Pentachoron
Pentachoron | |
---|---|
Rank | 4 |
Type | Regular |
Space | Spherical |
Bowers style acronym | Pen |
Info | |
Coxeter diagram | x3o3o3o |
Schläfli symbol | {3,3,3} |
Tapertopic notation | 1^{3} |
Symmetry | A4, order 120 |
Army | Pen |
Regiment | Pen |
Elements | |
Vertex figure | Tetrahedron, edge length 1 |
Cells | 5 tetrahedra |
Faces | 10 triangles |
Edges | 10 |
Vertices | 5 |
Measures (edge length 1) | |
Circumradius | |
Edge radius | |
Face radius | |
Inradius | |
Hypervolume | |
Dichoral angle | |
Heights | Point atop tet: |
Dyad atop perp trig: | |
Central density | 1 |
Euler characteristic | 0 |
Number of pieces | 5 |
Level of complexity | 1 |
Related polytopes | |
Dual | Pentachoron |
Conjugate | Pentachoron |
Properties | |
Convex | Yes |
Orientable | Yes |
Nature | Tame |
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.
Cross-sections[edit | edit source]
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 step prism construction, is given by:
together with reflections through the x=y and the z=w hyperplanes.
Surtope Angles[edit | edit source]
The surtope-angle represents the fraction of space occupied by an element.
- A2 :25.20.108 acos(1/4) dichoral or margin-angle
- A3 :07.71.42 1.5 * acos(1/4) - 1/4. edge-angle
- A4 :01.20.108 acos(1/4)-1/5. vertex-angle
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)
Segmentochoron display[edit | edit source]
Point atop tetrahedron
Variations[edit | edit source]
Besides the regular pentachora, 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
- 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.
Two of the seven regular polychoron compounds are composed of pentachora:
- Stellated decachoron (2)
- Medial hexacosichoron (120)
Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|
Pentachoron | pen | {3,3,3} | ||
Truncated pentachoron | tip | t{3,3,3} | ||
Rectified pentachoron | rap | r{3,3,3} | ||
Decachoron | deca | 2t{3,3,3} | ||
Rectified pentachoron | rap | r{3,3,3} | ||
Truncated pentachoron | tip | t{3,3,3} | ||
Pentachoron | pen | {3,3,3} | ||
Small rhombated pentachoron | srip | rr{3,3,3} | ||
Great rhombated pentachoron | grip | tr{3,3,3} | ||
Small rhombated pentachoron | srip | rr{3,3,3} | ||
Great rhombated pentachoron | grip | tr{3,3,3} | ||
Small prismatodecachoron | spid | t_{0,3}{3,3,3} | ||
Prismatorhombated pentachoron | prip | t_{0,1,3}{3,3,3} | ||
Prismatorhombated pentachoron | prip | t_{0,1,3}{3,3,3} | ||
Great prismatodecachoron | gippid | t_{0,1,2,3}{3,3,3} |
Isogonal derivatives[edit | edit source]
Substitution by vertices of these following elements will produce these convex isogonal polychora:
- Tetrahedron (5): Pentachoron
- Triangle (10): Rectified pentachoron
- Edge (10): Rectified pentachoron
External links[edit | edit source]
- Bowers, Jonathan. "Category 1: Regular Polychora" (#1).
- Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".
- Klitzing, Richard. "Pen".
- Quickfur. "The Pentachoron".
- Wikipedia Contributors. "5-cell".