5cell (4simplex)  

Type  Convex regular 4polytope 
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  A_{4}, [3,3,3] 
Dual  Selfdual 
Properties  convex, isogonal, isotoxal, isohedral 
Uniform index  1 
In geometry, the 5cell is the convex 4polytope with Schläfli symbol {3,3,3}. It is a 5vertex fourdimensional object bounded by five tetrahedral cells.^{[a]} It is also known as a C_{5}, pentachoron,^{[1]} pentatope, pentahedroid,^{[2]} or tetrahedral pyramid. It is the 4simplex (Coxeter's polytope),^{[3]} the simplest possible convex 4polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5cell is a 4dimensional pyramid with a tetrahedral base and four tetrahedral sides.
The regular 5cell is bounded by five regular tetrahedra, and is one of the six regular convex 4polytopes (the fourdimensional analogues of the Platonic solids). A regular 5cell 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 3dimensional space. The regular 5cell 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 matchsticks intersect one another. No solution exists in three dimensions.
The 5cell is the 4dimensional simplex, the simplest possible 4polytope. As such it is the first in the sequence of 6 convex regular 4polytopes (in order of size and complexity).^{[b]}
Regular convex 4polytopes  

Symmetry group  A_{4}  B_{4}  F_{4}  H_{4}  
Name  5cell Hypertetrahedron 
16cell Hyperoctahedron 
8cell Hypercube 
24cell

600cell Hypericosahedron 
120cell Hyperdodecahedron  
Schläfli symbol  {3, 3, 3}  {3, 3, 4}  {4, 3, 3}  {3, 4, 3}  {3, 3, 5}  {5, 3, 3}  
Coxeter mirrors  
Mirror dihedrals  𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2  𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2  𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2  𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2  
Graph  
Vertices  5 tetrahedral  8 octahedral  16 tetrahedral  24 cubical  120 icosahedral  600 tetrahedral  
Edges  10 triangular  24 square  32 triangular  96 triangular  720 pentagonal  1200 triangular  
Faces  10 triangles  32 triangles  24 squares  96 triangles  1200 triangles  720 pentagons  
Cells  5 tetrahedra  16 tetrahedra  8 cubes  24 octahedra  600 tetrahedra  120 dodecahedra  
Tori  1 5tetrahedron  2 8tetrahedron  2 4cube  4 6octahedron  20 30tetrahedron  12 10dodecahedron  
Inscribed  120 in 120cell  675 in 120cell  2 16cells  3 8cells  25 24cells  10 600cells  
Great polygons  2 squares x 3  4 rectangles x 4  4 hexagons x 4  12 decagons x 6  100 irregular hexagons x 4  
Petrie polygons  1 pentagon x 2  1 octagon x 3  2 octagons x 4  2 dodecagons x 4  4 30gons x 6  20 30gons x 4  
Long radius  
Edge length  
Short radius  
Area  
Volume  
4Content 
A 5cell 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 5cell, though not usually a regular 5cell. The regular 5cell is not found within any of the other regular convex 4polytopes except one: the 600vertex 120cell is a compound of 120 regular 5cells.^{[c]}
When a net of five tetrahedra is folded up in 4dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5cell 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 5cell is selfdual (as are all simplexes), and its vertex figure is the tetrahedron.^{[e]} Its maximal intersection with 3dimensional space is the triangular prism. Its dihedral angle is cos^{−1}(1/4), or approximately 75.52°.
The convex hull of two 5cells in dual configuration is the disphenoidal 30cell, dual of the bitruncated 5cell.
This configuration matrix represents the 5cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This selfdual polytope's matrix is identical to its 180 degree rotation.^{[7]} The kfaces can be read as rows left of the diagonal, while the kfigures are read as rows after the diagonal.^{[8]}
Element  kface  f_{k}  f_{0}  f_{1}  f_{2}  f_{3}  kfigs 

