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In geometry , the 5-CELL is a four-dimensional object bounded by 5 tetrahedral cells . It is also known as a C5, PENTACHORON, PENTATOPE, PENTAHEDROID, or TETRAHEDRAL PYRAMID . It is a 4-SIMPLEX , the simplest possible convex regular 4-polytope (four-dimensional analogue of a Platonic solid
Platonic solid
), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The pentachoron is a four dimensional pyramid with a tetrahedral base.

The REGULAR 5-CELL is bounded by regular tetrahedra , and is one of the six regular convex 4-polytopes , represented by Schläfli symbol {3,3,3}.

CONTENTS

* 1 Alternative names

* 2 Geometry
Geometry

* 2.1 Construction * 2.2 Boerdijk–Coxeter helix * 2.3 Projections

* 3 Irregular 5-cell
5-cell
* 4 Compound * 5 Related polytopes and honeycomb * 6 References * 7 External links

ALTERNATIVE NAMES

* Pentachoron * 4-simplex * Pentatope * Pentahedroid (Henry Parker Manning) * Pen (Jonathan Bowers: for pentachoron) * Hyperpyramid, tetrahedral pyramid

GEOMETRY

The 5-cell
5-cell
is self-dual , and its vertex figure is a tetrahedron. Its maximal intersection with 3-dimensional space is the triangular prism . Its dihedral angle is cos−1(1/4), or approximately 75.52°.

CONSTRUCTION

The 5-cell
5-cell
can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. (The 5-cell
5-cell
is essentially a 4-dimensional pyramid with a tetrahedral base.)

The simplest set of coordinates is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (τ,τ,τ,τ), with edge length 2√2, where τ is the golden ratio .

The Cartesian coordinates
Cartesian coordinates
of the vertices of an origin-centered regular 5-cell
5-cell
having edge length 2 are: ( 1 10 , 1 6 , 1 3 , 1 ) {displaystyle left({frac {1}{sqrt {10}}}, {frac {1}{sqrt {6}}}, {frac {1}{sqrt {3}}}, pm 1right)} ( 1 10 , 1 6 , 2 3 , 0 ) {displaystyle left({frac {1}{sqrt {10}}}, {frac {1}{sqrt {6}}}, {frac {-2}{sqrt {3}}}, 0right)} ( 1 10 , 3 2 , 0 , 0 ) {displaystyle left({frac {1}{sqrt {10}}}, -{sqrt {frac {3}{2}}}, 0, 0right)} ( 2 2 5 , 0 , 0 , 0 ) {displaystyle left(-2{sqrt {frac {2}{5}}}, 0, 0, 0right)}

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 2√2: ( 1 , 1 , 1 , 1 / 5 ) {displaystyle left(1,1,1,-1/{sqrt {5}}right)} ( 1 , 1 , 1 , 1 / 5 ) {displaystyle left(1,-1,-1,-1/{sqrt {5}}right)} ( 1 , 1 , 1 , 1 / 5 ) {displaystyle left(-1,1,-1,-1/{sqrt {5}}right)} ( 1 , 1 , 1 , 1 / 5 ) {displaystyle left(-1,-1,1,-1/{sqrt {5}}right)} ( 0 , 0 , 0 , 5 1 / 5 ) {displaystyle left(0,0,0,{sqrt {5}}-1/{sqrt {5}}right)}

The vertices of a 4-simplex (with edge √2) 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
5-orthoplex
or the rectified penteract .

BOERDIJK–COXETER HELIX

A 5-cell
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
Coxeter plane
projects the 5-cell
5-cell
into a regular pentagon and pentagram .

orthographic projections Ak Coxeter plane
Coxeter plane
A4 A3 A2

GRAPH

DIHEDRAL SYMMETRY

PROJECTIONS TO 3 DIMENSIONS

Stereographic projection
Stereographic projection
wireframe (edge projected onto a 3-sphere
3-sphere
)

A 3D projection of a 5-cell
5-cell
performing a simple rotation

The vertex-first projection of the 5-cell
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
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
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 projects 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
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-CELL

There are many lower symmetry forms, including these found in uniform polytope vertex figures :

SYMMETRY

Order 120 Order 24 Order 12 Order 6 + Order 10

NAME Regular 5-cell Tetrahedral pyramid Triangular-pyramidal pyramid Pentagonal hyperdisphenoid

SCHLäFLI SYMBOL {3,3,3} {3,3} ∨ ( ) {3} ∨ { }

Example Vertex figure 5-simplex
5-simplex
Truncated 5-simplex
5-simplex
Bitruncated 5-simplex
5-simplex
Cantitruncated 5-simplex
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 :

Symmetry , order 24 Schlegel diagram

Name Coxeter diagram { }×{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 5-polytopes have irregular 5-cell
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 , ORDER 12 , ORDER 6 , ORDER 8 , ORDER 4

Schlegel diagram

Name Coxeter diagram t12α5 t12γ5 t012α5 t012γ5 t123α5 t123γ5

SYMMETRY , ORDER 2 , ORDER 2 +, ORDER 1

Schlegel diagram

Name Coxeter diagram t0123α5 t0123γ5 t0123β5 t01234α5 t01234γ5

COMPOUND

The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane
Coxeter plane
projection, with a red and blue 5-cell
5-cell
vertices and edges. This compound has ] symmetry, order 240. The intersection of these two 5-cells is a uniform birectified 5-cell
5-cell
. = ∩ .

RELATED POLYTOPES AND HONEYCOMB

The pentachoron (5-cell) is the simplest of 9 uniform polychora constructed from the Coxeter group
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

SPACE FINITE EUCLIDEAN HYPERBOLIC

N 3 4 5 6 7 8 9 10

Coxeter group E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = E 8 {displaystyle {tilde {E}}_{8}} = E8+ E10 = T 8 {displaystyle {bar {T}}_{8}} = E8++

Coxeter diagram

Symmetry (order)

]

ORDER 12 120 192 103,680 2,903,040 696,729,600 ∞

GRAPH

- -

NAME 1−1,2 102 112 122 132 142 152 162

2K1 FIGURES IN N DIMENSIONS

SPACE FINITE EUCLIDEAN HYPERBOLIC

N 3 4 5 6 7 8 9 10

Coxeter group E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = E 8 {displaystyle {tilde {E}}_{8}} = E8+ E10 = T 8 {displaystyle {bar {T}}_{8}} = E8++

Coxeter diagram

SYMMETRY

]

ORDER 12 120 384 51,840 2,903,040 696,729,600 ∞

GRAPH

- -

NAME 2−1,1 201 211 221 231 241 251 261

It is in the sequence of regular polychora : the tesseract {4,3,3}, 120-cell
120-cell
{5,3,3}, of Euclidean 4-space, and hexagonal tiling honeycomb {6,3,3} of hyperbolic space. All of these have a tetrahedral vertex figure .

{P,3,3} POLYTOPES

SPACE S3 H3

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 similar to three regular polychora : the tesseract {4,3,3}, 600-cell
600-cell
{3,3,5} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cell.

{3,3,P} POLYTOPES

SPACE S3 H3

FORM FINITE PARACOMPACT NONCOMP