5cell
 REDIRECT Template:Infobox 4polytope
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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.^{[loweralpha 1]} It is also known as a C_{5}, pentachoron,^{[1]} pentatope, pentahedroid,^{[2]} or tetrahedral pyramid. It is the 4simplex (Coxeter's [math]\displaystyle{ \alpha_4 }[/math] 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 match sticks intersect one another. No solution exists in three dimensions.
Alternative names
 Pentachoron (5point 4polytope)
 Hypertetrahedron (4dimensional analogue of the tetrahedron)
 4simplex (4dimensional simplex)
 Tetrahedral pyramid (4dimensional hyperpyramid with a tetrahedral base)
 Pentatope
 Pentahedroid (Henry Parker Manning)
 Pen (Jonathan Bowers: for pentachoron)^{[4]}
Geometry
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).^{[loweralpha 2]}
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.^{[loweralpha 3]}
Structure
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.^{[loweralpha 5]} 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.
As a configuration
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.
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 2√2, where φ is the golden ratio.^{[9]} While these coordinates are not origincentered, subtracting [math]\displaystyle{ (1,1,1,1)/(2\tfrac{1}{\varphi}) }[/math] from each translates the 4polytope's circumcenter to the origin with radius [math]\displaystyle{ 2(\varphi1/(2\tfrac{1}{\varphi})) =\sqrt{\tfrac{16}{5}}\approx 1.7888 }[/math], with the following coordinates:
 [math]\displaystyle{ \left(\tfrac{2}{\varphi}3, 1, 1, 1\right)/(\tfrac{1}{\varphi}2) }[/math]
 [math]\displaystyle{ \left(1,\tfrac{2}{\varphi}3,1,1 \right)/(\tfrac{1}{\varphi}2) }[/math]
 [math]\displaystyle{ \left(1,1,\tfrac{2}{\varphi}3,1 \right)/(\tfrac{1}{\varphi}2) }[/math]
 [math]\displaystyle{ \left(1,1,1,\tfrac{2}{\varphi}3 \right)/(\tfrac{1}{\varphi}2) }[/math]
 [math]\displaystyle{ \left(\tfrac{2}{\varphi},\tfrac{2}{\varphi},\tfrac{2}{\varphi},\tfrac{2}{\varphi} \right)/(\tfrac{1}{\varphi}2) }[/math]
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:
 [math]\displaystyle{ \left( 1, 1, 1, \frac{1}\sqrt{5}\right) }[/math]
 [math]\displaystyle{ \left( 1,1,1,\frac{1}\sqrt{5} \right) }[/math]
 [math]\displaystyle{ \left(1, 1,1,\frac{1}\sqrt{5} \right) }[/math]
 [math]\displaystyle{ \left(1,1, 1,\frac{1}\sqrt{5} \right) }[/math]
 [math]\displaystyle{ \left( 0, 0, 0,\frac{4}\sqrt{5} \right) }[/math]
Scaling these or the previous set of coordinates by [math]\displaystyle{ \tfrac{\sqrt{5}}{4} }[/math] give unitradius origincentered regular 5cells with edge lengths [math]\displaystyle{ \sqrt{\tfrac{5}{2}} }[/math]. The hyperpyramid has coordinates:
 [math]\displaystyle{ \left( \sqrt{5}, \sqrt{5}, \sqrt{5}, 1 \right)/4 }[/math]
 [math]\displaystyle{ \left( \sqrt{5},\sqrt{5},\sqrt{5}, 1 \right)/4 }[/math]
 [math]\displaystyle{ \left(\sqrt{5}, \sqrt{5},\sqrt{5}, 1 \right)/4 }[/math]
 [math]\displaystyle{ \left(\sqrt{5},\sqrt{5}, \sqrt{5}, 1 \right)/4 }[/math]
 [math]\displaystyle{ \left( 0, 0, 0, 1 \right) }[/math]
Coordinates for the vertices of another origincentered regular 5cell with edge length 2 and radius [math]\displaystyle{ \sqrt{\tfrac{8}{5}}\approx 1.265 }[/math] are:
 [math]\displaystyle{ \left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right) }[/math]
 [math]\displaystyle{ \left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ 0 \right) }[/math]
