5-cell

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5-cell
(4-simplex)
Schlegel diagram
(vertices and edges)
Type Convex regular 4-polytope
Schläfli symbol {3,3,3}
Coxeter diagram
Cells 5 {3,3}
Faces 10 {3}
Edges 10
Vertices 5
Vertex figure
(tetrahedron)
Petrie polygon pentagon
Coxeter group A4, [3,3,3]
Dual Self-dual
Properties convex, isogonal, isotoxal, isohedral
Uniform index 1

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.

Contents

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]

Regular convex 4-polytopes
Symmetry group A4 B4 F4 H4
Name 5-cell

Hyper-tetrahedron
5-point

16-cell

Hyper-octahedron
8-point

8-cell

Hyper-cube
16-point

24-cell

24-point

600-cell

Hyper-icosahedron
120-point

120-cell

Hyper-dodecahedron
600-point

Schläfli symbol {3, 3, 3}{3, 3, 4}{4, 3, 3}{3, 4, 3}{3, 3, 5}{5, 3, 3}
Coxeter mirrors
Mirror dihedrals𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2𝝅/2𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2𝝅/2𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2𝝅/2𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2
Graph
Vertices5 tetrahedral8 octahedral16 tetrahedral24 cubical120 icosahedral600 tetrahedral
Edges10 triangular24 square32 triangular96 triangular720 pentagonal1200 triangular
Faces10 triangles32 triangles24 squares96 triangles1200 triangles720 pentagons
Cells5 tetrahedra16 tetrahedra8 cubes24 octahedra600 tetrahedra120 dodecahedra
Tori 1 5-tetrahedron 2 8-tetrahedron 2 4-cube4 6-octahedron 20 30-tetrahedron 12 10-dodecahedron
Inscribed120 in 120-cell675 in 120-cell2 16-cells3 8-cells25 24-cells10 600-cells
Great polygons 2 𝝅/2 squares x 34 𝝅/2 rectangles x 34 𝝅/3 hexagons x 412 𝝅/5 decagons x 650 𝝅/15 dodecagons x 4
Petrie polygons 1 pentagon 1 octagon 2 octagons 2 dodecagons 4 30-gons 20 30-gons
Isocline polygrams 1 octagram 3 4 2 octagram 344 hexagram 2 3 4 30-gram 2 1 20 30-gram 21
Long radius${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle 1}$
Edge length${\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}$${\displaystyle {\sqrt {2}}\approx 1.414}$${\displaystyle 1}$${\displaystyle 1}$${\displaystyle {\tfrac {1}{\phi }}\approx 0.618}$${\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}$
Short radius${\displaystyle {\tfrac {1}{4}}}$${\displaystyle {\tfrac {1}{2}}}$${\displaystyle {\tfrac {1}{2}}}$${\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}$${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$${\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}$
Area${\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}$${\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}$${\displaystyle 24}$${\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}$${\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}$${\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}$
Volume${\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}$${\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}$${\displaystyle 8}$${\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}$${\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}$${\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}$
4-Content${\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}$${\displaystyle {\tfrac {2}{3}}\approx 0.667}$${\displaystyle 1}$${\displaystyle 2}$${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}$${\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}$

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). Therefore any arbitrarily chosen five vertices of any 4-polytope 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. 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]

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

Coordinates

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 22, 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 22 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.

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 represent the Petrie polygon of the 5-cell.

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.

orthographic projections
Ak
Coxeter plane
A4A3A2
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.

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. 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 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). [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 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 5]

More generally, any regular polytope can be dissected into g instances of its characteristic orthoscheme that all meet at the regular polytope's center. [10] 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 6] 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 7] 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 8]

