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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
.........

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

$\{3^9\}$

$\{3^{10}\}$

Tridecahendon

$\{3^{11}\}$

$\{3^{12}\}$

$\{3^{13}\}$

$\{3^{14}\}$

$\{3^{15}\}$

... Omegasimplex
Cross

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

Square

$\{4\}$

Octahedron

$\{3, 4\}$

$\{3^2, 4\}$

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

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

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

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

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

... 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

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

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

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

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

... 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

$\mathbb B^{13}$

$\mathbb B^{14}$

$\mathbb B^{15}$
$\mathbb B^{16}$
$\mathbb B^{\aleph_0}$