5-cell

When a net of five tetrahedra is folded up in 4-dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5-cell has a total of 5 vertices, 10 edges and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge.

The 5-cell is self-dual (as are all simplexes), and its vertex figure is the 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 convex hull of two 5-cells in dual configuration is the disphenoidal 30-cell, dual of the bitruncated 5-cell.

This configuration matrix represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation.[7]

${\displaystyle {\begin{bmatrix}{\begin{matrix}5&4&6&4\\2&10&3&3\\3&3&10&2\\4&6&4&5\end{matrix}}\end{bmatrix}}}$

The simplest set of coordinates is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (φ,φ,φ,φ), with edge length 2√2, where φ is the golden ratio.[8]

The Cartesian coordinates of the vertices of an origin-centered regular 5-cell having edge length 2 and radius √1.6 are:

${\displaystyle \left({\frac {1}{\sqrt {10}}},\ {\frac {1}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)}$

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

${\displaystyle \left({\frac {1}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ 0,\ 0\right)}$

${\displaystyle \left(-2{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)}$

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 and radius √3.2:

${\displaystyle \left(1,1,1,{\frac {-1}{\sqrt {5}}}\right)}$

${\displaystyle \left(1,-1,-1,{\frac {-1}{\sqrt {5}}}\right)}$

${\displaystyle \left(-1,1,-1,{\frac {-1}{\sqrt {5}}}\right)}$

${\displaystyle \left(-1,-1,1,{\frac {-1}{\sqrt {5}}}\right)}$

${\displaystyle \left(0,0,0,{\frac {4}{\sqrt {5}}}\right)}$

The vertices of a 4-simplex (with edge √2 and radius 1) 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 A₄ Coxeter plane projects the 5-cell into a regular pentagon and pentagram. The A₃ Coxeter plane projection of the 5-cell is that of a square pyramid. The A₂ Coxeter plane projection of the regular 5-cell is that of a triangular bipyramid (two tetrahedra joined face-to-face) with the two opposite vertices centered.

orthographic projections

Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Projections to 3 dimensions
The vertex-first projection of the 5-cell into 3 dimensions has a tetrahedral projection envelope. The closest vertex of the 5-cell 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 5-cell 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 5-cell 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 project 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 5-cell 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.

Stereographic projection wireframe (edge projected onto a 3-sphere)

A 4-orthoscheme is a 5-cell where all 10 faces are right triangles.[lower-alpha 1] An orthoscheme is an irregular simplex that is the convex hull of a tree in which all edges are mutually perpendicular. In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a 3-orthoscheme, and each triangular face is a 2-orthoscheme (a right triangle).

Orthoschemes are the characteristic simplexes of the regular polytopes, because each regular polytope is generated by reflections in the bounding facets of its particular characteristic orthoscheme.[9] For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the 4-cube (also called the tesseract or 8-cell), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length √1, √2, √3, or √4, precisely the chord lengths of the unit 4-cube (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular polytope) can be dissected into instances of its characteristic orthoscheme.

A 3-cube dissected into six 3-orthoschemes. Three are left-handed and three are right handed. A left and a right meet at each square face.

A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a tetrahedral pyramid with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal √1 edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a √2 diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes).[lower-alpha 4] The third additional edge is a √3 diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a long diameter of the tesseract itself, of length √4. It reaches through the exact center of the tesseract to the antipodal vertex (a vertex of the opposing 3-cube), which is the apex. Thus the characteristic simplex of the 4-cube has four √1 edges, three √2 edges, two √3 edges, and one √4 edge.

The 4-cube can be dissected into 24 such 4-orthoschemes eight different ways, with six 4-orthoschemes surrounding each of four orthogonal √4 tesseract long diameters.[lower-alpha 5]

More generally, the characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope (in the right proportions).[lower-alpha 6] Because they are orthoschemes, they also possess all the requisite angles between elements (from 90 degrees on down). The characteristic simplexes are the genetic codes of polytopes: like a Swiss Army knife, they contain one of everything needed to construct the polytope by replication.

Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme. There is a 4-orthoscheme which is the characteristic simplex of the regular 5-cell. It is a tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. In a unit-radius 5-cell (edge length √2.5 ≈ 1.581) the 3-orthoscheme base has one longest √2.5 edge, two shortest √0.625 edges (half-edges of the regular 5-cell), two √1.875 edges, and one edge of length √5/2 = √1.118∼ (one of the three mid-edge to opposite mid-edge diameters of the regular tetrahedron). The apex adds four additional edges (one of each length), so the 4-orthoscheme has two longest √2.5 edges (opposite and perpendicular edges of the regular 5-cell), three shortest √0.625 edges (half-edges of the regular 5-cell), three √1.875 edges, and two √1.118∼ edges (mid-edge diameters of different regular tetrahedron cells). The path of 4 orthogonal edges is √0.625, √1.118∼, √0.625, √1.118∼ (along a half-edge, through a tetrahedron center, along another half-edge, and through a different tetrahedron center). The regular 5-cell can be dissected into ten such 4-orthoschemes four different ways, with six 4-orthoschemes surrounding their common regular 5-cell edge.[lower-alpha 7]

There are many lower symmetry forms of the 5-cell, including these found as uniform polytope vertex figures:

Symmetry	[3,3,3] Order 120	[3,3,1] Order 24	[3,2,1] Order 12	[3,1,1] Order 6	[5,2]+ Order 10
Name	Regular 5-cell	Tetrahedral pyramid	Triangular-pyramidal pyramid		Pentagonal hyperdisphenoid
Schläfli	{3,3,3}	{3,3} ∨ ( )	{3} ∨ { }	{3} ∨ ( ) ∨ ( )
Example Vertex figure	5-simplex	Truncated 5-simplex	Bitruncated 5-simplex	Cantitruncated 5-simplex	Omnitruncated 4-simplex honeycomb

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:

Symmetry [3,3,1], order 24

Schlegel diagram
Name Coxeter	{ }×{3,3,3}	{ }×{4,3,3}	{ }×{5,3,3}	t{3,3,3,3}	t{4,3,3,3}	t{3,4,3,3}

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.

Symmetry	[3,2,1], order 12		[3,1,1], order 6		[2+,4,1], order 8	[2,1,1], order 4
Schlegel diagram
Name Coxeter	t12α5	t12γ5	t012α5	t012γ5	t123α5	t123γ5

Symmetry	[2,1,1], order 2			[2+,1,1], order 2	[ ]+, order 1
Schlegel diagram
Name Coxeter	t0123α5	t0123γ5	t0123β5	t01234α5	t01234γ5