In geometry, a **pentagonal polytope** is a regular polytope in *n* dimensions constructed from the H_{n} 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, 3^{n− 2}} (dodecahedral) or {3^{n− 2}, 5} (icosahedral).

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.

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.

n | Coxeter group | Petrie polygon projection | Name Coxeter diagram Schläfli symbol | Facets | Elements | ||||
---|---|---|---|---|---|---|---|---|---|

Vertices | Edges | Faces | Cells | 4-faces | |||||

1 | [ ] (order 2) | Line segment { } | 2 vertices | 2 | |||||

2 | [5] (order 10) | Pentagon {5} | 5 edges | 5 | 5 | ||||

3 | [5,3] (order 120) | Dodecahedron {5, 3} | 12 pentagons | 20 | 30 | 12 | |||

4 | [5,3,3] (order 14400) | 120-cell {5, 3, 3} | 120 dodecahedra | 600 | 1200 | 720 | 120 | ||

5 | [5,3,3,3] (order ∞) | 120-cell honeycomb {5, 3, 3, 3} | ∞ 120-cells | ∞ | ∞ | ∞ | ∞ | ∞ |

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.

n | Coxeter group | Petrie polygon projection | Name Coxeter diagram Schläfli symbol | Facets | Elements | ||||
---|---|---|---|---|---|---|---|---|---|

Vertices | Edges | Faces | Cells | 4-faces | |||||

1 | [ ] (order 2) | Line segment { } | 2 vertices | 2 | |||||

2 | [5] (order 10) | Pentagon {5} | 5 Edges | 5 | 5 | ||||

3 | [5,3] (order 120) | Icosahedron {3, 5} | 20 equilateral triangles | 12 | 30 | 20 | |||

4 | [5,3,3] (order 14400) | 600-cell {3, 3, 5} | 600 tetrahedra | 120 | 720 | 1200 | 600 | ||

5 | [5,3,3,3] (order ∞) | Order-5 5-cell honeycomb {3, 3, 3, 5} | ∞ 5-cells | ∞ | ∞ | ∞ | ∞ | ∞ |

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.^{ [1] }

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.

- ↑ Coxeter, H. S. M.: Regular Polytopes (third edition), p. 107, p. 266

In geometry, a **regular icosahedron** is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.

In elementary geometry, a **polytope** is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions *n* as an *n*-dimensional polytope or ** n-polytope**. Flat sides mean that the sides of a (

A **regular polyhedron** is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

In geometry, the **Schläfli symbol** is a notation of the form {*p*,*q*,*r*,...} that defines regular polytopes and tessellations.

In mathematics, a **regular polytope** is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or *j*-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ *n*.

In geometry, the **120-cell** is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is also called a **C _{120}**,

In the geometry of hyperbolic 3-space, the **order-4 dodecahedral honeycomb** is one of four compact regular space-filling tessellations. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.

The **order-5 dodecahedral honeycomb** is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.

The **order-5 cubic honeycomb** is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

The **icosahedral honeycomb** is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

In geometry, the **grand 600-cell** or **grand polytetrahedron** is a regular star 4-polytope with Schläfli symbol {3,3,5/2}. It is one of 10 regular Schläfli-Hess polytopes. It is the only one with 600 cells.

In geometry, the **icosahedral 120-cell**, **polyicosahedron**, **faceted 600-cell** or **icosaplex** is a regular star 4-polytope with Schläfli symbol {3,5,5/2}. It is one of 10 regular Schläfli-Hess polytopes.

In geometry, a **quasiregular polyhedron** is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In mathematics, a **regular 4-polytope** is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

The **order-6 dodecahedral honeycomb** is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is *paracompact* because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

In the field of hyperbolic geometry, the **order-5 hexagonal tiling honeycomb** arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is *paracompact* because it has cells composed of an infinite number of faces. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere, a flat plane in hyperbolic space that approaches a single ideal point at infinity.

In the geometry of hyperbolic 3-space, the **dodecahedral-icosahedral honeycomb** is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron vertex figure.

**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, ISBN 978-0-471-01003-6- (Paper 10) H.S.M. Coxeter,
*Star Polytopes and the Schlafli Function f(α,β,γ)*[Elemente der Mathematik 44 (2) (1989) 25–36]

- (Paper 10) H.S.M. Coxeter,
- Coxeter,
*Regular Polytopes*, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Table I(ii): 16 regular polytopes {p, q,r} in four dimensions, pp. 292–293)

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