*The CRC Concise Encyclopedia of Mathematics* is a compendium of
mathematical definitions, formulas, figures, tabulations, and
references. It is written in an informal style intended to make it
accessible to a broad spectrum of readers with a wide range of
mathematical backgrounds and interests. Although mathematics is a
fascinating subject, it all too frequently is clothed in specialized
jargon and dry formal exposition that make many interesting and useful
mathematical results inaccessible to laypeople. This problem is often
further compounded by the difficulty in locating concrete and easily
understood examples. To give perspective to a subject, I find it
helpful to learn why it is useful, how it is connected to other areas
of mathematics and science, and how it is actually implemented. While
a picture may be worth a thousand words, explicit examples are worth at
least a few hundred! This work attempts to provide enough details to
give the reader a flavor for a subject without getting lost in
minutiae. While absolute rigor may suffer somewhat, I hope the
improvement in usefulness and readability will more than make up for
the deficiencies of this approach.

The format of this work is somewhere between a handbook, a dictionary, and an encyclopedia. It differs from existing dictionaries of mathematics in a number of important ways. First, the entire text and all the equations and figures are available in searchable electronic form on CD-ROM. Second, the entries are extensively cross-linked and cross-referenced, not only to related entries but also to many external sites on the Internet. This makes locating information very convenient. It also provides a highly efficient way to "navigate" from one related concept to another, a feature that is especially powerful in the electronic version. Standard mathematical references, combined with a few popular ones, are also given at the end of most entries to facilitate additional reading and exploration. In the interests of offering abundant examples, this work also contains a large number of explicit formulas and derivations, providing a ready place to locate a particular formula, as well as including the framework for understanding where it comes from.

The selection of topics in this work is more extensive than in most
mathematical dictionaries (e.g., Borowski and Borwein's
*HarperCollins Dictionary of Mathematics* and Jeans and Jeans'
*Mathematics Dictionary*). At the same time, the descriptions are
more accessible than in "technical" mathematical encyclopedias (e.g.,
Hazewinkel's *Encyclopaedia of Mathematics* and Iyanaga's
*Encyclopedic Dictionary of Mathematics*). While the latter
remain models of accuracy and rigor, they are not terribly useful to
the undergraduate, research scientist, or recreational mathematician.
In this work, the most useful, interesting, and entertaining (at least
to my mind) aspects of topics are discussed in addition to their
technical definitions. For example, in my entry for pi (), the definition in terms of the
diameter and circumference of a circle is supplemented by a great many
formulas and series for pi, including some of the amazing discoveries
of Ramanujan. These formulas are comprehensible to readers with only
minimal mathematical background, and are interesting to both those with
and without formal mathematics training. However, they have not
previously been collected in a single convenient location. For this
reason, I hope that, in addition to serving as a reference source, this
work has some of the same flavor and appeal of Martin Gardner's
delightful *Scientific American* columns.

Everything in this work has been compiled by me alone. I am an astronomer by training, but have picked up a fair bit of mathematics along the way. It never ceases to amaze me how mathematical connections weave their way through the physical sciences. It frequently transpires that some piece of recently acquired knowledge turns out to be just what I need to solve some apparently unrelated problem. I have therefore developed the habit of picking up and storing away odd bits of information for future use. This work has provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics. I have also found it very difficult to find clear yet accessible explanations of technical mathematics unless I already have some familiarity with the subject. I hope this encyclopedia will provide jumping-off points for people who are interested in the subjects listed here but who, like me, are not necessarily experts.

The encyclopedia has been compiled over the last 11 years or so,
beginning in my college years and continuing during graduate school.
The initial document was written in *Microsoft Word* on a Mac Plus
computer, and had reached about 200 pages by the time I started
graduate school in 1990. When Andrew Treverrow made his
OzT_{E}X program available for the Mac, I began the task of
converting all my documents to T_{E}X, resulting in a vast
improvement in readability. While undertaking the *Word* to
T_{E}X conversion, I also began cross-referencing entries,
anticipating that eventually I would be able to convert the entire
document to hypertext. This hope was realized beginning in 1995, when
the Internet explosion was in full swing and I learned of Nikos Drakos
and Ross Moore's excellent T_{E}X to HTML converter,
L^{A}T_{E}X2*HTML*. After some
additional effort, I was able to post an HTML version of my
encyclopedia to the World Wide Web, currently located at
www.treasure-troves.com/math/.

