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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.

Magic Square Lexicon: Illustrated
By Harvey D. Heinz and John R. Hendricks
ISBN 0-9687985-0-0, 228 pages 5 ½ x 8 ½, 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)

An example entry

Summations

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.

Magic Hypercube	Lowest Order	i- rows	n-agonals
Magic Hypercube	Lowest Order	i- rows	2	3	4	Total
Square
Regular	3	2m	2			2m + 2
Pandiagonal	4	2m	2m			4m
Cube
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
Tesseract
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.

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 <cashbacher@yahoo.com>
To:         hdheinz@istar.ca, magic-cubes@home.com

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
e-mail: harveyheinz@shaw.ca

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

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 http://www.magic-square.de/.
In addition, John has published over 40 articles and papers on these subjects as well as a half dozen books.

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.

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 harveyheinz@shaw.ca

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

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