Titlebook: Symmetry; A Mathematical Explo Kristopher Tapp Textbook 2021Latest edition Springer Nature Switzerland AG 2021 Cayley table.Euclidean space

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The Five Platonic Solids,cinated scientists and philosophers for thousands of years. We study their symmetry groups and their duality relationships. Counting their parts leads to Euler’s formula and the Euler characteristic, which is the starting point of an entire field of mathematics called ..

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The Algebra of Symmetry,ly enough! In this chapter, you’ll build the 8-by-8 table showing how to combine any pair of them. The resulting “group” viewpoint is a profound paradigm shift that’s crucial for understanding all remaining chapters, and that’s used in this chapter to prove theorems like this one: the number of impr

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The Classification Theorems,ry of their world using ever more precise language and methods. The classification theorems in this chapter represent some of the pinnacles of this journey. These theorems classify the ways in which (1) bounded objects, (2) border patterns, and (3) wallpaper patterns can be symmetric. The final “Ele

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Subgroups and Product Groups,o better understand symmetry groups. If we can recognize an object’s symmetry group as having been built out of smaller groups, then this realization might help us to much more clearly understand its underlying algebraic structure.

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Symmetries of 3D Objects,a classification of the ways in which bounded 3D objects can be symmetric. No matter what object I print on my 3D printer or sculpt from clay, you soon will be able to determine how it fits into the classification scheme.

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The Five Platonic Solids, we prove there are exactly five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. These five perfect shapes have fascinated scientists and philosophers for thousands of years. We study their symmetry groups and their duality relationships. Counting their parts leads