 [math]\displaystyle{ \left( \frac{1}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ 0,\ 0 \right) }[/math]
 [math]\displaystyle{ \left( 2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right) }[/math]
Scaling these by [math]\displaystyle{ \sqrt{\tfrac{5}{8}} }[/math] to unitradius and edge length [math]\displaystyle{ \sqrt{\tfrac{5}{2}} }[/math] gives:
 [math]\displaystyle{ \left(\sqrt{3}, \sqrt{5}, \sqrt{10},\pm\sqrt{30} \right)/(4\sqrt{3}) }[/math]
 [math]\displaystyle{ \left(\sqrt{3}, \sqrt{5}, \sqrt{40},0\right)/(4\sqrt{3}) }[/math]
 [math]\displaystyle{ \left(\sqrt{3},\sqrt{45},0,0\right)/(4\sqrt{3}) }[/math]
 [math]\displaystyle{ \left(1, 0, 0, 0 \right) }[/math]
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.
Geodesics and rotations
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.^{[loweralpha 4]} Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them.^{[loweralpha 7]} 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 exactly three different ways to make a circuit of the 5cell through all 5 vertices along 5 edges,^{[loweralpha 5]} so there are three discrete Hopf fibrations of the great digons of the 5cell. Each of the three fibrations corresponds to a distinct leftright pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5cell 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 5cell in the third fibration.^{[loweralpha 6]}
Boerdijk–Coxeter helix
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.^{[loweralpha 8]}
Projections
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.
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. 
Irregular 5cells
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.
Orthoschemes
A 4orthoscheme is a 5cell where all 10 faces are right triangles.^{[loweralpha 1]} An orthoscheme is an irregular simplex that is the convex hull of a tree in which all edges are mutually perpendicular.^{[loweralpha 9]} 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).^{[loweralpha 10]} 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.^{[loweralpha 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 mirrorsurfaced orthoscheme instance is reflected in its own facets.^{[loweralpha 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 5cell, has its characteristic orthoscheme.^{[loweralpha 13]} 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.^{[loweralpha 14]}
Characteristics of the regular 5cell^{[16]}  

edge^{[17]}  arc  dihedral^{[18]}  
𝒍  [math]\displaystyle{ \sqrt{\tfrac{5}{2}} \approx 1.581 }[/math]  104°30′40″  [math]\displaystyle{ \pi  2\text{𝜂} }[/math]  75°29′20″  [math]\displaystyle{ \pi  2\text{𝟁} }[/math] 
𝟀  [math]\displaystyle{ \sqrt{\tfrac{1}{10}} \approx 0.316 }[/math]  75°29′20″  [math]\displaystyle{ 2\text{𝜂} }[/math]  60°  [math]\displaystyle{ \tfrac{\pi}{3} }[/math] 
𝝉^{[loweralpha 15]}  [math]\displaystyle{ \sqrt{\tfrac{1}{30}} \approx 0.183 }[/math]  52°15′20″  [math]\displaystyle{ \tfrac{\pi}{2}\text{𝜂} }[/math]  60°  [math]\displaystyle{ \tfrac{\pi}{3} }[/math] 
𝟁  [math]\displaystyle{ \sqrt{\tfrac{2}{15}} \approx 0.103 }[/math]  52°15′20″  [math]\displaystyle{ \tfrac{\pi}{2}\text{𝜂} }[/math]  60°  [math]\displaystyle{ \tfrac{\pi}{3} }[/math] 
[math]\displaystyle{ _0R^3/l }[/math]  [math]\displaystyle{ \sqrt{\tfrac{3}{20}} \approx 0.387 }[/math]  75°29′20″  [math]\displaystyle{ 2\text{𝜂} }[/math]  90°  [math]\displaystyle{ \tfrac{\pi}{2} }[/math] 
[math]\displaystyle{ _1R^3/l }[/math]  [math]\displaystyle{ \sqrt{\tfrac{1}{20}} \approx 0.224 }[/math]  52°15′20″  [math]\displaystyle{ \tfrac{\pi}{2}\text{𝜂} }[/math]  90°  [math]\displaystyle{ \tfrac{\pi}{2} }[/math] 
