3-manifolds from Platonic solids

Abstract

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].

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.…”

“…In [14,15] the examination was extended also to the other homogeneous 3-spaces. The octahedron manifolds were discovered by J. M. Montesinos [8], B. Everitt [16] and M. Šarać, while the 3 icosahedron manifolds were enumerated by L. A. Best [9].…”

“…In the last line of Table 2 we can see the results for the hyperbolic dodecahedron tiling {5, 3, 6} in which the vertices are ideal points. Among them the 10 face identifications determining orientable non-compact manifolds were discovered independently by B. Everitt [16] (clearly with algebraic tools) and by the author [19], while the 68 face identifications yielding non-orientable non-compact hyperbolic manifolds were new results in [19].…”

Platonic polyhedra tune the three-sphere: II. Harmonic analysis on cubic spherical three-manifolds

“…Among the Platonic polyhedra it was applied in [9], [10] to the Poincare dodecahedron with H the binary icosahedral group. An algorithm due to Everitt in [7] describes homotopies for spherical 3-manifolds from five Platonic polyhedra. Following it we found and applied in [11] for the tetrahedron as H the cyclic group C 5 .…”

“…For the group action we start from the Coxeter group G < O(4, R) [8], [7] p. 254, with the diagram Fig. 1.…”