A Mathematical Picture Book

1. Polyhedral Models: Platonic Solids.- Duality and Symmetry.- Archimedean Solids.- Johnson and Catalan Solids.- The Geometry of the Soccer Ball.- Special Tetrahedra.- The Altitude Regulator.- The Art of Unfolding.- 2. Geometry in the Plane: The Pythagorean Theorem.- The Nine-Point Circle.- Concentric Circles.- Metric and Projective Scales.- The Fermat Point.- Morley's Theorem.- The Theorem of Fukuta and Cerin.- Maclaurin--Braikenridge Problems.- Derivation of the Addition Theorems.- Inscribed Squares and Equilateral Triangles.- Halving the Surface of a Triangle.- Every Angle Is a Right Angle? 3. Problems New and Old: Trisecting an Angle.- The Delian Cube Duplication Problem.- The Collatz Conjecture.- Dominoes on a Chessboard.- The Ham Sandwich Theorem.- Pick's Theorem.- Goldbach's Conjecture.- The Riemann Zeta Function.- 4. Formulas and the Integers: The Gauss Summation Formula.- Sums of Squares.- Sums of Fractions.- Pascal's Triangle.- Pascal and Fibonacci.- Pascal's Pyramids.- Estimating the Distribution of Prime Numbers.- Ulam's Prime Number Spiral.- How Many Integers Are There?.- Mad Formulas Involving I.- 5. Functions and Limits: Nondifferentiable Functions.- Taylor Series.- Fourier Series and Periodic Waves.- Total versus Partial Differentiability.- The Weierstrass A-Function.- Solitons.- The Volume of the Sphere.- The Brouwer Fixed-Point Theorem.- 6. Curves and Knots: Conic Sections--Defined Planimetrically and Spatially.- Spherical Conic Sections and Confocal Conic Sections.- Dandelin Spheres.- Apollonian Circles.- Cubic Curves.- The Cassini Oval.- The Astroid.- Conchoids.- Geodesic Curves and Straightest Lines.- Zoll Surfaces.- Geodesics on Polyhedra.- Knots.- Celtic Knots.- Borromean Rings.- 7. Geometry and Topology of Surfaces: Hyperboloids and Paraboloids.- Quadrics and Circular Sections.- The Clebsch Surface and Singular Cubics.- Dupin Cyclides.- Supercyclides.- Plucker's Conoid.- Helices and Spirals.- Rotoid Helicoids.- Collar Surfaces and Developable Strips.- The Pseudosphere.- The Kuen Surface.- The Csaszar Torus.- The Mobius Strip.- The Klein Bottle.- Models of the Projective Plane.- Seifert Surfaces.- Alexander's Horned Sphere.- Turning the Sphere Inside Out.- 8. Minimal Surfaces and Soap Bubbles: Minimal Surfaces and Soap Films.- Classical Minimal Surfaces.- The Gergonne Problem.- From Catenoid to Helicoid.- The Catenoid and Its Variations.- Periodic Minimal Surfaces.- Costa's Minimal Surface.- Discrete Minimal Surfaces.- Surfaces from Circle Patterns.- The Wente Surface.- Closed Soap Bubbles.- The Penta Surface.- 9. Tilings and Packings: Frieze Ornaments.- Ornamentation.- Aperiodic Tilings.- Kissing Number.- Space Tilings.- The Weaire--Phelan Foam and Optimal Space Packings.- Planar and Spatial Voronoi Diagrams.- 10. Space Forms and Dimension.- The Hyperbolic Plane.- Escher's Hyperbolic Plane.- Ideal Polyhedra in Hyperbolic Space.- The Shape of Space.- The Four-Dimensional Cube and Its Unfolding.- The Hyperdodecahedron.- 120 Cells and More!.- 11. Graphs and Incidence Geometry: Pascal's Theorem and Its Dual.- Desargues's Theorem.- Tangent Circles.- Escape into Space.- Systems of Curves Define Regions.- The Petersen Graph.- Hamiltonian and Eulerian Circuits.- Venn Diagrams.- Schlegel Diagrams.- Minimal Spanning Trees.- Counting Triangulations.- 12. Movable Forms: Elliptic Motion.- Movable Polyhedra.- Trajectories and Envelopes.- Constrained Spatial Motion.- Degrees of Freedom.- The Rolling Reuleaux Triangle.- The Gomboc.- 13. Fractals: The Pythagoras Tree.- Filling Space and the Plane with a Closed Curve.- Hilbert Curves on the Sphere.- Fractal Dimension.- The Menger Sponge.- Julia Sets and the Mandelbrot Set.- The Feigenbaum Diagram.- The Lorenz Attractor.- Curlicue Fractals.- Random Walks.- Percolation.- 14. Maps and Mappings: Isometric Maps.- Gnomonic Projection.- Inversion and Projection.- The Silhouette of a Sphere.- Mobius Transformations from Motions of the Sphere.- The Riemann Mapping Theorem.- The Schwarz--Christoffel Mapping.- Parameterization of Surfaces.- Space Collineation.- Zeros of Complex Functions.- The Riemann Sphere.- Domain Coloring.- The Szego Curve.- Polynomiography.- Zeros of Polynomials.- 15. Forms and Processes in Nature and Technology: Numbers in Motion.- The Karman Vortex Street.- The Topology of Currents.- Streamlines.- Electric Field Lines.- Smoothing of Three-Dimensional Scanned Data.- Vibrations.- The Traveling Salesman Problem.- Sorting Algorithms.- The DNA Double Helix.- Virtual Maxillary Surgery.- Radiolarians.- Epipolar Geometry.- From Photograph to Spatial Location.- Reflections.- Picture Credits.- Index