5-cell

Regular 5-cell
(pentachoron)
(4-simplex)

Schlegel diagram
(vertices and edges)
Type Convex regular 4-polytope
Schläfli symbol {3,3,3}
Coxeter-Dynkin diagram
Cells 5 {3,3}
Faces 10 {3}
Edges 10
Vertices 5
Vertex figure
(tetrahedron)
Petrie polygon pentagon
Coxeter group A4, [3,3,3]
Dual Self-dual
Properties convex, isogonal, isotoxal, isohedral
Uniform index 1

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as the pentachoron, pentatope, or hyperpyramid. It is a 4-simplex, the simplest possible convex regular 4-polytope (four-dimensional analogue of a polyhedron), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions.

The regular 5-cell is bounded by regular tetrahedra, and is one of the six regular convex polychora, represented by Schläfli symbol {3,3,3}.

Contents

Geometry

The 5-cell is self-dual, and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism. Its dihedral angle is cos−1(1/4), or approximately 75.52°.

Construction

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

The Cartesian coordinates of the vertices of an origin-centered regular 5-cell having edge length 2 are:

\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)
\left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)
\left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)
\left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right)

The vertices of a 4-simplex (with edge √2) can be more simply constructed on a hyperplane in 5-space, as permutations of (0,0,0,0,1) or (0,1,1,1,1); in these positions it is a facet of, respectively, the 5-orthoplex or the rectified penteract.

Projections

orthographic projections
Ak
Coxeter plane		 A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Projections to 3 dimensions

Stereographic projection wireframe (edge projected onto a 3-sphere)
A 3D projection of a 5-cell performing a simple rotation

The vertex-first projection of the pentachoron into 3 dimensions has a tetrahedral projection envelope. The closest vertex of the pentachoron projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex.
The edge-first projection of the pentachoron into 3 dimensions has a triangular dipyramidal envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope.

The face-first projection of the pentachoron into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face projects to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge-first projection.
The cell-first projection of the pentachoron into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here.

Alternative names

Related uniform polychora

The pentachoron (5-cell) is the simplest of 9 uniform polychora constructed from the [3,3,3] Coxeter group.

Name 5-cell truncated 5-cell rectified 5-cell cantellated 5-cell bitruncated 5-cell cantitruncated 5-cell runcinated 5-cell runcitruncated 5-cell omnitruncated 5-cell
Schläfli
symbol		 {3,3,3} t0,1{3,3,3} t1{3,3,3} t0,2{3,3,3} t1,2{3,3,3} t0,1,2{3,3,3} t0,3{3,3,3} t0,1,3{3,3,3} t0,1,2,3{3,3,3}
Coxeter-Dynkin
diagram
Schlegel
diagram
A4
Coxeter plane
Graph
A3 Coxeter plane
Graph
A2 Coxeter plane
Graph

Other forms

The 5-cell can also be considered a tetrahedral pyramid, constructed as a tetrahedron base in a 3-space hyperplane, and an apex point above the hyperplane. The four sides of the pyramid are made of tetrahedron cells.

References

External links

5-cell 8-cell 16-cell 24-cell 120-cell 600-cell

{3,3,3}
pentachoron

{4,3,3}
tesseract

{3,3,4}
hexadecachoron

{3,4,3}
icositetrachoron

{5,3,3}
hecatonicosachoron

{3,3,5}
hexacosichoron