It is also known as a C5, hypertetrahedron, pentachoron,[1] pentatope, pentahedroid,[2] tetrahedral pyramid, or 4-simplex (Coxeter's

[6] It is the first in the sequence of 6 convex regular 4-polytopes, in order of volume at a given radius or number of vertexes.

The rows and columns correspond to vertices, edges, faces, and cells.

[9] All these elements of the 5-cell are enumerated in Branko Grünbaum's Venn diagram of 5 points, which is literally an illustration of the regular 5-cell in projection to the plane.

It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell.

There are only two ways to make a circuit of the 5-cell through all 5 vertices along 5 edges, so there are two discrete Hopf fibrations of the great digons of the 5-cell.

Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour.

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.

These characteristic 5-cells are the fundamental domains of the different symmetry groups which give rise to the various 4-polytopes.

(The 5 vertices form 5 tetrahedral cells face-bonded to each other, with a total of 10 edges and 10 triangular faces.)

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.

[10] 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.

Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be dissected into instances of its characteristic orthoscheme.

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).

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).

The fourth additional edge (at the other end of the orthogonal path) is a long diameter of the tesseract itself, of length √4.

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.

The 4-cube can also be dissected into 384 smaller instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.

[11] The number g is the order of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a single mirror-surfaced orthoscheme instance is reflected in its own facets.

around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),[a] plus

There are many lower symmetry forms of the 5-cell, including these found as 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.

Many uniform 5-polytopes have tetrahedral pyramid vertex figures with Schläfli symbols ( )∨{3,3}.

The symmetry of a vertex figure of a uniform polytope is represented by removing the ringed nodes of the Coxeter diagram.

The purple edges form a regular pentagon which is the Petrie polygon of the 5-cell.

The blue edges connect every second vertex, forming a pentagram which is the Clifford polygon of the 5-cell.

gives: 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.

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.

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

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