Unformatted text preview: three pairs of opposite edges. Take the three lines joining the centers of the pairs of opposite edges. These three lines are mutually orthogonal; they are the axes of a cartesian coordinate system. There are ﬁve such coordinate systems that are permuted by the rotation group. Finally, given one such coordinate system, we can locate a cube whose faces are parallel to the coordinate planes and whose edges lie on the faces of the dodecahedron. Each edge of the cube is a diagonal of a face of the dodecahedron, and exactly one of the ﬁve diagonals of each face is an edge of the cube. There are ﬁve such cubes that are permuted by the rotation group. See Figure 4.2.5 on the next page . You are asked to show in Exercise 4.2.1 that the action of the rotation group on the set of ﬁve inscribed cubes is faithful; that is, the homomorphism of the rotation group into S 5 is injective....
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 Fall '08
 EVERAGE
 Algebra, Geometry, Euclidean geometry, Polar coordinate system, Coordinate systems

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