Characteristics of the regular 5-cell [14]
edge [15] arcdihedral [16]
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}}}$
𝝓${\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 _{0}R^{3}/l}$${\displaystyle {\sqrt {\tfrac {3}{20}}}\approx 0.387}$75°29′20″${\displaystyle 2{\text{𝜂}}}$90°${\displaystyle {\tfrac {\pi }{2}}}$
${\displaystyle _{1}R^{3}/l}$${\displaystyle {\sqrt {\tfrac {1}{20}}}\approx 0.224}$52°15′20″${\displaystyle {\tfrac {\pi }{2}}-{\text{𝜂}}}$90°${\displaystyle {\tfrac {\pi }{2}}}$
${\displaystyle _{2}R^{3}/l}$${\displaystyle {\sqrt {\tfrac {1}{60}}}\approx 0.129}$52°15′20″${\displaystyle {\tfrac {\pi }{2}}-{\text{𝜂}}}$90°${\displaystyle {\tfrac {\pi }{2}}}$
${\displaystyle _{0}R^{4}/l}$${\displaystyle {\sqrt {\tfrac {4}{25}}}=0.4}$
${\displaystyle _{1}R^{4}/l}$${\displaystyle {\sqrt {\tfrac {3}{50}}}\approx 0.245}$
${\displaystyle _{2}R^{4}/l}$${\displaystyle {\sqrt {\tfrac {2}{43}}}\approx 0.216}$
${\displaystyle _{3}R^{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 9] 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 ${\displaystyle {\sqrt {\tfrac {1}{10}}}}$, ${\displaystyle {\sqrt {\tfrac {1}{30}}}}$, ${\displaystyle {\sqrt {\tfrac {2}{15}}}}$ (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁), 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 10]

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
NameRegular 5-cellTetrahedral pyramid 3-2 fusil 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 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}∨( ){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.

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

Schläfli {3,3,3} t{3,3,3} r{3,3,3} rr{3,3,3} 2t{3,3,3} tr{3,3,3} t0,3{3,3,3} t0,1,3{3,3,3} t0,1,2,3{3,3,3}
Coxeter
Schlegel
1k2 figures in n dimensions
SpaceFiniteEuclideanHyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1E4=A4E5=D5 E6 E7 E8 E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter
diagram
Symmetry
(order)
[3−1,2,1][30,2,1][31,2,1][[32,2,1]][33,2,1][34,2,1][35,2,1][36,2,1]
Order 121201,920103,6802,903,040696,729,600
Graph --
Name 1−1,2 102 112 122 132 142 152 162
2k1 figures in n dimensions
SpaceFiniteEuclideanHyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1E4=A4E5=D5 E6 E7 E8 E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter
diagram
Symmetry [3−1,2,1][30,2,1][[31,2,1]][32,2,1][33,2,1][34,2,1][35,2,1][36,2,1]
Order 1212038451,8402,903,040696,729,600
Graph --
Name 2−1,1 201 211 221 231 241 251 261

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.

{p,3,3} polytopes
Space S3 H3
FormFiniteParacompactNoncompact
Name {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ...{,3,3}
Image
Cells
{p,3}

{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{,3}

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.

{3,3,p} polytopes
Space S3 H3
FormFiniteParacompactNoncompact
Name {3,3,3}
{3,3,4}

{3,3,5}
{3,3,6}

{3,3,7}
{3,3,8}

... {3,3,}

Image
Vertex
figure

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,}

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

{3,p,3} polytopes
Space S3 H3
FormFiniteCompactParacompactNoncompact
{3,p,3} {3,3,3} {3,4,3} {3,5,3} {3,6,3} {3,7,3} {3,8,3} ... {3,,3}
Image
Cells
{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,}
Vertex
figure

{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{,3}
{p,3,p} regular honeycombs
Space S3 Euclidean E3 H3
FormFiniteAffineCompactParacompactNoncompact
Name {3,3,3} {4,3,4} {5,3,5} {6,3,6} {7,3,7} {8,3,8} ...{,3,}
Image
Cells
{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}

{,3}
Vertex
figure

{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,}

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. 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.
5. 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.
6. 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. [11]
7. 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. [12] The interior tetrahedra and triangles thus formed will also be orthoschemes.
8. The 120 congruent [13] 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).
9. 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.
10. 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). [14] 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. 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.
5. Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
6. Coxeter 1973, p. 12, §1.8. Configurations.
7. Coxeter 1991, p. 30, §4.2. The Crystallographic regular polytopes.
8. Coxeter 1973, pp. 198–202, §11.7 Regular figures and their truncations.
9. Kim & Rote 2016, pp. 17–20, §10 The Coxeter Classification of Four-Dimensional Point Groups.
10. Coxeter 1973, pp. 130–133, §7.6 The symmetry group of the general regular polytope.
11. Coxeter 1973, p. 130, §7.6; "simplicial subdivision".
12. Coxeter 1973, §3.1 Congruent transformations.
13. Coxeter 1973, pp. 292–293, Table I(ii); "5-cell, 𝛼4".
14. Coxeter 1973, p. 139, §7.9 The characteristic simplex.
15. Coxeter 1973, p. 290, Table I(ii); "dihedral angles".

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References

Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds