Hypercube Book for Sale



Introducing a new reference book on magic squares, cubes, tesseracts, magic stars, etc.

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Front cover of the book

Magic Square Lexicon: Illustrated
By Harvey D. Heinz and John R. Hendricks

ISBN 0-9687985-0-0, 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 "how-to-do" 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 out-of-print

Download the free complete Lexicon eBook here. (2072 Kb).

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An Example Entry


The magic sum for an n-Dimensional Magic Hypercube of Order m is given by:

S = m(1 + mn)/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(r-1)n!m(n-1)/[r!(n-r)!]

  • Where: N is the number of r-agonals
  • 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










Regular 3 2m 2     2m + 2
Pandiagonal 4 2m 2m     4m
Regular 3 3m2   4   3m2 + 4
Diagonal 5 3m2 6m 4   3m2 + 6m+4
Pantriagonal 4 3m2   4m2   7m2
PantriagDiag 8? 3m2 6m 4m2   7m2 6m
Pandiagonal 7 3m2 6m2 4   9m2 + 4
Perfect 8 3m2 6m2 4m2   13m2
Regular 3 4m3     8 4m3 + 8
Diagonal ? 4m3 12m   8 4m3 + 12m+8
Pandiagonal ? 4m3 12m3   8 16m3 + 8
Pantriagonal ? 4m3   16m3 8 20m3 + 8
Panquadragonal 4 4m3     8m3 12m3
Pan2 + Pan3 ? 4m3 12m3 16m3 8 32m3 + 8
Pan2 +Pan4 ? 4m3 12m3   8m3 24m3
Pan3 + Pan4 ? 4m3   16m3 8m3 28m3
Perfect 16 4m3 12m3 16m3 8m3 40m3

162 - Hypercubes – number of correct summations.
Addendum: This table is out of date. There are actually 18 classes of magic tesseracts.

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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, magic-cubes@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), 2002-2003, pp59-60)

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 0-9687985-0-0.

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 well-done 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 square-like constructs.  This is one of the most impressive books I have ever read.

Reviewed by Charles Ashbacher
Editor, Journal of Recreational Mathematics

e-mail 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.

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July 2005: The second print run (v. 2) has all known errors corrected.

Unfortunately, the book is now out-of-print

Download the free complete Lexicon eBook here. (2072 Kb).

This page was originally posted December 2000
It was last updated March 04, 2013
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz