# 5-cell

Short description: Four-dimensional analogue of the tetrahedron
1. REDIRECT Template:Infobox 4-polytope
A 3D projection of a 5-cell performing a simple rotation

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells.[lower-alpha 1] It is also known as a C5, pentachoron,[1] pentatope, pentahedroid,[2] or tetrahedral pyramid. It is the 4-simplex (Coxeter's $\displaystyle{ \alpha_4 }$ polytope),[3] the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

The regular 5-cell is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes (the four-dimensional analogues of the Platonic solids). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and match sticks intersect one another. No solution exists in three dimensions.

## Alternative names

• Pentachoron (5-point 4-polytope)
• Hypertetrahedron (4-dimensional analogue of the tetrahedron)
• 4-simplex (4-dimensional simplex)
• Tetrahedral pyramid (4-dimensional hyperpyramid with a tetrahedral base)
• Pentatope
• Pentahedroid (Henry Parker Manning)
• Pen (Jonathan Bowers: for pentachoron)[4]

## Geometry

The 5-cell is the 4-dimensional simplex, the simplest possible 4-polytope. As such it is the first in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[lower-alpha 2]

A 5-cell is formed by any five points which are not all in the same hyperplane (as a tetrahedron is formed by any four points which are not all in the same plane, and a triangle is formed by any three points which are not all in the same line). Any such five points constitute a 5-cell, though not usually a regular 5-cell. The regular 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex 120-cell is a compound of 120 regular 5-cells.[lower-alpha 3]

### Structure

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.[lower-alpha 5] 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.

### As a configuration

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] The k-faces can be read as rows left of the diagonal, while the k-figures are read as rows after the diagonal.[8]

Grünbaum's rotationally symmetrical 5-set Venn diagram, 1975
Element k-face fk f0 f1 f2 f3 k-figs
( ) f0 5 4 6 4 {3,3}
{ } f1 2 10 3 3 {3}
{3} f2 3 3 10 2 { }
{3,3} f3 4 6 4 5 ( )

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.

### Coordinates

The simplest set of Cartesian 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 22, where φ is the golden ratio.[9] While these coordinates are not origin-centered, subtracting $\displaystyle{ (1,1,1,1)/(2-\tfrac{1}{\varphi}) }$ from each translates the 4-polytope's circumcenter to the origin with radius $\displaystyle{ 2(\varphi-1/(2-\tfrac{1}{\varphi})) =\sqrt{\tfrac{16}{5}}\approx 1.7888 }$, with the following coordinates:

$\displaystyle{ \left(\tfrac{2}{\varphi}-3, 1, 1, 1\right)/(\tfrac{1}{\varphi}-2) }$
$\displaystyle{ \left(1,\tfrac{2}{\varphi}-3,1,1 \right)/(\tfrac{1}{\varphi}-2) }$
$\displaystyle{ \left(1,1,\tfrac{2}{\varphi}-3,1 \right)/(\tfrac{1}{\varphi}-2) }$
$\displaystyle{ \left(1,1,1,\tfrac{2}{\varphi}-3 \right)/(\tfrac{1}{\varphi}-2) }$
$\displaystyle{ \left(\tfrac{2}{\varphi},\tfrac{2}{\varphi},\tfrac{2}{\varphi},\tfrac{2}{\varphi} \right)/(\tfrac{1}{\varphi}-2) }$

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 $\displaystyle{ \tfrac{\sqrt{5}}{4} }$ give unit-radius origin-centered regular 5-cells with edge lengths $\displaystyle{ \sqrt{\tfrac{5}{2}} }$. 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 $\displaystyle{ \sqrt{\tfrac{8}{5}}\approx 1.265 }$ are:

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

Scaling these by $\displaystyle{ \sqrt{\tfrac{5}{8}} }$ to unit-radius and edge length $\displaystyle{ \sqrt{\tfrac{5}{2}} }$ 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.

