A cross-polytope or orthoplex is a shape constructed by repeatedly taking the bipyramid of a point. Each hypercube has four simplices of the lower dimension surrounding each peak (e.g. an octahedron has four triangles around each vertex, a hexadecachoron has three tetrahedra around each vertex).
| D | Schläfli Symbol | Cross |
|---|---|---|
| 0 | Point | |
| 1 | Line segment | |
| 2 | Square | |
| 3 | Octahedron | |
| 4 | Hexadecachoron | |
| 5 | Pentacross | |
| 6 | Hexacross | |
| 7 | Heptacross | |
| 8 | Octacross | |
| 9 | Enneacross | |
| 10 | Dekacross |
See Also[]
| Dimensionality | Negative One | Zero | One | Two | Three | Four | Five | Six | Seven | Eight | Nine | Ten | Eleven | Twelve | Thirteen | Fourteen | Fifteen | Sixteen | Seventeen | ... | Aleph null |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Simplex
\(\{3^{n-1}\}\) |
Null polytope
\()(\) |
Point
\(()\) |
Line segment
\(\{\}\) |
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
\(\{3^{15}\}\) |
Octadecapedakon
\(\{3^{16}\}\) |
... | 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
\(\{3^{14}, 4\}\) |
Heptadekacross
\(\{3^{15}, 4\}\) |
... | 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
\(\{4, 3^{14}\}\) |
Heptadekeract
\(\{4, 3^{15}\}\) |
... | 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
\(\mathbb B^{16}\) |
Heptadekorb
\(\mathbb B^{17}\) |
... | Omegaball
\(\mathbb B^{\aleph_0}\) |