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A pentagonal polytope or rhodotope is a shape derived from a pentagon.

Cosmotopes

Cosmotopes or Dodecahedral polytopes are polytopes that are composed of cosmotopes of one less dimension by joining them by the vertex (e.g. a dodecahedron is composed of 12 pentagons with 3 at each vertex, a 120-cell is composed of 120 dodecahedra with 4 at each vertex).

DSchläfli SymbolHypercube
0$ () $Point
1$ \{\} $Line segment
2$ \{5\} $Pentagon
3$ \{5,3\} $Dodecahedron
4$ \{5,3,3\} $Hecatonicosachoron
5$ \{5,3,3,3\} $Order-3 hecatonicosachoric tetracomb
6$ \{5,3,3,3,3\} $Order-3-3 hecatonicosachoric pentacomb
.........

Hydrotopes

Hydrotopes or Icosahedral polytopes are polytopes that are composed of the most possible amount of simplices joined to a vertex (e.g. an icosahedron is composed of 20 triangles with 5 at each vertex, a 600-cell with 20 at each vertex).

DSchläfli SymbolCross
0$ () $Point
1$ \{\} $Line segment
2$ \{5\} $Pentagon
3$ \{3,5\} $Icosahedron
4$ \{3,3,5\} $Hexacosichoron
5$ \{3,3,3,5\} $Order-5 pentachoric tetracomb
6$ \{3,3,3,3,5\} $Order-5 hexateric pentacomb
.........


See Also

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
Simplex

$ \{3^{n-1}\} $

Null polytope

$ )( $
$ \emptyset $

Point

$ () $
$ \mathbb{B}^0 $

Line segment

$ \{\} $
$ \mathbb{B}^1 $

Triangle

$ \{3\} $

Tetrahedron

$ \{3^2\} $

Pentachoron

$ \{3^3\} $

Hexateron

$ \{3^4\} $

Heptapeton

$ \{3^5\} $

Octaexon

$ \{3^6\} $

Enneazetton

$ \{3^7\} $

Decayotton

$ \{3^8\} $

Hendecaxennon

$ \{3^9\} $

Dodecadakon

$ \{3^{10}\} $

Tridecahendon

$ \{3^{11}\} $

Tetradecadokon

$ \{3^{12}\} $

Pentadecatradakon

$ \{3^{13}\} $

Hexadecatedakon

$ \{3^{14}\} $

Heptadecapedakon

$ \{3^{15}\} $

... Omegasimplex
Cross

$ \{3^{n-2},4\} $

Square

$ \{4\} $

Octahedron

$ \{3, 4\} $

Hexadecachoron

$ \{3^2, 4\} $

Pentacross

$ \{3^3, 4\} $

Hexacross

$ \{3^4, 4\} $

Heptacross

$ \{3^5, 4\} $

Octacross

$ \{3^6, 4\} $

Enneacross

$ \{3^7, 4\} $

Dekacross

$ \{3^8, 4\} $

Hendekacross

$ \{3^9, 4\} $

Dodekacross

$ \{3^{10}, 4\} $

Tridekacross

$ \{3^{11}, 4\} $

Tetradekacross

$ \{3^{12}, 4\} $

Pentadekacross

$ \{3^{13}, 4\} $

Hexadekacross

$ \{3^{14}, 4\} $

... Omegacross
Hydrotopes

$ \{3^{n-2}, 5\} $

Pentagon

$ \{5\} $

Icosahedron

$ \{3, 5\} $

Hexacosichoron

$ \{3^2, 5\} $

Order-5 pentachoric tetracomb

$ \{3^3, 5\} $

Order-5 hexateric pentacomb

$ \{3^4, 5\} $

...
Hypercube

$ \{4, 3^{n-2}\} $

Square

$ \{4\} $

Cube

$ \{4, 3\} $

Tesseract

$ \{4, 3^2\} $

Penteract

$ \{4, 3^3\} $

Hexeract

$ \{4, 3^4\} $

Hepteract

$ \{4, 3^5\} $

Octeract

$ \{4, 3^6\} $

Enneract

$ \{4, 3^7\} $

Dekeract

$ \{4, 3^8\} $

Hendekeract

$ \{4, 3^9\} $

Dodekeract

$ \{4, 3^{10}\} $

Tridekeract

$ \{4, 3^{11}\} $

Tetradekeract

$ \{4, 3^{12}\} $

Pentadekeract

$ \{4, 3^{13}\} $

Hexadekeract

$ \{4, 3^{14}\} $

... Omegeract
Cosmotopes

$ \{5, 3^{n-2}\} $

Pentagon

$ \{5\} $

Dodecahedron

$ \{5, 3\} $

Hecatonicosachoron

$ \{5, 3^2\} $

Order-3 hecatonicosachoric tetracomb

$ \{5, 3^3\} $

Order-3-3 hecatonicosachoric pentacomb

$ \{5, 3^4\} $

...
Hyperball

$ \mathbb B^n $

Disk

$ \mathbb B^2 $

Ball

$ \mathbb B^3 $

Gongol

$ \mathbb B^4 $

Pentorb

$ \mathbb B^5 $

Hexorb

$ \mathbb B^6 $

Heptorb

$ \mathbb B^7 $

Octorb

$ \mathbb B^8 $

Enneorb

$ \mathbb B^9 $

Dekorb

$ \mathbb B^{10} $

Hendekorb

$ \mathbb B^{11} $

Dodekorb

$ \mathbb B^{12} $

Tridekorb

$ \mathbb B^{13} $

Tetradekorb

$ \mathbb B^{14} $

Pentadekorb

$ \mathbb B^{15} $

Hexadekorb

$ \mathbb B^{16} $

... Omegaball

$ \mathbb B^{\aleph_0} $

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