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? How can one better understand the structure of vortices and currents? In this book you will experience mathematics from the visual point of view, where fascinating images, many of them appearing in print for the first time, provide answers to the above questions. Every picture is accompanied by a brief explanatory text along with numerous references and web links for further reading. 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 further pursuit of topics in mathematics that may previously have seemed inaccessible. You will get to know mathematics from a totally new and colorful viewpoint. Enjoy the beauty and fascination of mathematics in over 300 pages richly illustrated with more than 1000 figures on the following topics: Polyhedral Models Geometry in the Plane Problems New and Old Formulas and Numbers Functions and Limits Curves and Knots Geometry and Topology of Surfaces Minimal Surfaces and Soap Bubbles Tilings and Packings Space Forms and Dimension Graphs and Incidence Geometry Flexible Shapes Fractals Maps and Mappings Forms and Processes in Nature and Technology

Hersteller:
Springer Verlag GmbH
juergen.hartmann@springer.com
Tiergartenstr. 17
DE 69121 Heidelberg

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 Ä¿erin.- 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 Ï¿.- 5. Functions and Limits: Nondifferentiable Functions.- Taylor Series.- Fourier Series and Periodic Waves.- Total versus Partial Differentiability.- The Weierstrass ÿ-Function.- Solitons.- The Volume of the Sphere.- The Brouwer Fixed-Point Theorem.- 6. Curves and Knots: Conic Sectionsa¿'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.- Plücker¿s Conoid.- Helices and Spirals.- Rotoid Helicoids.- Collar Surfaces and Developable Strips.- The Pseudosphere.- The Kuen Surface.- The Csaszar Torus.- The Möbius 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 Gömböc.- 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.- Möbius Transformations from Motions of the Sphere.- The Riemann Mappin ...