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Johnson D.L. Symmetries

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Johnson D.L. Symmetries
2nd printing. — Springer, 2003. — 209 p. — (SUMS) —ISBN 1852332700.
In many of the physical sciences a fundamental role is played by the concept of length: units of length are used to measure the distance between two points. In mathematics the idea of distance, as a function that assigns a real number to a given pair of points in some space, is formalised in terms of a few reasonablelooking properties, or axioms, and the result is called a metric on that space.
Having defined a structure such as this on a set, it is natural to study those transformations, or maps, of such sets which preserve that structure. The requirement that these maps be invertible then leads naturally into the theory of groups.
Many types of groups arise in this way. Important examples are permutation groups, linear groups, Galois groups and symmetry groups. The story of the last of these begins as follows.
Contents
Metric Spaces and their Groups
Isometries of the Plane
Some Basic Group Theory
Products of Reflections
Generators and Relations
Discrete Subgroups of the Euclidean Group
Plane Crystallographic Groups: OP Case
Plane Crystallographic Groups: OR Case
Tessellations of the Plane
Tessellations of the Sphere
Triangle Groups
Regular Polytopes
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