Euler's Gem by David S. Richeson download in ePub, pdf, iPad
Much later, Kepler constructed a magnificent but incorrect model of the solar system using the Platonic solids. Thus, a curved surface like the globe is approximated by a polyhedron with flat faces.
These chapters focus on the most famous class of polyhedra, the regular polyhedra. Advanced students will also glimpse the broad horizons of mathematics by reading and working through the book. Thus, with six squares we can make a cube like a dice. Recall that a regular polygon is a plane figure with all sides and angles equal, for example a square. This book will be valuable to every library with patrons looking for an awe-inspiring experience.
You will not be disappointed. If a particularly thorny argument is too difficult to grasp, jump to the next topic. It should aid scholars who would like to dig deeper into the subject matter. Plato used the regular polyhedra in an early atomic model and we still speak of the Platonic solids.
The examples, proofs, and historical anecdotes are interesting, informative, and useful for encouraging classroom discussions. Hexagonal meshes on a sphere have other serious applications.
While this book is accessible to a broad audience, I also wrote it for mathematicians. The fifth and final regular polyhedron is the dodecahedron, with twelve pentagonal faces. He is enthusiastic and humorous.
Although parts of this book overlap with other books, there is no single resource that contains all this information. In geometry, lengths and angles are fixed. Richeson is keenly aware of where the difficult twists and turns are located, and he covers them to satisfaction. Cauchy provided an alternative proof, using flattened polyhedra, and Lhuiler and others extended the result to some non-Eulerian polyhedra. One brand of golf ball has polygonal dimples, all hexagons but for twelve pentagons.
Well within the reach of the intelligent layperson, it is also a good book to use as a resource for a course where the students are required to make presentations. We can move shapes around but no distortion is permitted.
Four equilateral triangles make a tetrahedron, eight make an octahedron and twenty make an icosahedron. The book is not for everyone, but those who would not understand and appreciate the mathematics are precisely those who would never pick it up in the first place.