### Geodesics and rotations

The 5-cell has only digon central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell.[lower-alpha 4] Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them.[lower-alpha 7] The characteristic isoclinic rotation of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no 5-cell vertices.

There are exactly three different ways to make a circuit of the 5-cell through all 5 vertices along 5 edges,[lower-alpha 5] so there are three discrete Hopf fibrations of the great digons of the 5-cell. Each of the three fibrations corresponds to a distinct left-right pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5-cell has only three distinct period 5 isoclines (those circles through all 5 vertices), each of which acts as the single isocline of a right rotation and the single isocline of a left rotation in two different fibrations, and as the Petrie polygon of the 5-cell in the third fibration.[lower-alpha 6]

### Boerdijk–Coxeter helix

A 5-cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4-dimensional ring.[10] 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 the chords of the 5-cell's isocline, the circular rotational path its vertices take during an isoclinic rotation, also known as a Clifford displacement.[lower-alpha 8]

### Projections

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.

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)

## 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.[lower-alpha 1] An orthoscheme is an irregular simplex that is the convex hull of a tree in which all edges are mutually perpendicular.[lower-alpha 9] 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.[11] 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-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 10] 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.[lower-alpha 11]

More generally, any regular polytope can be dissected into g instances of its characteristic orthoscheme that all meet at the regular polytope's center.[12] 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.[lower-alpha 12] 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.[lower-alpha 13] 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.[lower-alpha 14]

Characteristics of the regular 5-cell[16]
edge[17] arc dihedral[18]
𝒍 $\displaystyle{ \sqrt{\tfrac{5}{2}} \approx 1.581 }$ 104°30′40″ $\displaystyle{ \pi - 2\text{𝜂} }$ 75°29′20″ $\displaystyle{ \pi - 2\text{𝟁} }$
𝟀 $\displaystyle{ \sqrt{\tfrac{1}{10}} \approx 0.316 }$ 75°29′20″ $\displaystyle{ 2\text{𝜂} }$ 60° $\displaystyle{ \tfrac{\pi}{3} }$
𝝉[lower-alpha 15] $\displaystyle{ \sqrt{\tfrac{1}{30}} \approx 0.183 }$ 52°15′20″ $\displaystyle{ \tfrac{\pi}{2}-\text{𝜂} }$ 60° $\displaystyle{ \tfrac{\pi}{3} }$
𝟁 $\displaystyle{ \sqrt{\tfrac{2}{15}} \approx 0.103 }$ 52°15′20″ $\displaystyle{ \tfrac{\pi}{2}-\text{𝜂} }$ 60° $\displaystyle{ \tfrac{\pi}{3} }$
$\displaystyle{ _0R^3/l }$ $\displaystyle{ \sqrt{\tfrac{3}{20}} \approx 0.387 }$ 75°29′20″ $\displaystyle{ 2\text{𝜂} }$ 90° $\displaystyle{ \tfrac{\pi}{2} }$
$\displaystyle{ _1R^3/l }$ $\displaystyle{ \sqrt{\tfrac{1}{20}} \approx 0.224 }$ 52°15′20″ $\displaystyle{ \tfrac{\pi}{2}-\text{𝜂} }$ 90° $\displaystyle{ \tfrac{\pi}{2} }$
$\displaystyle{ _2R^3/l }$ $\displaystyle{ \sqrt{\tfrac{1}{60}} \approx 0.129 }$ 52°15′20″ $\displaystyle{ \tfrac{\pi}{2}-\text{𝜂} }$ 90° $\displaystyle{ \tfrac{\pi}{2} }$
$\displaystyle{ _0R^4/l }$ $\displaystyle{ \sqrt{\tfrac{4}{25}} = 0.4 }$
$\displaystyle{ _1R^4/l }$ $\displaystyle{ \sqrt{\tfrac{3}{50}} \approx 0.245 }$
$\displaystyle{ _2R^4/l }$ $\displaystyle{ \sqrt{\tfrac{2}{43}} \approx 0.216 }$
$\displaystyle{ _3R^4/l }$ $\displaystyle{ \sqrt{\tfrac{1}{100}} = 0.1 }$
$\displaystyle{ \text{𝜼} }$ 37°44′40″ $\displaystyle{ \tfrac{\text{arc sec }4}{2} }$

