Manifolds of Spherical Tessellations
The spherical tessellations are represented by the platonic solids or regular polyhedra. Thus the set of positions of a platonic solid inscribed in the 2-sphere S2 is a closed 3-manifold. The fundamental group of such a manifold is an extension of ℤ2 by the group of isometries of the platonic solid. The subgroup ℤ2 is the center of the group. The manifold can be thought of as the spherical tangent bundle of a 2-dimensional spherical orbifold, or as a 3-dimensional spherical orbifold, i.e. the quotient of S3 under a finite subgroup of SO(4). Some material of this chapter was borrowed in part from [DuV].
KeywordsFundamental Domain Stereographic Projection Lens Space Solid Torus Finite Subgroup
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