Euler's Gem : The Polyhedron Formula and the Birth of Topology

Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. __Euler's Gem__ tells the illuminating story of this indispensable mathematical idea. From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of __V__ vertices, __E__ edges, and __F__ faces satisfies the equation __V__-__E__+__F__=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, __Euler's Gem__ will fascinate every mathematics enthusiast. Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. Euler's Gem tells the illuminating story of this indispensable mathematical idea. From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V - E + F =2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast. Leonhard Euler's Polyhedron Formula Describes The Structure Of Many Objects--from Soccer Balls And Gemstones To Buckminster Fuller's Buildings And Giant All-carbon Molecules. Yet Euler's Formula Is So Simple It Can Be Explained To A Child. Euler's Gem Tells The Illuminating Story Of This Indispensable Mathematical Idea --front Jacket Flap. Leonhard Euler And His Three Great Friends -- What Is A Polyhedron? -- The Five Perfect Bodies -- The Pythagorean Brotherhood And Plato's Atomic Theory -- Euclid And His Elements -- Kepler's Polyhedral Universe -- Euler's Gem -- Platonic Solids, Gold Balls, Fullerenes, And Geodesic Domes -- Scooped By Descartes? -- Legendre Gets It Right -- A Stroll Through Königsberg -- Cauchy's Flattened Polyhedra -- Planar Graphs, Geoboards, And Brussels Sprouts -- It's A Colorful World -- New Problems And New Proofs -- Rubber Sheets, Hollow Doughnuts, And Crazy Bottles -- Are They The Same, Or Are They Different? -- A Knotty Problem -- Combing The Hair On A Coconut -- When Topology Controls Geometry -- The Topology Of Curvy Surfaces -- Navigating In N Dimensions -- Henri Poincaré And The Ascendance Of Topology -- The Million-dollar Question. David S. Richeson. Includes Bibliographical References (p. [295]-308) And Index. FM.pdf......Page 2 ack.pdf......Page 286 Introduction.pdf......Page 16 chapter1.pdf......Page 25 chapter2.pdf......Page 42 chapter3.pdf......Page 46 chapter4.pdf......Page 51 chapter5.pdf......Page 59 chapter6.pdf......Page 66 chapter7.pdf......Page 78 chapter8.pdf......Page 90 chapter9.pdf......Page 96 chapter10.pdf......Page 102 chapter11.pdf......Page 115 chapter12.pdf......Page 127 chapter13.pdf......Page 134 chapter14.pdf......Page 145 chapter15.pdf......Page 160 chapter16.pdf......Page 171 chapter17.pdf......Page 188 chapter18.pdf......Page 201 chapter19.pdf......Page 217 chapter20.pdf......Page 234 chapter21.pdf......Page 246 chapter22.pdf......Page 256 chapter23.pdf......Page 268 epilogue.pdf......Page 280 AppendixA.pdf......Page 288 AppendixB.pdf......Page 298 endnote.pdf......Page 302 references.pdf......Page 310 index.pdf......Page 324 Leonhard Euler's polyhedron formula describes the structure of many objects - from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. This work celebrates the discovery of Euler's beloved polyhedron formula and impact on topology, the study of shapes.