Pentachoron

From Wikipedia, the free encyclopedia

Pentachoron (5-cell)
Schlegel diagram
Type	Regular polychoron
Cells	5 (3.3.3)
Faces	10 {3}
Edges	10
Vertices	5
Vertex figure	4 (3.3.3) (tetrahedron)
Schläfli symbol	{3,3,3}
Symmetry group	A₄, [3,3,3]
Dual	self-dual
Properties	convex

Vertex figure: tetrahedron

Four orthographic projections

The pentachoron, or 5-cell, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). It is an analog of the planar triangle and solid tetrahedron.

1 Geometry
- 1.1 Construction
- 1.2 Projections
2 See also
3 External links

[edit] Geometry

The pentatope consists of five cells, all tetrahedra, and is self-dual. Its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism.

The Schläfli symbol of the pentatope is {3,3,3}.

[edit] Construction

The pentatope can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant with all the other vertices of the tetrahedron. (Essentially, the pentatope is a 4-dimensional pyramid with a tetrahedral base.)

[edit] Projections

One of the possible projections of the pentachoron into 2 dimensions is the pentagram inscribed inside a pentagon.

Both the vertex-first and cell-first parallel projection of the pentachoron into 3 dimensions have a tetrahedral envelope. The closest or farthest vertex of the pentachoron, respectively, projects to the center of the tetrahedron. The farthest/closest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the center.

The edge-first and face-first projections of the pentachoron into 3 dimensions have a triangular dipyramidal envelope. Two of the cells project to the upper and lower halves of the dipyramid, while the remaining 3 project to 3 non-regular tetrahedral volumes arranged around the central axis of the dipyramid at 120 degrees to each other.