cover -- copyright page -- title page -- Introduction -- Contents -- Part I Visualizing Mathematics by Creating Pictures -- 1 Representing Numbers by Graphical Elements -- 1.1 Sums of odd integers -- 1.2 Sums of integers -- 1.3 Alternating sums of squares -- 1.4 Challenges -- 2 Representing Numbers by Lengths of Segments -- 2.1 Inequalities among means -- 2.2 The mediant property -- 2.3 A Pythagorean inequality -- 2.4 Trigonometric functions -- 2.5 Numbers as function values -- 2.6 Challenges -- 3 Representing Numbers by Areas of Plane Figures -- 3.1 Sums of integers revisited -- 3.2 The sum of terms in arithmetic progression -- 3.3 Fibonacci numbers -- 3.4 Some inequalities -- 3.5 Sums of squares -- 3.6 Sums of cubes -- 3.7 Challenges -- 4 Representing Numbers by Volumes of Objects -- 4.1 From two dimensions to three -- 4.2 Sums of squares of integers revisited -- 4.3 Sums of triangular numbers -- 4.4 A double sum -- 4.5 Challenges -- 5 Identifying Key Elements -- 5.1 On the angle bisectors of a convex quadrilateral -- 5.2 Cyclic quadrilaterals with perpendicular diagonals -- 5.3 A property of the rectangular hyperbola -- 5.4 Challenges -- 6 Employing Isometry -- 6.1 The Chou pei suan ching proof of the Pythagorean theorem -- 6.2 A theorem of Thales -- 6.3 Leonardo da Vinci's proof of the Pythagorean theorem -- 6.4 The Fermat point of a triangle -- 6.5 Viviani's theorem -- 6.6 Challenges -- 7 Employing Similarity -- 7.1 Ptolemy's theorem -- 7.2 The golden ratio in the regular pentagon -- 7.3 The Pythagorean theorem-again -- 7.4 Area between sides and cevians of a triangle -- 7.5 Challenges -- 8 Area-preserving Transformations -- 8.1 Pappus and Pythagoras -- 8.2 Squaring polygons -- 8.3 Equal areas in a partition of a parallelogram -- 8.4 The Cauchy-Schwarz inequality -- 8.5 A theorem of Gaspard Monge -- 8.6 Challenges.

9 Escaping from the Plane -- 9.1 Three circles and six tangents -- 9.2 Fair division of a cake -- 9.3 Inscribing the regular heptagon in a circle -- 9.4 The spider and the fly -- 9.5 Challenges -- 10 Overlaying Tiles -- 10.1 Pythagorean tilings -- 10.2 Cartesian tilings -- 10.3 Quadrilateral tilings -- 10.4 Triangular tilings -- 10.5 Tiling with squares and parallelograms -- 10.6 Challenges -- 11 Playing with Several Copies -- 11.1 From Pythagoras to trigonometry -- 11.2 Sums of odd integers revisited -- 11.3 Sums of squares again -- 11.4 The volume of a square pyramid -- 11.5 Challenges -- 12 Sequential Frames -- 12.1 The parallelogram law -- 12.2 An unknown angle -- 12.3 Determinants -- 12.4 Challenges -- 13 Geometric Dissections -- 13.1 Cutting with ingenuity -- 13.2 The "Smart Alec" puzzle -- 13.3 The area of a regular dodecagon -- 13.4 Challenges -- 14 Moving Frames -- 14.1 Functional composition -- 14.2 The Lipschitz condition -- 14.3 Uniform continuity -- 14.4 Challenges -- 15 Iterative Procedures -- 15.1 Geometric series -- 15.2 Growing a figure iteratively -- 15.3 A curve without tangents -- 15.4 Challenges -- 16 Introducing Colors -- 16.1 Domino tilings -- 16.2 L-Tetromino tilings -- 16.3 Alternating sums of triangular numbers -- 16.4 In space, four colors are not enough -- 16.5 Challenges -- 17 Visualization by Inclusion -- 17.1 The genuine triangle inequality -- 17.2 The mean of the squares exceeds the square of the mean -- 17.3 The arithmetic mean-geometric mean inequality for three numbers -- 17.4 Challenges -- 18 Ingenuity in 3D -- 18.1 From 3D with love -- 18.2 Folding and cutting paper -- 18.3 Unfolding polyhedra -- 18.4 Challenges -- 19 Using 3D Models -- 19.1 Platonic secrets -- The dodecahedron's secret: a cube with six roofs -- The icosahedron's secret: three golden rectangles inside.

The octahedron's secret: a structure for a table -- The cube's secret: a union of triangular pyramids -- The tetrahedron's secret: a better kite -- Polyhedral dice -- 19.2 The rhombic dodecahedron -- 19.3 The Fermat point again -- 19.4 Challenges -- 20 Combining Techniques -- 20.1 Heron's formula -- 20.2 The quadrilateral law -- 20.3 Ptolemy's inequality -- 20.4 Another minimal path -- 20.5 Slicing cubes -- 20.6 Vertices, faces, and polyhedra -- 20.7 Challenges -- Part II Visualization in the Classroom -- Mathematical drawings: a short historical perspect ive -- On visual thinking -- Visualization in the classroom -- On the role of hands-on materials -- Everyday life objects as resources -- Polyhedra -- Polygons -- Curves -- Quadric surfaces -- Making models of polyhedra -- Using soap bubbles -- Lighting results -- Mirror images -- Towards creativity -- Part III Hints and Solutions to the Challenges -- Chapter 1 -- Chapter 2 -- Chapter 3 -- Chapter 4 -- Chapter 5 -- Chapter 6 -- Chapter 7 -- Chapter 8 -- Chapter 9 -- Chapter 10 -- Chapter 11 -- Chapter 12 -- Chapter 13 -- Chapter 14 -- Chapter 15 -- Chapter 16 -- Chapter 17 -- Chapter 18 -- Chapter 19 -- Chapter 20 -- References -- Index -- About the Authors.