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).[lower-alpha 16] If the regular 5-cell has unit radius and edge length 𝒍 = $\displaystyle{ \sqrt{\tfrac{5}{2}} }$, its characteristic 5-cell's ten edges have lengths $\displaystyle{ \sqrt{\tfrac{1}{10}} }$, $\displaystyle{ \sqrt{\tfrac{1}{30}} }$, $\displaystyle{ \sqrt{\tfrac{2}{15}} }$ around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),[lower-alpha 15] plus $\displaystyle{ \sqrt{\tfrac{3}{20}} }$, $\displaystyle{ \sqrt{\tfrac{1}{20}} }$, $\displaystyle{ \sqrt{\tfrac{1}{60}} }$ (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus $\displaystyle{ \sqrt{\tfrac{4}{25}} }$, $\displaystyle{ \sqrt{\tfrac{3}{50}} }$, $\displaystyle{ \sqrt{\tfrac{2}{43}} }$, $\displaystyle{ \sqrt{\tfrac{1}{100}} }$ (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is $\displaystyle{ \sqrt{\tfrac{1}{30}} }$, $\displaystyle{ \sqrt{\tfrac{2}{15}} }$, $\displaystyle{ \sqrt{\tfrac{1}{60}} }$, $\displaystyle{ \sqrt{\tfrac{1}{100}} }$, 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.[lower-alpha 17]

### Isometries

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
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 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.

Symmetry [3,2,1], order 12 [3,1,1], order 6 [2+,4,1], order 8 [2,1,1], order 4
Schläfli {3}∨{  } {3}∨( )∨( ) { }∨{ }∨( )
Schlegel
diagram
Name
Coxeter
t12α5
t12γ5
t012α5
t012γ5
t123α5
t123γ5
Symmetry [2,1,1], order 2 [2+,1,1], order 2 [ ]+, order 1
Schläfli { }∨( )∨( )∨( ) ( )∨( )∨( )∨( )∨( )
Schlegel
diagram
Name
Coxeter
t0123α5
t0123γ5
t0123β5
t01234α5
t01234γ5

## 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.[lower-alpha 5]

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.

