 239 terms defined, 2
appendices of bibliographies.
This book defines 239 terms associated
with magic squares, cubes, tesseracts, stars, etc. Many of these
terms have been in use hundreds of years while some were coined
in the last several years. While meant as a reference book, it
should be ideal for casual browsing with its almost 200
illustrations and tables, 171 of which are captioned.
While this book is not meant as a
"howtodo" book, it should be a source of inspiration for
anyone interested in this fascinating subject. Many tables
compare characteristics between orders or dimensions. The
illustrations were chosen, where possible, to demonstrate
additional features besides the particular definition.
(from the back cover) 

Unfortunately, the book is now outofprint
Download the free
complete Lexicon eBook here. (2072
Kb).

An Example Entry Summations
The magic sum
for an nDimensional Magic Hypercube of Order m is given by:
S = m(1 + m^{n})/2
In a magic object, there are many lines that produce the
magic sum. The table below, shows the minimum requirement of the number of
lines for various types of magic hypercubes and is derived from the
following equation:
N = 2^{(r1)}n!m^{(n1)}/[r!(nr)!]
 Where: N is the number of ragonals
 n is the dimension of the hypercube
 m is the order of the hypercube, and
 r is the dimension of the hyperplane.
When r = 1, the number of orthogonals is given by N. As
well, shown is the smallest order for the various classifications of
pandiagonal, pantriagonal, etc. which is known. for each dimension. Some of
the tesseracts are not known yet and some of these varieties have not been
constructed yet.
This table provides the minimum requirements for each
category. Usually, there are some extra lines which may sum the magic sum,
but not a complete set so as to change the category.
It is possible that when the tesseract is explored more fully, some
additional classifications will be found.
In the case of the cube, John Hendricks missed the Diagonal and the
PantriagDiag. 
Magic
Hypercube 
Lowest
Order 
i
rows 
nagonals 

2 
3 
4 
Total 
Square 






Regular 
3 
2m 
2 


2m + 2 
Pandiagonal 
4 
2m 
2m 


4m 
Cube 






Regular 
3 
3m^{2} 

4 

3m^{2}
+ 4 
Diagonal 
5 
3m^{2} 
6m 
4 

3m^{2}
+ 6m+4 
Pantriagonal 
4 
3m^{2} 

4m^{2} 

7m^{2} 
PantriagDiag 
8? 
3m^{2} 
6m 
4m^{2} 

7m^{2}
6m 
Pandiagonal 
7 
3m^{2} 
6m^{2} 
4 

9m^{2}
+ 4 
Perfect 
8 
3m^{2} 
6m^{2} 
4m^{2} 

13m^{2} 
Tesseract 






Regular 
3 
4m^{3} 


8 
4m^{3}
+ 8 
Diagonal 
? 
4m^{3} 
12m 

8 
4m^{3}
+ 12m+8 
Pandiagonal 
? 
4m^{3} 
12m^{3} 

8 
16m^{3}
+ 8 
Pantriagonal 
? 
4m^{3} 

16m^{3} 
8 
20m^{3}
+ 8 
Panquadragonal 
4 
4m^{3} 


8m^{3} 
12m^{3} 
Pan2 + Pan3 
? 
4m^{3} 
12m^{3} 
16m^{3} 
8 
32m^{3}
+ 8 
Pan2 +Pan4 
? 
4m^{3} 
12m^{3} 

8m^{3} 
24m^{3} 
Pan3 + Pan4 
? 
4m^{3} 

16m^{3} 
8m^{3} 
28m^{3} 
Perfect 
16 
4m^{3} 
12m^{3} 
16m^{3} 
8m^{3} 
40m^{3} 
162  Hypercubes – number of
correct summations.
Addendum: This table is out of date. There are actually 18
classes of magic tesseracts.

Book Review
Subject: Your book, Magic Square Lexicon:
Date: Fri, 22 Dec 2000 16:36:33 0800 (PST)
From: Charles Ashbacher <cashbacher@yahoo.com>
To: hdheinz@istar.ca,
magiccubes@home.com (hh note: email addresses now
changed)
Thank you for sending me a copy of your wonderful book.
The following review will appear in the book reviews column of 30(4) of JRM.
(It appeared in JRM 31(1), 20022003, pp5960)
Review of:
Magic Square Lexicon: Illustrated, by H. D. Heinz and J. R. Hendricks
Published by Harvey D. Heinz, Surrey, BC, 2000.
174 pages, $25.00(paper). ISBN 0968798500.
Book Review
While magic squares have a long history, until I read this book, I had no
idea how much has been done in the last few decades. The basic principles
that make up a magic square can be used to create an enormous number of
similar objects. There are magic cubes, tesseracts, stars, circles,
triangular regions, hexagons and just about every other shape in
existence. Further complicating the mix are additional features such as
using only prime numbers or numbers whose squares also make the structure
magic.
The purpose of this book is to introduce and explain
these results. Designed in the format of a dictionary, the topics are in
alphabetical order for easy reference. Profusely illustrated, nearly every
topic is accompanied by an illustration, all of which are welldone and
make the topic completely unambiguous.
There is no doubt that magic squares will still be a
popular field of mathematics one hundred years from now. To me, it is also
clear that at that time the publication of this book will be considered a
major event in the history of magic squarelike constructs. This is one
of the most impressive books I have ever read.
Reviewed by Charles Ashbacher
Editor, Journal of Recreational Mathematics
email and book review quoted and used by
permission
Magic Square Lexicon:
Illustrated is all about magic squares, magic cubes, magic
tesseracts, magic hypercubes, magic stars, magic circles, etc. It gives
you definitions, limits, examples, illustrations, tables, terminology and
everything that you want to know about magic objects.
It is written by two men who have spent a
lifetime studying the subject and who have pooled
their knowledge and experience in order to produce this book.
The contents of this web site show a fair
representation of Harvey Heinz’s work and interests.
The work of John Hendricks, may be seen at
his
page on this site.
In addition, John has published over 40 articles and papers on these subjects as
well as a half dozen books.
July 2005: The second
print run (v. 2) has all known errors corrected.
Unfortunately, the book is now outofprint
Download the free
complete Lexicon eBook here. (2072 Kb).