[math]\displaystyle{ _2R^3/l }[/math]  [math]\displaystyle{ \sqrt{\tfrac{1}{60}} \approx 0.129 }[/math]  52°15′20″  [math]\displaystyle{ \tfrac{\pi}{2}\text{𝜂} }[/math]  90°  [math]\displaystyle{ \tfrac{\pi}{2} }[/math] 
[math]\displaystyle{ _0R^4/l }[/math]  [math]\displaystyle{ \sqrt{\tfrac{4}{25}} = 0.4 }[/math]  
[math]\displaystyle{ _1R^4/l }[/math]  [math]\displaystyle{ \sqrt{\tfrac{3}{50}} \approx 0.245 }[/math]  
[math]\displaystyle{ _2R^4/l }[/math]  [math]\displaystyle{ \sqrt{\tfrac{2}{43}} \approx 0.216 }[/math]  
[math]\displaystyle{ _3R^4/l }[/math]  [math]\displaystyle{ \sqrt{\tfrac{1}{100}} = 0.1 }[/math]  
[math]\displaystyle{ \text{𝜼} }[/math]  37°44′40″  [math]\displaystyle{ \tfrac{\text{arc sec }4}{2} }[/math] 
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).^{[loweralpha 16]} If the regular 5cell has unit radius and edge length 𝒍 = [math]\displaystyle{ \sqrt{\tfrac{5}{2}} }[/math], its characteristic 5cell's ten edges have lengths [math]\displaystyle{ \sqrt{\tfrac{1}{10}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{30}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{2}{15}} }[/math] around its exterior righttriangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),^{[loweralpha 15]} plus [math]\displaystyle{ \sqrt{\tfrac{3}{20}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{20}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{60}} }[/math] (the other three edges of the exterior 3orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus [math]\displaystyle{ \sqrt{\tfrac{4}{25}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{3}{50}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{2}{43}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{100}} }[/math] (edges which are the characteristic radii of the regular 5cell). The 4edge path along orthogonal edges of the orthoscheme is [math]\displaystyle{ \sqrt{\tfrac{1}{30}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{2}{15}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{60}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{100}} }[/math], 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.^{[loweralpha 17]}
Isometries
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} 
Compound
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.
Related polytopes and honeycombs
The pentachoron (5cell) 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 120cell {5,3,3} of Euclidean 4space, and the hexagonal tiling honeycomb {6,3,3} of hyperbolic space.^{[loweralpha 5]}
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.
It is selfdual like the 24cell {3,4,3}, having a palindromic {3,p,3} Schläfli symbol.
Notes
 ↑ ^{1.0} ^{1.1} A 5cell's 5 vertices form 5 tetrahedral cells facebonded to each other, with a total of 10 edges and 10 triangular faces.
 ↑ The convex regular 4polytopes can be ordered by size as a measure of 4dimensional 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 4simplex (5cell) is the limit smallest case, and the 120cell 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 5cell is the 5point 4polytope: first in the ascending sequence that runs to the 600point 4polytope.
 ↑ The regular 120cell has a curved 3dimensional boundary surface consisting of 120 regular dodecahedron cells. It also has 120 disjoint regular 5cells inscribed in it.^{[6]} These are not 3dimensional cells but 4dimensional objects which share the 120cell's center point, and collectively cover all 600 of its vertices.