## Notes

1. A 5-cell's 5 vertices form 5 tetrahedral cells face-bonded to each other, with a total of 10 edges and 10 triangular faces.
2. The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content[5] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 5-cell is the 5-point 4-polytope: first in the ascending sequence that runs to the 600-point 4-polytope.
3. The regular 120-cell has a curved 3-dimensional boundary surface consisting of 120 regular dodecahedron cells. It also has 120 disjoint regular 5-cells inscribed in it.[6] These are not 3-dimensional cells but 4-dimensional objects which share the 120-cell's center point, and collectively cover all 600 of its vertices.
4. In a polytope with a tetrahedral vertex figure,[lower-alpha 5] a geodesic path along edges does not lie on an ordinary great circle in a single central plane: each successive edge lies in a different central plane than the previous edge. Nonetheless the edge-path Clifford polygon is the skew chord set of a true geodesic great circle, circling through four dimensions rather than through only two dimensions: but it is not an ordinary "flat" great circle of circumference 2𝝅𝑟, it is an isocline.[lower-alpha 6]
5. The Schlegel diagram of the 5-cell (at the top of this article) illustrates its tetrahedral vertex figure. Six of the 5-cell's 10 edges are the bounding edges of the Schlegel regular tetrahedron. The other four edges converge at the fifth vertex, at the center of volume of the tetrahedron. Consider any circular geodesic (shortest) path along edges. Arriving at that fifth "central" vertex along an edge, the path must bend to follow another edge departing from the vertex, and there is a choice of three such departing edges (three "directions" in which to bend). The 5-cell has exactly three distinct pentagonal geodesic circles in it, each corresponding to making that choice the same way at all 5 vertices of the circle. These three geodesic skew pentagons are the 5-cell's three distinct Petrie polygons. Each is a different sequence of 5 of the 10 edges, and there are only three such distinct sequences.[lower-alpha 4]
6. The 5-cell (4-simplex) is unique among regular 4-polytopes in that its isocline chords[lower-alpha 8] are its own edges. In the other regular 4-polytopes, the isocline chord is the longer edge of another regular polytope that is inscribed. Another aspect of this uniqueness is that the 5-cell's isocline Clifford polygon (the skew pentagram) and its zig-zag Petrie polygon (the skew pentagon) are exactly the same object; in the other regular 4-polytopes they are quite different.
7. Each edge intersects 6 others (3 at each end) and is disjoint from the other 3, to which it is orthogonal as the edge of a tetrahedron to its opposite edge.
8. Each isocline chord (blue pentagram edge) runs from one of the 5 vertices, through the interior volume of one of the 5 tetrahedral cells, through the cell's triangular face opposite the vertex, and then straight on through the volume of the neighboring cell that shares the face, to its vertex opposite the face. The isocline chord is a straight line between the two vertices through the volume of the two cells. As you can see in the illustration, the blue isocline chord does not pass through the exact center of the shared face, but rather through a point closer to one face vertex. There are in fact three different isocline pentagrams in the 5-cell, one of which appears as the blue pentagram in the illustration. Each of these three Clifford pentagrams is a different circular sequence of 5 of the 5-cell's 10 edges.[lower-alpha 5] All 10 edges are present in each of the 5 tetrahedral cells: each cell is bounded by 6 of the 10 edges, and has the other 4 of the 10 edges running through its volume as isocline chords, from its 4 vertices and through their 4 opposite faces.[lower-alpha 6]
9. A right triangle is a 2-dimensional orthoscheme; orthoschemes are the generalization of right triangles to n dimensions. A 3-dimensional orthoscheme is a tetrahedron with 4 right triangle faces (not necessarily similar).
10. The 4-cube (tesseract) contains eight 3-cubes (so it is also called the 8-cell). Each 3-cube is face-bonded to six others (that entirely surround it), but entirely disjoint from the one other 3-cube which lies opposite and parallel to it on the other side of the 8-cell.
11. The dissection of the 4-cube into 384 4-orthoschemes is 16 of the dissections into 24 4-orthoschemes. First, each 4-cube edge is divided into 2 smaller edges, so each square face is divided into 4 smaller squares, each cubical cell is divided into 8 smaller cubes, and the entire 4-cube is divided into 16 smaller 4-cubes. Then each smaller 4-cube is divided into 24 4-orthoschemes that meet at the center of the original 4-cube.
12. For a regular k-polytope, the Coxeter-Dynkin diagram of the characteristic k-orthoscheme is the k-polytope's diagram without the generating point ring. The regular k-polytope is subdivided by its symmetry (k-1)-elements into g instances of its characteristic k-orthoscheme that surround its center, where g is the order of the k-polytope's symmetry group.[13]
13. A regular polytope of dimension k has a characteristic k-orthoscheme, and also a characteristic (k-1)-orthoscheme. A regular 4-polytope has a characteristic 5-cell (4-orthoscheme) into which it is subdivided by its (3-dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3-orthoscheme) into which its surface is subdivided by its cells' (2-dimensional) planes of symmetry. After subdividing its (3-dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4-dimensional) interior can be subdivided into characteristic 5-cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4-polytope's center.[14] The interior tetrahedra and triangles thus formed will also be orthoschemes.
14. The 120 congruent[15] 4-orthoschemes of the regular 5-cell occur in two mirror-image forms, 60 of each. Each 4-orthoscheme is cell-bonded to 4 others of the opposite chirality (by the 4 of its 5 tetrahedral cells that lie in the interior of the regular 5-cell). If the 60 left-handed 4-orthoschemes are colored red and the 60 right-handed 4-orthoschemes are colored black, each red 5-cell is surrounded by 4 black 5-cells and vice versa, in a pattern 4-dimensionally analogous to a checkerboard (if checkerboards had triangles instead of squares).
15. (Coxeter 1973) uses the greek letter 𝝓 (phi) to represent one of the three characteristic angles 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the golden ratio constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.
16. 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. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.
17. If the regular 5-cell has radius $\displaystyle{ \sqrt{\tfrac{2}{5}} \approx 0.632 }$ and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths $\displaystyle{ \sqrt{\tfrac{1}{3}} \approx 0.577 }$, $\displaystyle{ \sqrt{\tfrac{1}{4}} {{=}} 0.5 }$, $\displaystyle{ \sqrt{\tfrac{1}{12}} \approx 0.289 }$ (the exterior right triangle face, the characteristic triangle), plus $\displaystyle{ \sqrt{\tfrac{3}{8}} \approx 0.612 }$, $\displaystyle{ \sqrt{\tfrac{1}{8}} \approx 0.354 }$, $\displaystyle{ \sqrt{\tfrac{1}{24}} \approx 0.204 }$ (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron), plus $\displaystyle{ \sqrt{\tfrac{4}{10}} \approx 0.632 }$, $\displaystyle{ \sqrt{\tfrac{3}{20}} \approx 0.387 }$, $\displaystyle{ \sqrt{\tfrac{1}{8.6}} \approx 0.341 }$, $\displaystyle{ \sqrt{\tfrac{1}{40}} \approx 0.158 }$ (edges that are the characteristic radii of the regular 5-cell).[16] The 4-edge path along orthogonal edges of the orthoscheme is $\displaystyle{ \sqrt{\tfrac{1}{4}} }$, $\displaystyle{ \sqrt{\tfrac{1}{12}} }$, $\displaystyle{ \sqrt{\tfrac{1}{24}} }$, $\displaystyle{ \sqrt{\tfrac{1}{40}} }$.

