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Icons of Mathematics: An Exploration of Twenty Key Images

Claudi Alsina and Roger B. Nelsen
Mathematical Association of America
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 45
[Reviewed by
Cheryl J. McAllister
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Icons of Mathematics is #45 in the Dolciani Mathematical Expositions series, and it certainly lives up to the charge of the Dolciani series, providing highly readable discussions of a selection of 20 images that are as recognizable to the general public as they are to mathematicians. Each chapter focuses on one of the icons and includes a standard set of items: quotes from a wide variety of sources (Zhang Zai to Ralph Waldo Emerson), a short history of the figure including examples of uses of the image in mathematical and non-mathematical settings, examples of the figure in mathematical proofs of theorems or solutions to problems, and problems for the reader to solve. Outlines of the solutions to the problems are in the back of the book.

The images range from the traditional figure used to either prove or explain the Pythagorean Theorem (a right triangle with a square built on each side) to Yin and Yang and tatami mats (an arrangement of rectangles used in the design of traditional floor coverings in Japanese homes). The mathematics explored includes the expected geometry and trigonometry as well as number theory and recreational mathematics. The content is appropriate for advanced high school students and college undergraduates who have taken courses in geometry, trigonometry, and algebra. Mathematics majors and graduate students can be enriched and entertained by the interesting historical background of the topics and the variety of different proofs provided. Even the general reader, who may not follow all of the mathematical explanations, will appreciate the background information on each icon and some of the more basic mathematical ideas that are supported by clear diagrams and easy to follow explanations.

The book would be an excellent supplemental text for a history of mathematics course, a high school or college Euclidean geometry course, or a course on the mathematics of art. It could be used to design an independent study on geometry and visual proof or a problem solving course. Any of the chapters in the book could serve as the basis for a presentation to a math club or as a springboard for a student preparing a paper for an undergraduate research conference. It could be used as the main text for an elective course for students preparing to teach high school mathematics or as a professional development course for middle school and high school mathematics teachers. Teachers of any of these courses should have this book on their shelf.

Cheryl J. McAllister taught high school for 4 years and has been on the faculty of Southeast Missouri State for 20 years. Using what she learned as an undergraduate from her minor in art, she teaches a freshman seminar on “The Mathematics of Art.” You may contact her at:

Twenty Key Icons of Mathematics
1. The Bride's Chair
2. Zhou Bi Suan Jing
3. Garfield's Trapezoid
4. The semicircle
5. Similar Figures
6. Cevians
7. The Right Triangle 8. Napoleon's Triangles
9. Arcs and Angles
10. Polygons with Circles
11. Two Circles
12. Venn Diagrams
13. Overlapping Figures
14. Yin and Yang
15. Polygonal Lines
16. Star Polygons
17. Self-similar Figures
18. Tatami
19. The Rectangular Hyperbola
20. Tiling
Solutions to the Challenges References