 ↑ ^{4.0} ^{4.1} In a polytope with a tetrahedral vertex figure,^{[loweralpha 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 edgepath 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.^{[loweralpha 6]}
 ↑ ^{5.0} ^{5.1} ^{5.2} ^{5.3} ^{5.4} The Schlegel diagram of the 5cell (at the top of this article) illustrates its tetrahedral vertex figure. Six of the 5cell'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 5cell 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 5cell's three distinct Petrie polygons. Each is a different sequence of 5 of the 10 edges, and there are only three such distinct sequences.^{[loweralpha 4]}
 ↑ ^{6.0} ^{6.1} ^{6.2} The 5cell (4simplex) is unique among regular 4polytopes in that its isocline chords^{[loweralpha 8]} are its own edges. In the other regular 4polytopes, the isocline chord is the longer edge of another regular polytope that is inscribed. Another aspect of this uniqueness is that the 5cell's isocline Clifford polygon (the skew pentagram) and its zigzag Petrie polygon (the skew pentagon) are exactly the same object; in the other regular 4polytopes they are quite different.
 ↑ 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.
 ↑ ^{8.0} ^{8.1} 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 5cell, 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 5cell's 10 edges.^{[loweralpha 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.^{[loweralpha 6]}
 ↑ A right triangle is a 2dimensional orthoscheme; orthoschemes are the generalization of right triangles to n dimensions. A 3dimensional orthoscheme is a tetrahedron with 4 right triangle faces (not necessarily similar).
 ↑ The 4cube (tesseract) contains eight 3cubes (so it is also called the 8cell). Each 3cube is facebonded to six others (that entirely surround it), but entirely disjoint from the one other 3cube which lies opposite and parallel to it on the other side of the 8cell.
 ↑ The dissection of the 4cube into 384 4orthoschemes is 16 of the dissections into 24 4orthoschemes. First, each 4cube 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 4cube is divided into 16 smaller 4cubes. Then each smaller 4cube is divided into 24 4orthoschemes that meet at the center of the original 4cube.
 ↑ For a regular kpolytope, the CoxeterDynkin diagram of the characteristic korthoscheme is the kpolytope's diagram without the generating point ring. The regular kpolytope is subdivided by its symmetry (k1)elements into g instances of its characteristic korthoscheme that surround its center, where g is the order of the kpolytope's symmetry group.^{[13]}
 ↑ A regular polytope of dimension k has a characteristic korthoscheme, and also a characteristic (k1)orthoscheme. A regular 4polytope has a characteristic 5cell (4orthoscheme) into which it is subdivided by its (3dimensional) hyperplanes of symmetry, and also a characteristic tetrahedron (3orthoscheme) into which its surface is subdivided by its cells' (2dimensional) planes of symmetry. After subdividing its (3dimensional) surface into characteristic tetrahedra surrounding each cell center, its (4dimensional) interior can be subdivided into characteristic 5cells by adding radii joining the vertices of the surface characteristic tetrahedra to the 4polytope's center.^{[14]} The interior tetrahedra and triangles thus formed will also be orthoschemes.
 ↑ The 120 congruent^{[15]} 4orthoschemes of the regular 5cell occur in two mirrorimage forms, 60 of each. Each 4orthoscheme is cellbonded to 4 others of the opposite chirality (by the 4 of its 5 tetrahedral cells that lie in the interior of the regular 5cell). If the 60 lefthanded 4orthoschemes are colored red and the 60 righthanded 4orthoschemes are colored black, each red 5cell is surrounded by 4 black 5cells and vice versa, in a pattern 4dimensionally analogous to a checkerboard (if checkerboards had triangles instead of squares).
 ↑ ^{15.0} ^{15.1} (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.
 ↑ The four edges of each 4orthoscheme which meet at the center of a regular 4polytope are of unequal length, because they are the four characteristic radii of the regular 4polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4orthoscheme always include one regular 4polytope vertex, one regular 4polytope edge center, one regular 4polytope face center, one regular 4polytope cell center, and the regular 4polytope 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 4orthoscheme. The 4orthoscheme has five dissimilar 3orthoscheme facets.
