2 nd ed. CRC Concise Encyclopedia of Mathematics

1. 2nd ed. CRC Concise Encyclopedia of Mathematics LIBR 150 Reference Services Kamal Lombardi Call No. R510.3

2. Bibliographic Citation • Weisstein, Eric W. ed. 2nd ed. CRC Concise Encyclopedia of Mathematics. Boca Raton: Chapman & Hall/CRC. 2003 • Available at Kirkland Library, Reference, KCLS system, Kirkland, WA.

3. Indexing Arrangement • 3242 pages of 2 columned text with accompanying black and white diagrams. • There is no indexing as far as I can see. Entries are in alphabetical order. There are “see also” lists of further references provided. Wolfram Mathworld (explained later) has search indexing features which allow one to search for a specific entry. The Butterfly fractal is explained in the encyclopedia as “the fractal like curve generated by the 2d function F(x, y) = [(x2 – y2) sin (x + y)/ a) x2 + y2 .” Picture from Wolfram Mathworld: http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math.htm

4. Scope • The book is an alphabetical guide to different mathematical concepts and ideas. It covers all kinds of math from Numerals, at the beginning of the book to the last entry which is “Z-Transform and Z-Transform (population). Many interesting kinds of math are described and covered including fractals, inverse chessboards, different types of polyhedrons, compound cubes, other interesting geometrical shapes, elementary functions, Johnson solids, Archimedan spirals, and interesting mathematical oddities like magic circles, to name just a few.

5. Currency (and Wolfram Mathworld) • It is up to date and current as of 2003, when the last edition was published. Since that time there is also a web page that has been put up called Wolfram Mathworld which is based off of the information Eric Weisstein has compiled over the years, which is available through the web at http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math.htm. For the most current information on the mathematical topics included, one might want to first look in the book, then consult the web resource to see if more current information is available. This presentation is primarily based on what I found in the book as are the diagrams, although some diagrams herein are used gotten from Wolfram Mathworld with proper citation given to the page from which I got them.

6. Intended Purpose • Its purpose is to serve as a reference for looking up alphabetically all different kinds of math topics and concepts and ideas. • Dendrite Fractal: A Dendrite Fractal is defined in both 2nd Ed. CRC Concise Encyclopedia of Mathematics and Wolfram Mathworld as “A Julia set with constant chosen at the boundary of the Mandelbrot set (Branner 1989; Dufner et al. 1998, p. 225). The image above was computed using c = i.” (2nd Ed. CRC Concise Encyclopedia of Mathematics, p. 701), (Weisstein, Eric W. "Dendrite Fractal." From Mathworld --A Wolfram Web Resource. http://mathworld.wolfram.com/DendriteFractal.html) • IMG url= DendriteFractal_1000.gif, http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math.htm)

7. Additional Purposes Served • The text can serve as a reference for students, mathematicians, and those just interested in searching for interesting mathematical concepts and ideas. A student might use it to prepare for a quiz or look up interesting mathematical ideas not covered in class. A mathematician might use it to get information on the most current information regarding interesting mathematical ideas. Others might find it interesting because of its diagrams and interesting topics and equations.

8. Echidnahedron: This is a type of Icosahedron, which is itself created through a stellation process (#4 in the list). Both the 2nd Ed. CRC Concise Encyclopedia... and Wolfram Mathworld agree on these facts. Weisstein, Eric W. "Echidnahedron." From Mathworld --A Wolfram Web Resource. http://mathworld.wolfram.com/Echidnahedron.html • IMG = “Icos04_500.gif”, http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math.htm • Convex Polyhedron: “… a Polyhedron for which a line connecting any two (noncoplanar) points on the surface always lies in the interior of the polyhedron. The 92 convex polyhedra having only Regular Polygons as faces are called the Johnson solids, which include the Platonic Solids and Archimedean Solids. No method is known for computing the Volume of a general convex polyhedron (Ogilvy 1990, p. 173). • See also Archimedean Solid, Deltahedrron, Johnson Solid, Kepler-Poinsot Solid, Platonic Solid, Regular Polygon • Weisstein, Eric W. “Convex Polyhedron.” From Mathworld – A wolfram Web Resource http://mathworld.wolfram.com/ConvexPolyhedron.html • IMG = PolyhedronConvex_1000.gif” http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math.htm

9. Format (s) • It is available in book, a CD-Rom version, “… written in ISO 9660 format using plain HTML with Java … readable by all computers using a Java-capable web browser. … In this CD-ROM, links to pages on the CD-ROM itself are shown in blue (or your browser's default color), while links to external pages requiring an Internet connection are shown in red.” (Weisstein, Eric W., “Concise Encyclopedia of Mathematics CD-ROM”. Wolfram Mathworld. (on the web)), and electronic form (as Wolfram Mathworld on the www at http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math.htm.

