A Mathematical Picture Book.pdf

A Mathematical Picture Book.pdf


The consistently excellent graphical images provide good examples of how to construct a clean and elegant argument. May this book seduce many readers into mathematics." c't 17/09One can browse in this book to one's heart's content. For while the individual mathematical tidbits are arranged thematically, they do not depend on what has come previously. Therefore this book-amazingly for a mathematics text-is suitable even for the night table. Deutschlandradio Kultur<

Georg Glaeser is professor of mathematics and geometry at the University of Applied Art in Vienna and the author of numerous books on computational geometry. Konrad Polthier is professor of mathematics at the Free University of Berlin and the DFG Research Center Matheon. Springer has published a number of his books on mathematical visualization as well as entertaining videos on mathematics, including the prizewinning MESH and the collection of films from the MathFilm Festival 2008.

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 pi.- 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

How can one visualize a curve that fills the entire plane or all of space? Can a polyhedron be smoothly turned inside out? What is the projective plane? What does four-dimensional space look like? Can soap bubbles exist that are not spherical? And how can one better understand the structure of vortices and currents? In this book you will experience mathematics from the visual point of view, discovering fascinating and never previously published images that offer illustrative examples to the above questions. Every picture is accompanied by a brief explanatory text, references to further reading, and a number of web links where you can obtain further information. This book is intended for all friends of mathematics - students, teachers, amateurs, and professionals - who want to see something beyond dry text and endless formulas. It will provide inspiration for pursuing further one or another topic that may previously have seemed inaccessible. You will get to know mathematics from a totally new and colorful viewpoint.


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