The selection of topics included in this compendium is not based on any
fixed set of criteria, but rather reflects my own random walk through
mathematics. In truth, there is no good way of selecting topics in
such a work. The mathematician James Sylvester may have summed up the
situation most aptly. According to Sylvester (as quoted in the
introduction to Ian Stewart's book *From Here to Infinity*),
"Mathematics is not a book confined within a cover and bound between
brazen clasps, whose contents it needs only patience to ransack; it is
not a mine, whose treasures may take long to reduce into possession,
but which fill only a limited number of veins and lodes; it is not a
soil, whose fertility can be exhausted by the yield of successive
harvests; it is not a continent or an ocean, whose area can be mapped
out and its contour defined; it is as limitless as that space which it
finds too narrow for its aspiration; its possibilities are as infinite
as the worlds which are forever crowding in and multiplying upon the
astronomer's gaze; it is as incapable of being restricted within
assigned boundaries or being reduced to definitions of permanent
validity, as the consciousness of life."

Several of Sylvester's points apply particularly to this undertaking.
As he points out, mathematics itself cannot be confined to the pages of
a book. The results of mathematics, however, are shared and passed on
primarily through the printed (and now electronic) medium. While there
is no danger of mathematical results being lost through lack of
dissemination, many people miss out on fascinating and useful
mathematical results simply because they are not aware of them. Not
only does collecting many results in one place provide a single
starting point for mathematical exploration, but it should also lessen
the aggravation of encountering explanations for new concepts which
themselves use unfamiliar terminology. In this work, the reader is
only a cross-reference (or a mouse click) away from the necessary
background material. As to Sylvester's second point, the very fact
that the quantity of mathematics is so great means that any attempt to
catalog it with any degree of completeness is doomed to failure. This
certainly does not mean that it's not worth trying. Strangely, except
for relatively small works usually on particular subjects, there do not
appear to have been any substantial attempts to collect and display in
a place of prominence the treasure trove of mathematical results that
have been discovered (invented?) over the years (one notable exception
being Sloane and Plouffe's *Encyclopedia of Integer Sequences*).
This work, the product of the "gazing" of a single astronomer,
attempts to fill that omission.

Finally, a few words about logistics. Because of the alphabetical listing of entries in the encyclopedia, neither table of contents nor index are included. In many cases, a particular entry of interest can be located from a cross-reference (indicated in SMALL CAPS TYPEFACE in the text) in a related article. In addition, most articles are followed by a "see also" list of related entries for quick navigation. This can be particularly useful if you are looking for a specific entry (say, "Zeno's Paradoxes"), but have forgotten the exact name. By examining the "see also" list at bottom of the entry for "Paradox," you will likely recognize Zeno's name and thus quickly locate the desired entry.

The alphabetization of entries contains a few peculiarities which need mentioning. All entries beginning with a numeral are ordered by increasing value and appear before the first entry for "A." In multiple-word entries containing a space or dash, the space or dash is treated as a character which precedes "a," so entries appear in the following order: "Sum," "Sum P...," "Sum-P...," and "Summary." One exception is that in a series of entries where a trailing "s" appears in some and not others, the trailing "s" is ignored in the alphabetization. Therefore, entries involving Euclid would be alphabetized as follows: "Euclid's Axioms," "Euclid Number," "Euclidean Algorithm." Because of the non-standard nomenclature that ensues from naming mathematical results after their discoverers, an important result such as the "Pythagorean Theorem" is written variously as "Pythagoras's Theorem," the "Pythagoras Theorem," etc. In this encyclopedia, I have endeavored to use the most widely accepted form. I have also tried to consistently give entry titles in the singular (e.g., "Knot" instead of "Knots").

In cases where the same word is applied in different contexts, the
context is indicated in parentheses or appended to the end. Examples
of the first type are
"Crossing Number (Graph)" and
"Crossing Number (Link)."
Examples of the second type are
"Convergent Sequence" and
"Convergent Series."
In the case of an entry like "Euler Theorem," which
may describe one of three or four different formulas, I have taken the
liberty of adding descriptive words ("Euler's *Something*
Theorem") to all variations, or kept the standard name for the most
commonly used variant and added descriptive words for the others. In
cases where specific examples are derived from a general concept, em
dashes (--) are used (for example,
"Fourier Series,"
"Fourier Series--Power Series,"
"Fourier Series--Square Wave,"
"Fourier Series--Triangle").
The decision to put a possessive 's at the end of a name or to use a
lone trailing apostrophe is based on whether the final "s" is
voiced.
"Gauss's Theorem" is therefore written
out, whereas
"Archimedes' Recurrence Formula"
is not. Finally, given the absence of a definitive
stylistic convention, plurals of numerals are written without an
apostrophe (e.g., 1990s instead of 1990's).

