In geometry, the **5-cell** is a four-dimensional object bounded by 5 tetrahedral cells. It is also known as a **C**_{5}, **pentachoron**, **pentatope**, **pentahedroid**, or **tetrahedral pyramid**. It is a **4-simplex**, the simplest possible convex regular 4-polytope (four-dimensional analogue of a Platonic solid), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base.

The **regular 5-cell** is bounded by regular tetrahedra, and is one of the six regular convex 4-polytopes, represented by Schläfli symbol {3,3,3}.

Pentachoron
4-simplex
Pentatope
Pentahedroid (Henry Parker Manning)
Pen (Jonathan Bowers: for pentachoron)
Hyperpyramid, tetrahedral pyramid
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°.

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 simplest set of coordinates is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (τ,τ,τ,τ), with edge length 2√2, where τ is the golden ratio.

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

(
1
10
,
1
6
,
1
3
,
±
1
)
(
1
10
,
1
6
,
−
2
3
,
0
)
(
1
10
,
−
3
2
,
0
,
0
)
(
−
2
2
5
,
0
,
0
,
0
)

Another set of origin-centered coordinates in 4-space can be seen as a hyperpyramid with a regular tetrahedral base in 3-space, with edge length 2√2:

(
1
,
1
,
1
,
−
1
/
5
)
(
1
,
−
1
,
−
1
,
−
1
/
5
)
(
−
1
,
1
,
−
1
,
−
1
/
5
)
(
−
1
,
−
1
,
1
,
−
1
/
5
)
(
0
,
0
,
0
,
5
−
1
/
5
)
The vertices of a 4-simplex (with edge √2) can be more simply constructed on a hyperplane in 5-space, as (distinct) 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.

A *5-cell* can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, although folding into 4-dimensions causes edges to coincide. The purple edges represent the Petrie polygon of the 5-cell.

The A_{4} Coxeter plane projects the 5-cell into a regular pentagon and pentagram.

There are many lower symmetry forms, including these found in uniform polytope vertex figures:

The **tetrahedral pyramid** is a special case of a **5-cell**, a polyhedral pyramid, constructed as a regular 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.

Many uniform 5-polytopes have **tetrahedral pyramid** vertex figures:

Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram.

The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and blue 5-cell vertices and edges. This compound has [[3,3,3]] symmetry, order 240. The intersection of these two 5-cells is a uniform birectified 5-cell. = ∩ .

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

It is in the sequence of regular polychora: the tesseract {4,3,3}, 120-cell {5,3,3}, of Euclidean 4-space, and hexagonal tiling honeycomb {6,3,3} of hyperbolic space. All of these have a tetrahedral vertex figure.

It is similar to three regular polychora: the tesseract {4,3,3}, 600-cell {3,3,5} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cell.