In geometry, a pentagonal polytope is a regular polytope in n dimensions constructed from the Hn Coxeter group. The family was named by H. S. M. Coxeter, because the two-dimensional pentagonal polytope is a pentagon. It can be named by its Schläfli symbol as {5, 3n − 2} (dodecahedral) or {3n − 2, 5} (icosahedral).
Family members The family starts as 1-polytopes and ends with n = 5 as infinite tessellations of 4-dimensional hyperbolic space.
There are two types of pentagonal polytopes; they may be termed the dodecahedral and icosahedral types, by their three-dimensional members. The two types are duals of each other.
Dodecahedral The complete family of dodecahedral pentagonal polytopes are: Line segment, { } Pentagon, {5} Dodecahedron, {5, 3} (12 pentagonal faces) 120-cell, {5, 3, 3} (120 dodecahedral cells) Order-3 120-cell honeycomb, {5, 3, 3, 3} (tessellates hyperbolic 4-space (∞ 120-cell facets)
The facets of each dodecahedral pentagonal polytope are the dodecahedral pentagonal polytopes of one less dimension. Their vertex figures are the simplices of one less dimension. + Dodecahedral pentagonal polytopesnCoxeter groupPetrie polygonprojectionNameCoxeter diagramSchläfli symbolFacetsElementsVerticesEdgesFacesCells4-faces1[ ](order 2)Line segment{ }2 vertices22(order 10)Pentagon{5}5 edges553[5,3](order 120)Dodecahedron{5, 3}12 pentagons2030124[5,3,3](order 14400)120-cell{5, 3, 3}120 dodecahedra60012007201205[5,3,3,3](order ∞)120-cell honeycomb{5, 3, 3, 3}∞ 120-cells∞∞∞∞∞
Icosahedral The complete family of icosahedral pentagonal polytopes are: Line segment, { } Pentagon, {5} Icosahedron, {3, 5} (20 triangular faces) 600-cell, {3, 3, 5} (600 tetrahedron cells) Order-5 5-cell honeycomb, {3, 3, 3, 5} (tessellates hyperbolic 4-space (∞ 5-cell facets)
The facets of each icosahedral pentagonal polytope are the simplices of one less dimension. Their vertex figures are icosahedral pentagonal polytopes of one less dimension. + Icosahedral pentagonal polytopesnCoxeter groupPetrie polygonprojectionNameCoxeter diagramSchläfli symbolFacetsElementsVerticesEdgesFacesCells4-faces1[ ](order 2)Line segment{ }2 vertices22(order 10)Pentagon{5}5 Edges553[5,3](order 120)Icosahedron{3, 5}20 equilateral triangles1230204[5,3,3](order 14400)600-cell{3, 3, 5}600 tetrahedra12072012006005[5,3,3,3](order ∞)Order-5 5-cell honeycomb{3, 3, 3, 5}∞ 5-cells∞∞∞∞∞
Related star polytopes and honeycombs The pentagonal polytopes can be stellated to form new star regular polytopes: In two dimensions, we obtain the pentagram {5/2}, In three dimensions, this forms the four Kepler–Poinsot polyhedra, {3,5/2}, {5/2,3}, {5,5/2}, and {5/2,5}. In four dimensions, this forms the ten Schläfli–Hess polychora: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5,3,5/2}, {5/2,3,5}, {5/2,5,5/2}, {5,5/2,3}, {3,5/2,5}, {3,3,5/2}, and {5/2,3,3}. In four-dimensional hyperbolic space there are four regular star-honeycombs: {5/2,5,3,3}, {3,3,5,5/2}, {3,5,5/2,5}, and {5,5/2,5,3}.
In some cases, the star pentagonal polytopes are themselves counted among the pentagonal polytopes.Coxeter, H. S. M.: Regular Polytopes (third edition), p. 107, p. 266
Like other polytopes, regular stars can be combined with their duals to form compounds;
In two dimensions, a decagrammic star figure {10/2} is formed, In three dimensions, we obtain the compound of dodecahedron and icosahedron, In four dimensions, we obtain the compound of 120-cell and 600-cell.
Star polytopes can also be combined.
Notes
References Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, (Paper 10) H.S.M. Coxeter, Star Polytopes and the Schlafli Function f(α,β,γ) [Elemente der Mathematik 44 (2) (1989) 25–36] Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Table I(ii): 16 regular polytopes {p, q, r} in four dimensions, pp. 292–293)