5-cell

Irregular 5-cells

In the case of simplexes such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These characteristic 5-cells are the fundamental domains of the different symmetry groups which give rise to the various 4-polytopes.

Orthoschemes

A 4-orthoscheme is a 5-cell where all 10 faces are right triangles. (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. 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. 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 convex polytope) can be dissected into instances of its characteristic orthoscheme.

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). 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 5-cell 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. 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.

More generally, any regular polytope can be dissected into g instances of its characteristic orthoscheme that all meet at the regular polytope's center. 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. More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. 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 5-cell of the regular 5-cell. It is a tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 5-cell can be dissected into 120 instances of this characteristic 4-orthoscheme just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.

The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell). The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. If the regular 5-cell has unit radius and edge length ${\displaystyle \sqrt{\frac{5}{2}}}$, its characteristic 5-cell's ten edges have lengths ${\displaystyle \sqrt{\frac{1}{10}}}$, ${\displaystyle \sqrt{\frac{1}{30}}}$, ${\displaystyle \sqrt{\frac{2}{15}}}$ around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁), plus ${\displaystyle \sqrt{\frac{3}{20}}}$, ${\displaystyle \sqrt{\frac{1}{20}}}$, ${\displaystyle \sqrt{\frac{1}{60}}}$ (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus ${\displaystyle \sqrt{1}}$, ${\displaystyle \sqrt{\frac{3}{8}}}$, ${\displaystyle \sqrt{\frac{1}{6}}}$, ${\displaystyle \sqrt{\frac{1}{16}}}$ (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is ${\displaystyle \sqrt{\frac{1}{30}}}$, ${\displaystyle \sqrt{\frac{2}{15}}}$, ${\displaystyle \sqrt{\frac{1}{60}}}$, ${\displaystyle \sqrt{\frac{1}{16}}}$, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.

Isometries

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. The four sides of the pyramid are made of triangular pyramid cells.

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

Construction

As a Boerdijk–Coxeter helix

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 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. The pentagram's blue edges are chords of the 5-cell's isocline, the circular rotational path its vertices take during an isoclinic rotation, also known as a Clifford displacement.

Net

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. This makes the six tetrahedron as its cell.

Coordinates

The simplest set of Cartesian coordinates is $(2,0,0,0),(0,2,0,0),(0,0,2,0),(0,0,0,2),(\varphi ,\varphi ,\varphi ,\varphi ),$ with edge length ${\displaystyle 2\sqrt{2}}$, where ${\displaystyle \varphi }$ is the golden ratio. While these coordinates are not origin-centered, subtracting ${\displaystyle (1,1,1,1)/\left(2-\frac{1}{\varphi }\right)}$ from each translates the 4-polytope's circumcenter to the origin with radius ${\displaystyle 2\left(\varphi -1/\left(2-\frac{1}{\varphi }\right)\right)=\sqrt{\frac{16}{5}}\approx 1.7888}$, with the following coordinates: $${\displaystyle \begin{array}{cc}\left(\frac{2}{\varphi }-3,1,1,1\right)/\left(\frac{1}{\varphi }-2\right),& \left(1,\frac{2}{\varphi }-3,1,1\right)/\left(\frac{1}{\varphi }-2\right)\\ \left(1,1,\frac{2}{\varphi }-3,1\right)/\left(\frac{1}{\varphi }-2\right)& \left(1,1,1,\frac{2}{\varphi }-3\right)/\left(\frac{1}{\varphi }-2\right)\\ \left(\frac{2}{\varphi },\frac{2}{\varphi },\frac{2}{\varphi },\frac{2}{\varphi }\right)/\left(\frac{1}{\varphi }-2\right)\end{array}}$$

The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a regular tetrahedral base in 3-space:

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

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

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

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

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

Scaling these or the previous set of coordinates by ${\displaystyle \frac{\sqrt{5}}{4}}$ give unit-radius origin-centered regular 5-cells with edge lengths ${\displaystyle \sqrt{\frac{5}{2}}}$. The hyperpyramid has coordinates:

${\displaystyle \left(\sqrt{5},\sqrt{5},\sqrt{5},-1\right)/4}$

${\displaystyle \left(\sqrt{5},-\sqrt{5},-\sqrt{5},-1\right)/4}$

${\displaystyle \left(-\sqrt{5},\sqrt{5},-\sqrt{5},-1\right)/4}$

${\displaystyle \left(-\sqrt{5},-\sqrt{5},\sqrt{5},-1\right)/4}$

${\displaystyle \left(0,0,0,1\right)}$

Coordinates for the vertices of another origin-centered regular 5-cell with edge length 2 and radius ${\displaystyle \sqrt{\frac{8}{5}}\approx 1.265}$ are:

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

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

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

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

Scaling these by ${\displaystyle \sqrt{\frac{5}{8}}}$ to unit-radius and edge length ${\displaystyle \sqrt{\frac{5}{2}}}$ gives:

${\displaystyle \left(\sqrt{3},\sqrt{5},\sqrt{10},\pm \sqrt{30}\right)/(4\sqrt{3})}$

${\displaystyle \left(\sqrt{3},\sqrt{5},-\sqrt{40},0\right)/(4\sqrt{3})}$

${\displaystyle \left(\sqrt{3},-\sqrt{45},0,0\right)/(4\sqrt{3})}$

${\displaystyle \left(-1,0,0,0\right)}$

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.

Compound

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 bitruncated 5-cell. = ∩ .

This compound can be seen as the 4D analogue of the 2D hexagram {⁠6/2⁠} and the 3D compound of two tetrahedra.

Related polytopes and honeycombs

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.

It is one of three {3,3,p} regular 4-polytopes with tetrahedral cells, along with the 16-cell {3,3,4} and 600-cell {3,3,5}. The order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space also has tetrahedral cells.

It is self-dual like the 24-cell {3,4,3}, having a palindromic {3,p,3} Schläfli symbol.