In an endeavor of this magnitude, errors and typographical mistakes are inevitable. The blame for these lies with me alone. Although the current length makes extensive additions in a printed version problematic, I plan to continue updating, correcting, and improving the work.

Eric Weisstein

Charlottesville, Virginia

May 20, 1999

Although I alone have compiled and typeset this work, many people have
contributed indirectly and directly to its creation. I have not yet
had the good fortune to meet Donald Knuth of Stanford University, but
he is unquestionably the person most directly responsible for making
this work possible. Before his mathematical typesetting program
T_{E}X, it would have been impossible for a single
individual to compile such a work as this. Had Prof. Bateman
owned a personal computer equipped with T_{E}X, perhaps his
shoe box of notes would not have had to await the labors of Erdelyi,
Magnus, Oberhettinger, and Tricomi to become a three-volume work on mathematical
functions. Andrew Trevorrow's shareware implementation of
T_{E}X for the Macintosh, OzT_{E}X
(www.kagi.com/authors/akt/oztex.html),
was also of fundamental importance. Nikos Drakos and Ross Moore have
provided another building block for this work by developing the
L^{A}T_{E}X2*HTML* program
(www-dsed.llnl.gov/files/programs/unix/latex2html/manual/manual.html),
which has allowed me to easily maintain and update an on-line version
of the encyclopedia long before it existed in book form.

I would like to thank Steven Finch of MathSoft, Inc., for his
interesting on-line essays about mathematical constants
(www.mathsoft.com/asolve/constant/constant.html),
and also for his kind permission to reproduce excerpts from some of
these essays. I hope that Steven will someday publish his detailed
essays in book form. Thanks also to Neil Sloane and Simon Plouffe for
compiling and making available the printed and on-line
(www.research.att.com/~njas/sequences/)
versions of the *Encyclopedia of Integer Sequences*, an immensely
valuable compilation of useful information which represents a truly
mind-boggling investment of labor.

Thanks to Robert Dickau, Simon Plouffe, and Richard Schroeppel for
reading portions of the manuscript and providing a number of helpful
suggestions and additions. Thanks also to algebraic topologist Ryan
Budney for sharing some of his expertise, to Charles Walkden for his
helpful comments about dynamical systems theory, and to Lambros Lambrou
for his contributions. Thanks to David W. Wilson for a number of
helpful comments and corrections. Thanks to Dale Rolfsen, compiler
James Bailey, and artist Ali Roth for permission to reproduce their
beautiful knot and link diagrams. Thanks to Gavin Theobald for
providing diagrams of his masterful polygonal dissections. Thanks to
Wolfram Research, not only for creating an indispensable mathematical
tool in *Mathematica*, but also for permission to include figures
from the *Mathematica* book and *MathSource* repository for
the
braid,
conical spiral,
Enneper's Minimal Surface,
Hadamard matrix,
helicoid,
helix,
Henneberg's minimal surface,
hyperbolic polyhedra,
Klein bottle,
Maeder's owl minimal surface,
Penrose tiles,
polyhedron,
and
Scherk's minimal surfaces entries.

Particular thanks to Tim Buchowski, Patrick De Geest, Robert Dickau, Dewi Jones, Lloyd Rowsey, and Paul Stanford for their help in correcting typographical errors from the first hardcover printing of this work.

Many thanks to Joshua Kempner for design and layout of the book's cover illustration and for layout and graphic design on the CD-ROM.

Sincere thanks to Judy Schroeder for her skill and diligence in the
monumental task of proofreading the entire document for syntax. Thanks
also to Bob Stern, my executive editor from *CRC Press*, for his
encouragement, and to Mimi Williams of CRC Press for her careful
reading of the manuscript for typographical and formatting errors. As
this encyclopedia's entry on
Proofreading Mistakes shows, the
number of mistakes that are expected to remain after three independent
proofreadings is much lower than the original number, but unfortunately
still nonzero. Many thanks to the library staff at the University of
Virginia, who have provided invaluable assistance in tracking down many
an obscure citation. Finally, I would like to thank the hundreds of
people who took the time to e-mail me comments and suggestions while
this work was in its formative stages. Your continued comments and
feedback are very welcome.

© 1999 Eric W. Weisstein and Chapman & Hall/CRCnetBASE

comments@treasure-troves.com

May 20, 1999