|Bowers style acronym||Pen|
|Coxeter diagram||x3o3o3o ()|
|Vertex figure||Tetrahedron, edge length 1|
|Edge figure||tet 3 tet 3 tet 3|
|Petrie polygons||12 pentagonal-pentagrammic coils|
|Measures (edge length 1)|
|Heights||Point atop tet:|
|Dyad atop perp trig:|
|Number of external pieces||5|
|Level of complexity||1|
|Abstract & topological properties|
|Symmetry||A4, order 120|
The pentachoron, or pen, also commonly called the 5-cell or the 4-simplex, is the simplest possible non-degenerate polychoron. The full symmetry version has 5 regular tetrahedra as cells, joining 3 to an edge and 4 to a vertex, and is one of the 6 convex regular polychora. It is the 4-dimensional simplex.
In addition, it can also be considered to be the regular-faced pyramid of the tetrahedron, or the pyramid product of a triangle and a dyad. This makes it the simplest segmentochoron as well, and it is designated K-4.1 in Richard Klitzing's list of convex segmentochora. It is also the 5-2 step prism and gyrochoron.
Gallery[edit | edit source]
Wireframe, cell, net
Vertex coordinates[edit | edit source]
The vertices of a regular pentachoron of edge length 1, centered at the origin, are given by:
Much simpler coordinates can be given in five dimensions, as all permutations of:
A further set of coordinates, derived from the 5-2 step prism construction, is given by:
together with reflections through the and the hyperplanes.
Representations[edit | edit source]
A pentachoron has the following Coxeter diagrams:
- x3o3o3o () (full symmetry)
- ox3oo3oo&#x (A3 axial, as tetrahedral pyramid)
- xo ox3oo&#x (A2×A1 axial, as triangle-dyad disphenoid)
- oox3ooo&#x (A2 axial, as triangular scalene)
- oxo oox&#x (A1×A1 axial, as disphenoidal pyramid)
- ooox&#x (bilateral symmetry only)
- ooooo&#x (no symmetry)
Variations[edit | edit source]
Besides the regular pentachoron, various other types of pentachora exist, some of which occur in vertex figures. These include:
- Tetrahedral pyramid - 1 regular tetrahedron, 4 triangular pyramids, tetrahedral symmetry
- Triangular scalene - 2 triangular pyramids, 3 digonal disphenoids, triangular prismatic axial symmetry
- Triangular pyramidal pyramid - 2 different triangular pyramids, 3 sphenoids, triangular symmetry
- Tetragonal disphenoidal pyramid - 1 tetragonal disphenoid, 4 sphenoids, digonal antiprismatic axial symmetry
- Rhombic disphenoidal pyramid - 1 rhombic disphenoid, 4 irregular tetrahedra, chiral digonal prismatic axial symmetry
- Digonal disphenoidal pyramid - 1 digonal disphenoid, 2 pairs of sphenoids, digonal axial symmetry
- Sphenoidal pyramid - 2 identical irregular tetrahedra, 3 different sphenoids
- Phyllic disphenoidal pyramid - 1 phyllic dissphenoid, 2 pairs of identical irregular tetrahedra, bilateral symmetry
- Irregular pentachoron - no symmetry, 5 different irregular tetrahedra
- 5-2 step prism (or gyrochoron) noble, 5 phyllic disphenoids
Related polychora[edit | edit source]
Two pentachora can be attached at a common cell to form the tetrahedral tegum.
Uniform polychoron compounds composed of pentachora include:
- Stellated decachoron (2)
- Gyrosimplexifissal icosachoron (4)
- Chirosimplexifissal icosipentachoron (5)
- Simplexifissial disicosipentachoron (10)
- Compound of 12 pentachora (12)
- Medial hexacosichoron (120)
- Compound of 720 pentachora (720)
- An infinite family of gyrochoron-symmetric compounds
Isogonal derivatives[edit | edit source]
[edit | edit source]
- Bowers, Jonathan. "Category 1: Regular Polychora" (#1).
- Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".
- Bowers, Jonathan. "Pennic and Decaic Isogonals".
- Klitzing, Richard. "Pen".
- Quickfur. "The Pentachoron".