pentagonal polytope

{{Short description|Regular polytope whose 2D form is a pentagon}}

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).

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.

class="wikitable" \|+ Dodecahedral pentagonal polytopes
rowspan=2\|n !rowspan=2\|Coxeter group !rowspan=2\|Petrie polygon projection !rowspan=2\|Name Coxeter diagram Schläfli symbol !rowspan=2\|Facets !colspan=5\|Elements
Vertices !Edges !Faces !Cells !4-faces
align=center \|1 \| $H_1$ [ ] (order 2) \|80px \|Line segment {{CDD\|node_1}} { } \|2 vertices \|2 \| \| \| \|
align=center \|2 \| $H_2$ [5] (order 10) \|80px \|Pentagon {{CDD\|node_1\|5\|node}} {5} \|5 edges \|5 \|5 \| \| \|
align=center \|3 \| $H_3$ [5,3] (order 120) \|80px \|Dodecahedron {{CDD\|node_1\|5\|node\|3\|node}} {5, 3} \|12 pentagons 80px \|20 \|30 \|12 \| \|
align=center \|4 \| $H_4$ [5,3,3] (order 14400) \|80px \|120-cell {{CDD\|node_1\|5\|node\|3\|node\|3\|node}} {5, 3, 3} \|120 dodecahedra 80px \|600 \|1200 \|720 \|120 \|
align=center \|5 \| ${\bar{H}}_4$ [5,3,3,3] (order ∞) \| \|120-cell honeycomb {{CDD\|node_1\|5\|node\|3\|node\|3\|node\|3\|node}} {5, 3, 3, 3} \|∞ 120-cells 80px \|∞ \|∞ \|∞ \|∞ \|∞

class="wikitable"

Dodecahedral pentagonal polytopes

rowspan=2|n

!rowspan=2|Coxeter group

!rowspan=2|Petrie polygon
projection

!rowspan=2|Name
Coxeter diagram
Schläfli symbol

!rowspan=2|Facets

!colspan=5|Elements

!4-faces

align=center

| $H_1$
[ ]
(order 2)

|80px

|Line segment
{{CDD|node_1}}
{ }

|2 vertices

align=center

| $H_2$
[5]
(order 10)

|80px

|Pentagon
{{CDD|node_1|5|node}}
{5}

|5 edges

align=center

| $H_3$
[5,3]
(order 120)

|80px

|Dodecahedron
{{CDD|node_1|5|node|3|node}}
{5, 3}

|12 pentagons
80px

|20

|30

|12

align=center

| $H_4$
[5,3,3]
(order 14400)

|80px

|120-cell
{{CDD|node_1|5|node|3|node|3|node}}
{5, 3, 3}

|120 dodecahedra
80px

|600

|1200

|720

|120

align=center

| ${\bar{H}}_4$
[5,3,3,3]
(order ∞)

|120-cell honeycomb
{{CDD|node_1|5|node|3|node|3|node|3|node}}
{5, 3, 3, 3}

|∞ 120-cells
80px

|∞

=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.

class="wikitable" \|+ Icosahedral pentagonal polytopes
rowspan=2\|n !rowspan=2\|Coxeter group !rowspan=2\|Petrie polygon projection !rowspan=2\|Name Coxeter diagram Schläfli symbol !rowspan=2\|Facets !colspan=5\|Elements
Vertices !Edges !Faces !Cells !4-faces
align=center \|1 \| $H_1$ [ ] (order 2) \|80px \|Line segment {{CDD\|node_1}} { } \|2 vertices \|2 \| \| \| \|
align=center \|2 \| $H_2$ [5] (order 10) \|80px \|Pentagon {{CDD\|node_1\|5\|node}} {5} \|5 Edges \|5 \|5 \| \| \|
align=center \|3 \| $H_3$ [5,3] (order 120) \|80px \|Icosahedron {{CDD\|node_1\|3\|node\|5\|node}} {3, 5} \|20 equilateral triangles 80px \|12 \|30 \|20 \| \|
align=center \|4 \| $H_4$ [5,3,3] (order 14400) \|80px \|600-cell {{CDD\|node_1\|3\|node\|3\|node\|5\|node}} {3, 3, 5} \|600 tetrahedra 80px \|120 \|720 \|1200 \|600 \|
align=center \|5 \| ${\bar{H}}_4$ [5,3,3,3] (order ∞) \| \|Order-5 5-cell honeycomb {{CDD\|node_1\|3\|node\|3\|node\|3\|node\|5\|node}} {3, 3, 3, 5} \|∞ 5-cells 80px \|∞ \|∞ \|∞ \|∞ \|∞

class="wikitable"

Icosahedral pentagonal polytopes

rowspan=2|n

!rowspan=2|Coxeter group

!rowspan=2|Petrie polygon
projection

!rowspan=2|Name
Coxeter diagram
Schläfli symbol

!rowspan=2|Facets

!colspan=5|Elements

!4-faces

align=center

| $H_1$
[ ]
(order 2)

|80px

|Line segment
{{CDD|node_1}}
{ }

|2 vertices

align=center

| $H_2$
[5]
(order 10)

|80px

|Pentagon
{{CDD|node_1|5|node}}
{5}

|5 Edges

align=center

| $H_3$
[5,3]
(order 120)

|80px

|Icosahedron
{{CDD|node_1|3|node|5|node}}
{3, 5}

|20 equilateral triangles
80px

|12

|30

|20

align=center

| $H_4$
[5,3,3]
(order 14400)

|80px

|600-cell
{{CDD|node_1|3|node|3|node|5|node}}
{3, 3, 5}

|600 tetrahedra
80px

|120

|720

|1200

|600

align=center

| ${\bar{H}}_4$
[5,3,3,3]
(order ∞)

|Order-5 5-cell honeycomb
{{CDD|node_1|3|node|3|node|3|node|5|node}}
{3, 3, 3, 5}

|∞ 5-cells
80px

|∞

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, {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
(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. {{isbn|0-486-61480-8}}. (Table I(ii): 16 regular polytopes {p, q, r} in four dimensions, pp. 292–293)

Category:Regular polytopes

Category:Multi-dimensional geometry