 ↑ If the regular 5cell has radius [math]\displaystyle{ \sqrt{\tfrac{2}{5}} \approx 0.632 }[/math] and edge length 𝒍 = 1, its characteristic 5cell's ten edges have lengths [math]\displaystyle{ \sqrt{\tfrac{1}{3}} \approx 0.577 }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{4}} {{=}} 0.5 }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{12}} \approx 0.289 }[/math] (the exterior right triangle face, the characteristic triangle), plus [math]\displaystyle{ \sqrt{\tfrac{3}{8}} \approx 0.612 }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{8}} \approx 0.354 }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{24}} \approx 0.204 }[/math] (the other three edges of the exterior 3orthoscheme facet the characteristic tetrahedron), plus [math]\displaystyle{ \sqrt{\tfrac{4}{10}} \approx 0.632 }[/math], [math]\displaystyle{ \sqrt{\tfrac{3}{20}} \approx 0.387 }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{8.6}} \approx 0.341 }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{40}} \approx 0.158 }[/math] (edges that are the characteristic radii of the regular 5cell).^{[16]} The 4edge path along orthogonal edges of the orthoscheme is [math]\displaystyle{ \sqrt{\tfrac{1}{4}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{12}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{24}} }[/math], [math]\displaystyle{ \sqrt{\tfrac{1}{40}} }[/math].
Citations
 ↑ N.W. Johnson: Geometries and Transformations, (2018) ISBN:9781107103405 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
 ↑ Matila Ghyka, The geometry of Art and Life (1977), p.68
 ↑ Coxeter 1973, p. 120, §7.2. see illustration Fig 7.2A.
 ↑ Category 1: Regular Polychora
 ↑ Coxeter 1973, pp. 292293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.
 ↑ Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
 ↑ Coxeter 1973, p. 12, §1.8. Configurations.
 ↑ "Pen". https://bendwavy.org/klitzing/incmats/pen.htm.
 ↑ Coxeter 1991, p. 30, §4.2. The Crystallographic regular polytopes.
 ↑ Banchoff 2013.
 ↑ Coxeter 1973, pp. 198202, §11.7 Regular figures and their truncations.
 ↑ Kim & Rote 2016, pp. 1720, §10 The Coxeter Classification of FourDimensional Point Groups.
 ↑ Coxeter 1973, pp. 130133, §7.6 The symmetry group of the general regular polytope.
 ↑ Coxeter 1973, p. 130, §7.6; "simplicial subdivision".
 ↑ Coxeter 1973, §3.1 Congruent transformations.
 ↑ ^{16.0} ^{16.1} Coxeter 1973, pp. 292293, Table I(ii); "5cell, 𝛼_{4}".
 ↑ Coxeter 1973, p. 139, §7.9 The characteristic simplex.
 ↑ Coxeter 1973, p. 290, Table I(ii); "dihedral angles".
References
 T. Gosset: On the Regular and SemiRegular 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 ndimensions (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, WileyInterscience Publication, 1995, ISBN:9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover.
 Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, The Symmetries of Things 2008, ISBN:9781568812205 (Chapter 26. pp. 409: Hemicubes: 1_{n1})
 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 4space". in Senechal, Marjorie. Shaping Space. Springer New York. pp. 257–266. doi:10.1007/9780387927145_20. ISBN 9780387927138. https://archive.org/details/shapingspaceexpl00sene.
External links
 Weisstein, Eric W.. "Pentatope". http://mathworld.wolfram.com/Pentatope.html.
 Klitzing, Richard. "4D uniform polytopes (polychora) x3o3o3o  pen". https://bendwavy.org/klitzing/dimensions/polychora.htm.
 Der 5Zeller (5cell) Marco Möller's Regular polytopes in R^{4} (German)
 Jonathan Bowers, Regular polychora
 Java3D Applets
 pyrochoron
Original source: https://en.wikipedia.org/wiki/5cell.
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