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

Download the free Lexicon_sample.pdf (549 Kb). This is a faithful reproduction of about the first one-quarter of the book.

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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)

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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
Panquadrag-onal 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.

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Book Review

(e-mail quoted and used by permission HH note: My address is now changed)
Subject:  Your book, Magic Square Lexicon:
Date:      Fri, 22 Dec 2000 16:36:33 -0800 (PST)
From:     Charles Ashbacher <>

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.

Contact information: Harvey Heinz, 22-5688 152 Street, Surrey, BC  V3S 3K2

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

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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 and also at
In addition, John has published over 40 articles and papers on these subjects as well as a half dozen books.

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Learn about the geometry of n-dimensional space through the medium of magic squares and cubes. The new concept of ‘perfect’ is shown applied consistently to magic squares, cubes, tesseracts, etc. Many tables provide comparisons between orders or dimensions.

This is a reference book. There is no other book on the market like it. There are many references to the latest up-to-date research in this field. Did you even know that such mathematical items as inlaid magic cubes existed, or inlaid magic tesseracts?.

July 2005: The second print run (now selling) has all known errors corrected.

Need more information?
Download the free Lexicon_sample.pdf (549 Kb). This is a faithful reproduction of about the first one-quarter of the book.

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Copies of Magic Square Lexicon: Illustrated are available postage paid (in North America) by sending either a postal Money Order or a bank draft for $32.00 Canadian, or $25.00 U.S. funds per copy to:

H. D. Heinz, #22 - 5688 152 Street, Surrey, B.C.,V3S 3K2,Canada.

For additional information email

Order a copy of this book today, for yourself or a friend !

Copies mailed outside North America are $30.00 U. S, including airmail postage.
Possibly the simplest and least expensive method is to send $45.00 Canadian funds via Western Union
(see your local money store). You must notify me of this because Western Union does not contact the recipient).

NEW! And now the cheapest and simplest way!
You can pay with a credit card over a Secure server on the Internet .

For Shipments within North America.

For shipments outside of North America.

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This page last updated December 18, 2005
Harvey Heinz
Copyright 2000, 2001 by Harvey D. Heinz