Four dodecahedral spaces

On Maximal Homogeneous 3-Geometries—A Polyhedron Algorithm for Space Tilings

“…Best [9]. The spherical dodecahedron spaces were discovered by H. Poincaré, P. J. Lorimer [17] and B. Everitt [16]. Table 2.…”

3-manifolds from Platonic solids

“…In each of the Tables, the column headed N indexes the manifolds M i carrying the indicated geometric structure. The columns F I and EI give the face and edge identifications in the form of an encoded string of letters and ± signs to be read in conjunction with where e indicates that edges 9,11,17,20,26 and 29 are identified, and e(+-+--) means edge 9 is identified with edge 11 so that the identifications match, with edge 17 so they are reversed, with edge 20 so they match, and so on. From the data in these two columns one may reconstruct the side pairing transformations {γ S } s∈Σ .…”

“…1 abcdefefbcda a(-+)b(-+)c(-+)d(-+)e(-+)f(-+)g(-+) 00000 h(-+)i(-+)j(-+)idjefagbhcghijfeabcd 2 abcdefbdcfea a(-+)b(-+)c(-+)d(-+)e(-+)f(++)g(++) (15)0000 h(++)i(++)j(++)ajcgbfeidhfhgjieabcd TABLE 1. The spherical manifolds arising from a dodecahedron with dihedral angle 2π/3, [11].…”

“…In this paper we formulate the problem in terms of classifying certain subgroups of rank four Coxeter groups. This approach is implicit in [11,17], and this paper should be viewed as completing their work in the spherical and hyperbolic cases. It follows an earlier, oft quoted but flawed attempt in [2].…”

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The problem of classifying, upto isometry (or similarity), the orientable spherical, Euclidean and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. In the spherical and hyperbolic cases, this allows us to complete the classification begun by Lorimer [11], Richardson and Rubinstein [17] and Best [2].

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