10. Format(s): Where available • Available in KCLS system: • Kent Reference • Kirkland Reference • Woodinville Reference Available in SPL System: • Central Library, Reference, Levels 7-9 University of Washington Libraries • Math Reference • Odegaard Reference • Link provided in UW Library searching to connect online to Wolfram Mathworld. Trim size of 2nd Edition: 29 cm.

11. Special Features • In the print version, there is an introduction to the 1st edition and a preface to the 2nd edition. Online, there is supposed to be an Errata to the Hardcover version, but it does not seem to be currently reachable. The Introduction to the CD-Rom edition is on the web. Defined as: “An Archimedan Spiral with m = -2 , having polar equation. Lituus means a ``crook,'' in the sense of a bishop's crosier. The lituus curve originated with Cotes in 1722. Maclaurin used the term lituus in his book Harmonia Mensurarum in 1722 (MacTutor Archive). The lituus is the locus of the point moving such that the Area of a circular Sector remains constant.” (2nd Ed. CRC Concise Encyclopedia of Mathematics, p. 1785).

12. Authority or credentials (review opinions) • On the website itself: “The author has been compiling scientific encyclopedias for more than ten years, beginning as a high school student and continuing through his college, graduate school, and post-graduate years. His mathematics encyclopedia is part of a larger project to collect and make available detailed mathematical and scientific information in a way most accessible to laypeople. The author hopes this encyclopedia and the others currently under development will provide jumping-off points for people who are interested in scientific subjects but are not necessarily experts.” (“About the Author. Wolfram Mathworld) • From the Preface to the Second Edition: “The long awaited second edition of this Encyclopedia is finished… Heavily revised by the author Eric Weisstein over the past three years, it contains… approximately 3, 600 new entries … many illustrations, and updated the bibliographies that follow each entry to include the most recent references … this edition integrates the use of Mathmatica software into many of its entries, presenting the precise commands that allow you to implement the formulas presented, perform many different calculations, construct graphical displays of your results, and generate remarkable mathematical illustrations.” (2nd Edition CRC Concise Encyclopedia of Mathematics, “Preface to the New Edition”.)

13. Authority or credentials cont. (review opinions) • From the Introduction to the 1st Edition: “Everything in this work has been compiled by me alone. I am an astronomer by training, but I 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 thereby developed the habit of picking up and storing away odd bits of information for further use. This work has provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics. 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 here but who, like me, are not necessarily experts.” (Weisstein, Eric W. “Introduction to the First Edition”, 2nd Edition CRC Concise encyclopedia of Mathematics.

14. Authority or credentials (Review Opinions) • Book Review: There is a review of the book by Emma Previato and Robert E. O Malley, Jr. in SIAM Review, 2004, Vol 46, No. 2, pp. 349-372. available at http://moe.ic.highline.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true&AuthType=ip&db=a9h&AN=13376916&site=ehost-live&scope=site Prevatio and O Malley (2004) after initially giving several extensive definitions of what an encyclopedia is, turn to the ways in which Weisstein’s text is different from what came before, stating: “Weinstein has produced something new as well. It seems to me that his encyclopedia inaugurates a new era for two reasons: one, as the author says this book lives and breathes hand-in-hand with the internet. This is truly the new culture of science. Electronic archives are becoming the best resource to learn about a research topic up to the current most sophisticated point. (see, for example, http://www.aimath.org/rmt.html) Moreover, we are the B generation, or at least among the young you are: the generation that blogs have enabled to post their life’s findings on a webpage. Professional expertise is becoming a shared experience: “mathematics is not a lonely profession!”…

15. Authority or credentials (Review Opinions) (cont.) • (cont.) Present times are so overloaded with information that the ability to manage knowledge has become a criterion more valued than problem-solving (according to a review of How would you move Mt. Fuji? by William Poundstone). Channeling information efficiently and satisfactorily is one of the crown achievements of the Google search engine-such a unique feature that Sergey Brin, president and cofounder of Google, Inc. At an after-dinner address to the members of the MSRI community in Berkeley (March 2001), refused to disclose the recipe for success, even while entertaining all other questions. (A random quote from a website: ‘MOUNTAIN VIEW, Calif-Dec 1, 2000. The quarterly NPD study, which gathered data from 19,357 internet users, [found Google] No. 1 for Relevant and Accurate Search Results, No. 1 for Meeting the Needs of Users, No. 1 for Ease of Use and Navigation…’ Sounds like Mathworld, the encyclopedia’s online version!) This book’s ease of reference, despite a lack of index due to size limitation, and interactive features place it squarely and successfully in this new age. Weisstein has been adamant, despite litigation that nearly wiped him out (see his statement on the mathworld site) about putting his encyclopedia online and having it used and enriched by you via the internet. This is not the place for opining whether electronic publishing makes the printed page useless (what about the Britannica anyway?) but I can’t imagine anybody feeling that way. (Prevatio, 2004, p. 350) • Prevatio (2004) then goes on to look at Weisstein’s work in terms of “Coverage”, “References”, “History”, “Audience”, “Clarity”, “Pictures”, and “Accuracy”, (Prevatio, 2004, pgs. 352-354.). She goes into much detail regarding whether or not the mathematics are accurate and up to date pointing out that the only major flaw