( )  f_{0}  5  4  6  4  {3,3}  
{ }  f_{1}  2  10  3  3  {3}  
{3}  f_{2}  3  3  10  2  { }  
{3,3}  f_{3}  4  6  4  5  ( ) 
All these elements of the 5cell are enumerated in Branko Grünbaum's Venn diagram of 5 points, which is literally an illustration of the regular 5cell in projection to the plane.
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 2√2, where 𝜙 is the golden ratio.^{[9]} While these coordinates are not origincentered, subtracting from each translates the 4polytope's circumcenter to the origin with radius , with the following coordinates:
The following set of origincentered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a regular tetrahedral base in 3space:
Scaling these or the previous set of coordinates by give unitradius origincentered regular 5cells with edge lengths . The hyperpyramid has coordinates:
Coordinates for the vertices of another origincentered regular 5cell with edge length 2 and radius are:
Scaling these by to unitradius and edge length gives:
The vertices of a 4simplex (with edge √2 and radius 1) can be more simply constructed on a hyperplane in 5space, 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 5orthoplex or the rectified penteract.
The 5cell 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 5cell.^{[d]} Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them.^{[h]} The characteristic isoclinic rotation of the 5cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0gon planes which intersect no 5cell vertices.
There are only two ways to make a circuit of the 5cell through all 5 vertices along 5 edges,^{[e]} so there are two discrete Hopf fibrations of the great digons of the 5cell. Each of the two fibrations corresponds to a leftright pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5cell has only two 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.^{[g]}
Below, a spinning 5cell is visualized with the fourth dimension squashed and displayed as colour. The Clifford torus is depicted in its rectangular (wrapping) form.
A 5cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4dimensional 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 4dimensions causes edges to coincide. The purple edges form a regular pentagon which is the Petrie polygon of the 5cell. The blue edges connect every second vertex, forming a pentagram which is the Clifford polygon of the 5cell. The pentagram's blue edges are the chords of the 5cell's isocline, the circular rotational path its vertices take during an isoclinic rotation, also known as a Clifford displacement.^{[i]}
The A_{4} Coxeter plane projects the 5cell into a regular pentagon and pentagram. The A_{3} Coxeter plane projection of the 5cell is that of a square pyramid. The A_{2} Coxeter plane projection of the regular 5cell is that of a triangular bipyramid (two tetrahedra joined facetoface) with the two opposite vertices centered.
A_{k} Coxeter plane 
A_{4}  A_{3}  A_{2} 

Graph  
Dihedral symmetry  [5]  [4]  [3] 
Projections to 3 dimensions  

The vertexfirst projection of the 5cell into 3 dimensions has a tetrahedral projection envelope. The closest vertex of the 5cell 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 edgefirst projection of the 5cell 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 facefirst projection of the 5cell 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 edgefirst projection. 
The cellfirst projection of the 5cell 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. 
In the case of simplexes such as the 5cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5cells cannot fill 4space or the regular 4polytopes, there are irregular 5cells which do. These characteristic 5cells are the fundamental domains of the different symmetry groups which give rise to the various 4polytopes.
A 4orthoscheme is a 5cell where all 10 faces are right triangles.^{[a]} An orthoscheme is an irregular simplex that is the convex hull of a tree in which all edges are mutually perpendicular.^{[j]} In a 4dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three rightangled 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 4orthoscheme is a 3orthoscheme, and each triangular face is a 2orthoscheme (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 4orthoscheme with equallength perpendicular edges is the characteristic orthoscheme of the 4cube (also called the tesseract or 8cell), the 4dimensional analogue of the 3dimensional cube. If the three perpendicular edges of the 4orthoscheme are of unit length, then all its edges are of length √1, √2, √3, or √4, precisely the chord lengths of the unit 4cube (the lengths of the 4cube's edges and its various diagonals). Therefore this 4orthoscheme fits within the 4cube, and the 4cube (like every regular convex polytope) can be dissected into instances of its characteristic orthoscheme.
A 3orthoscheme is easily illustrated, but a 4orthoscheme is more difficult to visualize. A 4orthoscheme is a tetrahedral pyramid with a 3orthoscheme as its base. It has four more edges than the 3orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5cell). Pick out any one of the 3orthoschemes of the six shown in the 3cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3edge path that makes two rightangled turns. Imagine that this 3orthoscheme is the base of a 4orthoscheme, so that from each of those four vertices, an unseen 4orthoscheme edge connects to a fifth apex vertex (which is outside the 3cube 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 3edge 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 3cube, but of another of the tesseract's eight 3cubes).^{[k]} The third additional edge is a √3 diagonal of a 3cube (again, not the original illustrated 3cube). 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 3cube), which is the apex. Thus the characteristic 5cell of the 4cube has four √1 edges, three √2 edges, two √3 edges, and one √4 edge.
The 4cube can be dissected into 24 such 4orthoschemes eight different ways, with six 4orthoschemes surrounding each of four orthogonal √4 tesseract long diameters. The 4cube can also be dissected into 384 smaller instances of this same characteristic 4orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4orthoschemes that all meet at the center of the 4cube.^{[l]}
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 mirrorsurfaced orthoscheme instance is reflected in its own facets.^{[m]} 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 5cell, has its characteristic orthoscheme.^{[n]} There is a 4orthoscheme which is the characteristic 5cell of the regular 5cell. It is a tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 5cell can be dissected into 120 instances of this characteristic 4orthoscheme just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4orthoschemes that all meet at the center of the regular 5cell.^{[o]}
Characteristics of the regular 5cell^{[16]}  

