Alsina C., Nelsen R.B. Icons of Mathematics: An Exploration of Twenty Key Images

New York: Mathematical Association of America, 2011. — 347 p. — (Dolciani Mathematical Expositions). — ISBN: 9780883853528.The authors present twenty icons of mathematics, that is, geometrical shapes such as the right triangle, the Venn diagram, and the yang and yin symbol and explore mathematical results associated with them. As with their previous books (Charming Proofs, When Less is More, Math Made Visual) proofs are visual whenever possible.
The results require no more than high-school mathematics to appreciate and many of them will be new even to experienced readers. Besides theorems and proofs, the book contains many illustrations and it gives connections of the icons to the world outside of mathematics. There are also problems at the end of each chapter, with solutions provided in an appendix.
The book could be used by students in courses in problem solving, mathematical reasoning, or mathematics for the liberal arts. It could also be read with pleasure by professional mathematicians, as it was by the members of the Dolciani editorial board, who unanimously recommend its publication.Twenty Key Icons of Mathematics
The Bride’s Chair
The Pythagorean theorem—Euclid’s proof and more
The Vecten configuration
The law of cosines
Grebe’s theorem and van Lamoen’s extension
Pythagoras and Vecten in recreational mathematics
Challenges
Zhou Bi Suan Jing
The Pythagorean theorem—a proof from ancient China
Two classical inequalities
Two trigonometric formulas
Challenges
Garfield’s Trapezoid
The Pythagorean theorem—the Presidential proof
Inequalities and Garfield’s trapezoid
Trigonometric formulas and identities
Challenges
The Semicircle
Thales’ triangle theorem
The right triangle altitude theorem and the geometric mean
Queen Dido’s semicircle
The semicircles of Archimedes
Pappus and the harmonic mean
More trigonometric identities
Areas and perimeters of regular polygons
Euclid’s construction of the five Platonic solids
Challenges
Similar Figures
Thales’ proportionality theorem
Menelaus’s theorem
Reptiles
Homothetic functions
Challenges
Cevians
The theorems of Ceva and Stewart
Medians and the centroid
Altitudes and the orthocenter
Angle-bisectors and the incenter
Circumcircle and circumcenter
Non-concurrent cevians
Ceva’s theorem for circles
Challenges
The Right Triangle
Right triangles and inequalities
The incircle, circumcircle, and excircles
Right triangle cevians
A characterization of Pythagorean triples
Some trigonometric identities and inequalities
Challenges
Napoleon’s Triangles
Napoleon’s theorem
Fermat’s triangle problem
Area relationships among Napoleon’s triangles
Escher’s theorem
Challenges
Arcs and Angles
Angles and angle measurement
Angles intersecting circles
The power of a point
Euler’s triangle theorem
The Taylor circle
The Monge circle of an ellipse
Challenges
Polygons with Circles
Cyclic quadrilaterals
Sangaku and Carnot’s theorem
Tangential and bicentric quadrilaterals
Fuss’s theorem
The butterfly theorem
Challenges
Two Circles
The eyeball theorem
Generating the conics with circles
Common chords
Vesica piscis
The vesica piscis and the golden ratio
Lunes
The crescent puzzle
Mrs. Miniver’s problem
Concentric circles
Challenges
Venn Diagrams
Three-circle theorems
Triangles and intersecting circles
Reuleaux polygons
Challenges
Overlapping Figures
The carpets theorem
The irrationality of √2 and √3
Another characterization of Pythagorean triples
Inequalities between means
Chebyshev’s inequality
Sums of cubes
Challenges
Yin and Yang
The great monad
Combinatorial Yin and Yang
Integration via the symmetry of Yin and Yang
Recreational Yin and Yang
Challenges
Polygonal Lines
Lines and line segments
Polygonal numbers
Polygonal lines in calculus
Convex polygons
Polygonal cycloids
Polygonal cardioids
Challenges
Star Polygons
The geometry of star polygons
The pentagram
The star of David
The star of Lakshmi and the octagram
Star polygons in recreational mathematics
Challenges
Self-similar Figures
Geometric series
Growing figures iteratively
Folding paper in half twelve times
The spira mirabilis
The Menger sponge and the Sierpinski carpet
Challenges
Tatami
The Pythagorean theorem—Bhaskara’s proof
Tatami mats and Fibonacci numbers
Tatami mats and representations of squares
Tatami inequalities
Generalized tatami mats
Challenges
The Rectangular Hyperbola
One curve, many definitions
The rectangular hyperbola and its tangent lines
Inequalities for natural logarithms
The hyperbolic sine and cosine
The series of reciprocals of triangular numbers
Challenges
Tiling
Lattice multiplication
Tiling as a proof technique
Tiling a rectangle with rectangles
The Pythagorean theorem—infinitely many proofs
Challenges
Solutions to the ChallengesAbout the Authors