Mathematics for the Liberal Arts by Erickson Martin J. Bindner Donald Hemmeter Joe & Martin J. Erickson & Joe Hemmeter

Mathematics for the Liberal Arts by Erickson Martin J. Bindner Donald Hemmeter Joe & Martin J. Erickson & Joe Hemmeter

Author:Erickson, Martin J., Bindner, Donald, Hemmeter, Joe & Martin J. Erickson & Joe Hemmeter
Language: eng
Format: epub
Publisher: Wiley
Published: 2014-08-13T16:00:00+00:00


Groups and Geometry

Recall that a group is a set with an operation satisfying certain properties. Groups can describe the symmetries of an object.

Here is a small example. Consider the equilateral triangle in Figure 3.13, whose vertices are numbered 1, 2, and 3.

An equilateral triangle can be rotated or flipped over and occupy the same space. There are six symmetries, including all rotations and flipping over the triangle. The group consists of these six symmetries, where the operation is performing one symmetry after another (this is called composition). The group contains a smaller group of three symmetries, namely, the three rotations. (One of the rotations is the identity element, which leaves the triangle unmoved.) The smaller group is a subgroup of the larger group.

The five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) have symmetry groups. Let’s consider the dodecahedron (Figure 3.14). It has twelve faces which are regular pentagons. The dodecahedron can be picked up and set down on any of its twelve faces. Then it can be rotated into any one of five positions. After these maneuvers, the dodecahedron occupies its original space. Since there are twelve choices for which face goes on the bottom, and five choices for the rotation, there are altogether 12 · 5 = 60 symmetries of the dodecahedron. This 60-element group is “simple,” meaning that it cannot be broken down into smaller symmetry groups.

Figure 3.14 The dodecahedron has 60 symmetries.



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