## Citations

1. N.W. Johnson: Geometries and Transformations, (2018) ISBN:978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
2. Matila Ghyka, The geometry of Art and Life (1977), p.68
3. Coxeter 1973, p. 120, §7.2. see illustration Fig 7.2A.
4. Category 1: Regular Polychora
5. Coxeter 1973, pp. 292-293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.
6. Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
7. Coxeter 1973, p. 12, §1.8. Configurations.
8. Coxeter 1991, p. 30, §4.2. The Crystallographic regular polytopes.
9. Coxeter 1973, pp. 198-202, §11.7 Regular figures and their truncations.
10. Kim & Rote 2016, pp. 17-20, §10 The Coxeter Classification of Four-Dimensional Point Groups.
11. Coxeter 1973, pp. 130-133, §7.6 The symmetry group of the general regular polytope.
12. Coxeter 1973, p. 130, §7.6; "simplicial subdivision".
13. Coxeter 1973, §3.1 Congruent transformations.
14. Coxeter 1973, pp. 292-293, Table I(ii); "5-cell, 𝛼4".
15. Coxeter 1973, p. 139, §7.9 The characteristic simplex.
16. Coxeter 1973, p. 290, Table I(ii); "dihedral angles".

## References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• H.S.M. Coxeter:
• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
• p. 120, §7.2. see illustration Fig 7.2A
• p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Coxeter, H.S.M. (1991), Regular Complex Polytopes (2nd ed.), Cambridge: Cambridge University Press
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN:978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN:978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". in Senechal, Marjorie. Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8.