edge^{[17]}  arc  dihedral^{[18]}  
𝒍  104°30′40″  75°29′20″  
𝟀  75°29′20″  60°  
𝝉^{[p]}  52°15′20″  60°  
𝟁  52°15′20″  60°  
75°29′20″  90°  
52°15′20″  90°  
52°15′20″  90°  
37°44′40″ 
The characteristic 5cell (4orthoscheme) of the regular 5cell has four more edges than its base characteristic tetrahedron (3orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4orthoscheme, at the center of the regular 5cell).^{[q]} If the regular 5cell has unit radius and edge length , its characteristic 5cell's ten edges have lengths , , around its exterior righttriangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),^{[p]} plus , , (the other three edges of the exterior 3orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus , , , (edges which are the characteristic radii of the regular 5cell). The 4edge path along orthogonal edges of the orthoscheme is , , , , first from a regular 5cell vertex to a regular 5cell edge center, then turning 90° to a regular 5cell face center, then turning 90° to a regular 5cell tetrahedral cell center, then turning 90° to the regular 5cell center.^{[r]}
There are many lower symmetry forms of the 5cell, 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 5cell  Tetrahedral pyramid  Triangular pyramidal pyramid  
Schläfli  {3,3,3}  {3,3}∨( )  {3}∨{ }  {3}∨( )∨( )  
Example Vertex figure 
5simplex 
Truncated 5simplex 
Bitruncated 5simplex 
Cantitruncated 5simplex 
Omnitruncated 4simplex honeycomb 
The tetrahedral pyramid is a special case of a 5cell, a polyhedral pyramid, constructed as a regular tetrahedron base in a 3space hyperplane, and an apex point above the hyperplane. The four sides of the pyramid are made of triangular pyramid cells.
Many uniform 5polytopes have tetrahedral pyramid vertex figures with Schläfli symbols ( )∨{3,3}.
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 5polytopes have irregular 5cell 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 
t_{12}α_{5} 
t_{12}γ_{5} 
t_{012}α_{5} 
t_{012}γ_{5} 
t_{123}α_{5} 
t_{123}γ_{5} 
Symmetry  [2,1,1], order 2  [2^{+},1,1], order 2  [ ]^{+}, order 1  

Schläfli  { }∨( )∨( )∨( )  ( )∨( )∨( )∨( )∨( )  
Schlegel diagram 

Name Coxeter 
t_{0123}α_{5} 
t_{0123}γ_{5} 
t_{0123}β_{5} 
t_{01234}α_{5} 
t_{01234}γ_{5} 
The compound of two 5cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and blue 5cell vertices and edges. This compound has [[3,3,3]] symmetry, order 240. The intersection of these two 5cells is a uniform bitruncated 5cell. = ∩ .
This compound can be seen as the 4D analogue of the 2D hexagram {6/2} and the 3D compound of two tetrahedra.
The pentachoron (5cell) 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}  t_{0,3}{3,3,3}  t_{0,1,3}{3,3,3}  t_{0,1,2,3}{3,3,3} 

Coxeter  
Schlegel 
1_{k2} figures in n dimensions  

Space  Finite  Euclidean  Hyperbolic  
n  3  4  5  6  7  8  9  10  
Coxeter group 
E_{3}=A_{2}A_{1}  E_{4}=A_{4}  E_{5}=D_{5}  E_{6}  E_{7}  E_{8}  E_{9} = = E_{8}^{+}  E_{10} = = E_{8}^{++}  
Coxeter diagram 

Symmetry (order) 
[3^{−1,2,1}]  [3^{0,2,1}]  [3^{1,2,1}]  [[3^{2,2,1}]]  [3^{3,2,1}]  [3^{4,2,1}]  [3^{5,2,1}]  [3^{6,2,1}]  
Order  12  120  1,920  103,680  2,903,040  696,729,600  ∞  
Graph      
Name  1_{−1,2}  1_{02}  1_{12}  1_{22}  1_{32}  1_{42}  1_{52}  1_{62} 
2_{k1} figures in n dimensions  

Space  Finite  Euclidean  Hyperbolic  
n  3  4  5  6  7  8  9  10  
Coxeter group 
E_{3}=A_{2}A_{1}  E_{4}=A_{4}  E_{5}=D_{5}  E_{6}  E_{7}  E_{8}  E_{9} = = E_{8}^{+}  E_{10} = = E_{8}^{++}  
Coxeter diagram 

Symmetry  [3^{−1,2,1}]  [3^{0,2,1}]  [[3^{1,2,1}]]  [3^{2,2,1}]  [3^{3,2,1}]  [3^{4,2,1}]  [3^{5,2,1}]  [3^{6,2,1}]  
Order  12  120  384  51,840  2,903,040  696,729,600  ∞  
Graph      
Name  2_{−1,1}  2_{01}  2_{11}  2_{21}  2_{31}  2_{41}  2_{51}  2_{61} 
It is in the {p,3,3} sequence of regular polychora with a tetrahedral vertex figure: the tesseract {4,3,3} and 120cell {5,3,3} of Euclidean 4space, and the hexagonal tiling honeycomb {6,3,3} of hyperbolic space.^{[e]}
{p,3,3} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Paracompact  Noncompact  
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 4polytopes with tetrahedral cells, along with the 16cell {3,3,4} and 600cell {3,3,5}. The order6 tetrahedral honeycomb {3,3,6} of hyperbolic space also has tetrahedral cells.
{3,3,p} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Paracompact  Noncompact  
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 selfdual like the 24cell {3,4,3}, having a palindromic {3,p,3} Schläfli symbol.
{3,p,3} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Compact  Paracompact  Noncompact  
{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  S^{3}  Euclidean E^{3}  H^{3}  
Form  Finite  Affine  Compact  Paracompact  Noncompact  
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,∞} 