Pentagonal polytope

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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).

Contents

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:

  1. Line segment, { }
  2. Pentagon, {5}
  3. Dodecahedron, {5, 3} (12 pentagonal faces)
  4. 120-cell, {5, 3, 3} (120 dodecahedral cells)
  5. 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 polytopes
n Coxeter group Petrie polygon
projection
Name
Coxeter diagram
Schläfli symbol
Facets Elements
Vertices Edges Faces Cells 4-faces
1
[ ]
(order 2)
Cross graph 1.svg Line segment
CDel node 1.png
{ }
2 vertices 2
2
[5]
(order 10)
Regular polygon 5.svg Pentagon
CDel node 1.pngCDel 5.pngCDel node.png
{5}
5 edges 55
3
[5,3]
(order 120)
Dodecahedron H3 projection.svg Dodecahedron
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
{5, 3}
12 pentagons
Regular polygon 5.svg
203012
4
[5,3,3]
(order 14400)
120-cell graph H4.svg 120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{5, 3, 3}
120 dodecahedra
Dodecahedron H3 projection.svg
6001200720120
5
[5,3,3,3]
(order ∞)
120-cell honeycomb
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{5, 3, 3, 3}
120-cells
120-cell graph H4.svg

Icosahedral

The complete family of icosahedral pentagonal polytopes are:

  1. Line segment, { }
  2. Pentagon, {5}
  3. Icosahedron, {3, 5} (20 triangular faces)
  4. 600-cell, {3, 3, 5} (600 tetrahedron cells)
  5. 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 polytopes
n Coxeter group Petrie polygon
projection
Name
Coxeter diagram
Schläfli symbol
Facets Elements
Vertices Edges Faces Cells 4-faces
1
[ ]
(order 2)
Cross graph 1.svg Line segment
CDel node 1.png
{ }
2 vertices 2
2
[5]
(order 10)
Regular polygon 5.svg Pentagon
CDel node 1.pngCDel 5.pngCDel node.png
{5}
5 Edges 55
3
[5,3]
(order 120)
Icosahedron H3 projection.svg Icosahedron
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{3, 5}
20 equilateral triangles
Regular polygon 3.svg
123020
4
[5,3,3]
(order 14400)
600-cell graph H4.svg 600-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{3, 3, 5}
600 tetrahedra
3-simplex t0.svg
1207201200600
5
[5,3,3,3]
(order ∞)
Order-5 5-cell honeycomb
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{3, 3, 3, 5}
5-cells
4-simplex t0.svg

The pentagonal polytopes can be stellated to form new star regular polytopes:

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;

Star polytopes can also be combined.

Notes

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

Related Research Articles

Regular icosahedron Platonic solid

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 (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

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.

Schläfli symbol Notation that defines regular polytopes and tessellations

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

Regular polytope

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.

120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.

Order-4 dodecahedral honeycomb

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.

Order-5 dodecahedral 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.

Order-5 cubic honeycomb

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.

Icosahedral 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.

Grand 600-cell

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.

Icosahedral 120-cell

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.

Regular 4-polytope

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.

Order-6 dodecahedral honeycomb Regular geometrical object in hyperbolic space

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.

Order-5 hexagonal tiling honeycomb

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.

Dodecahedral-icosahedral honeycomb

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.

References

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds