Pentachoron

A **pentachoron** is a 4-dimensional simplex. It is also called the **pyrochoron** under the elemental naming scheme. Its Bowers acronym is "**pen**".

## Symbols

The pentachoron can be given by these Dynkin symbols:

* x3o3o3o (regular)
* ox3oo3oo&#x (tetraherdral pyramid)
* xo ox3oo&#x (trigonal disphenoid)
* oxo3ooo&#x (triangular scalene)
* oxo oox&#xt (disphenoid pyramid)
* ooox&#x (line tettene)
* ooooo&#x (irregular pen)

## Structure and Sections

### Structure

The pentachoron is the pyramid of the tetrahedron, with 4 tetrahedra on each vertex.

### Hypervolumes

* vertex count =

5

{\displaystyle 5}

* edge length =

10
l

{\displaystyle 10l}

* surface area =

5

3

2

l

2

{\displaystyle \frac { 5\sqrt { 3 } }{ 2 } { l }^{ 2 }}

* surcell volume =

2

3

l

3

{\displaystyle \frac { \sqrt { 2 } }{ 3 } { l }^{ 3 }}

* surteron bulk =

5

96

l

4

{\displaystyle \frac { \sqrt { 5 } }{ 96 } { l }^{ 4 }}

### Subfacets

* 5 points (0D)
* 10 line segments (1D)
* 10 triangles (2D)
* 5 tetrahedra (3D)

### Radii

* Vertex radius:

10

5

l

{\displaystyle \frac{\sqrt{10}}{5}l}

* Edge radius:

15

10

l

{\displaystyle \frac{\sqrt{15}}{10}l}

* Face radius:

15

15

l

{\displaystyle \frac{\sqrt{15}}{15}l}

* Cell radius:

10

20

l

{\displaystyle \frac{\sqrt{10}}{20}l}

### Angles

* Dichoral angle:

arccos
⁡
(

1
4

)

{\displaystyle \arccos(\frac{1}{4})}

### Vertex coordinates

The verticeds of a pentachoron can best be represented as a facet of the pentacross, as all permutations of (√2,0,0,0,0). The coordinates of a pentachoron in a 4D space can be given by:

* (±1,-√3/3,-√6/6,-√10/10)
* (0,2√3/3,-√6/6,-√10/10)
* (0,0,√6/2,-√10/10)
* (0,0,0,2√10/5)

### Notations

* Tapertopic notation:

1
3

{\displaystyle 1^3}

### Related shapes

* Dual: Self dual
* Vertex figure: Tetrahedron, side length 1

## See also

**Self-intersecting regular polychora:** fix · gohi · gahi · sishi · gaghi · gishi · gashi · gofix · gax · gogishi
**Tesseractihemioctachoron:** tho

{
2
,
3
,
3
}

{\displaystyle \{2,3,3\}}

{
3
,
3
,
3
}

{\displaystyle \{3,3,3\}}

{
4
,
3
,
3
}

{\displaystyle \{4, 3, 3\}}

{
5
,
3
,
3
}

{\displaystyle \{5,3,3\}}

{
6
,
3
,
3
}

{\displaystyle \{6,3,3\}}

{
3
,
3
,
2
}

{\displaystyle \{3,3,2\}}

{
3
,
3
,
3
}

{\displaystyle \{3,3,3\}}

{
3
,
3
,
4
}

{\displaystyle \{3, 3, 4\}}

{
3
,
3
,
5
}

{\displaystyle \{3,3,5\}}

{
3
,
3
,
6
}

{\displaystyle \{3,3,6\}}

t
0

{
3
,
3
,
3
}

{\displaystyle t\_0 \{3,3,3\}}

t
1

{
3
,
3
,
3
}

{\displaystyle t\_1 \{3,3,3\}}

t
2

{
3
,
3
,
3
}

{\displaystyle t\_2 \{3,3,3\}}

t
3

{
3
,
3
,
3
}

{\displaystyle t\_3 \{3,3,3\}}

t

0
,
1

{
3
,
3
,
3
}

{\displaystyle t\_{0,1} \{3,3,3\}}

t

1
,
2

{
3
,
3
,
3
}

{\displaystyle t\_{1,2} \{3,3,3\}}

t

2
,
3

{
3
,
3
,
3
}

{\displaystyle t\_{2,3} \{3,3,3\}}

t

0
,
2

{
3
,
3
,
3
}

{\displaystyle t\_{0,2} \{3,3,3\}}

t

1
,
3

{
3
,
3
,
3
}

{\displaystyle t\_{1,3} \{3,3,3\}}

t

0
,
1
,
2

{
3
,
3
,
3
}

{\displaystyle t\_{0,1,2} \{3,3,3\}}

t

1
,
2
,
3

{
3
,
3
,
3
}

{\displaystyle t\_{1,2,3} \{3,3,3\}}

t

0
,
3

{
3
,
3
,
3
}

{\displaystyle t\_{0,3} \{3,3,3\}}

t

0
,
2
,
3

{
3
,
3
,
3
}

{\displaystyle t\_{0,2,3} \{3,3,3\}}

t

0
,
1
,
3

{
3
,
3
,
3
}

{\displaystyle t\_{0,1,3} \{3,3,3\}}

t

0
,
1
,
2
,
3

{
3
,
3
,
3
}

{\displaystyle t\_{0,1,2,3} \{3,3,3\}}