16. Authority or credentials (Review Opinions) cont. • (cont.) seems to be “inconsistency”. (p. 352). Prevatio points out that, “Under Poncelet’s porism you will not find Berger’s Geometry, cited elsewhere in the book, nor will you find Baker’s An Introduction to Plane Geometry , whereas it is cited under Fuerbach’s theorem. The lavish and intriguing Visual Mind: Art and Mathematics edited by M. Emmer is not present under, say, polytope, but it is under Dyck’s surface.” (Prevatio, p. 352). • Magic Circles: “A set of magic circles is a numbering of the intersections of the circles such that the sum over all intersections is the same constant for all circles. The above sets of three and four magic circle A set of magic circles is a numbering of the intersections of the circles such that the sum over all intersections is the same constant for all circles. The above sets of three and four magic circles have magic constants 14 and 39 (Madachy 1979). s have magic constants 14 and 39 (Madachy 1979).”),Weisstein Eric W. . "Magic Circles." From Mathworld--A Wolfram Web Resource. http://mathworld.wolfram.com/MagicCircles.html • “Another type of magic circle arranges the number 1, 2, ..., in a number of rings, which each ring containing the same number of elements and corresponding elements being connected with radial lines. ne of the numbers (which is subsequently ignored) is placed at the center. In a magic circle arrangement, the rings have equal sums and this sum is also equal to the sum of elements along each diameter (excluding the central number). Three magic circles using the numbers 1 to 33 are illustrated above (Hung).” (Ibid.)

17. Personal Impression Putting the question of whether the electronic version supersedes the print version in this case aside, just judging the book on its own merits one finds a good guide to mathematical topics presented in an easy to read and follow format. One can understand this book usually without being an expert in mathematics, nor having had to read accompanying literature to get a general gist of the ideas presented in it. It is thus, a good text for the layman, student, or non-specialist interested in the topic. • The online version of the book, extends the features of the text version in many ways, but still, having the print version on hand is in my eyes useful. The online version has both searchability features and the ability to scroll alphabetically, but such an approach presupposes you know exactly what you’re looking for, rather than browsing through the topics and accompanying diagrams. Because of this, I see the other versions of the book on CD-Rom or electronically not so much intended as replacements or further updates beyond the existing print edition, but rather as supplements and extensions of the reference item’s use and purpose.

18. Examples typifying reference uses • Student or interested patron: What’s known about fractals? What can you tell me about a polyhedron? What is a magic circle? • Mathematicians: How many interesting cubic structures are there beyond the first? What is Elementary Symmetric Function? • Non-Specialist: I heard about these interesting shapes. Can you tell me about or show me Johnson Solids?

19. Complementary Sources • From the Introduction to the CD-Rom Edition: “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.” (“Concise Encyclopedia of Mathematics CD-ROM: A Treasure Trove of Mathematical Formulas, Facts, Figures, and Fun” Introduction to the CD-Rom edition, CRC Concise Encyclopedia of Mathematics, on the web at http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math.htm

20. Complementary Sources • I also turned up an interesting web article by Weisstein on how the book and online version is used by the TV show Numbers. It is called The Math(ematica) behind Television’s Crime Drama NUMB3RS. • Weisstein points out “So how do we help NUMB3RS? We improve the mathematics, catch potential inconsistencies, correct arithmetic on occasion, and in general answer a wide variety of questions from the writers, directors, and actors. We research and derive many of the equations and diagrams that appear in the show. We also enrich the show with modern topics in math and science, including some so current that they may have first appeared in research literature as little as a few weeks earlier. Finally, we provide the NUMB3RS props department with T-shirts, posters, and books for the show. In fact, past episodes of the show have sported The Mathematica Book, Stephen Wolfram’s A New Kind of Science, Michael Trott’s Mathematica Guidebooks, and my own Concise Encyclopedia of Mathematics. The keen-eyed viewer may also have noticed the Mathematica-generated Rieman zeta-function poster in one episode and a nested triangles t-shirt donned by the arch-villain in another.

21. Some more images (Johnson Solids), That’s it.