Hexadecachoron

A **hexadecachoron** (also called **tetracross**) is the four-dimensional cross polytope and the dual of a tesseract. It is also called a **16-cell**, due to its 16 tetrahedral cells. It has the schläfli symbol

{
3
,
3
,
4
}

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

, meaning that 4 tetrahedra meet at each edge. Its Bowers acronym is "**hex**". It is also called an **aerochoron** in the elemental naming scheme. It is one of three regular polychora that can tile 4-dimensional space, leading to the hexadecachoric tetracomb.

## Properties

The hexadecachoron is the 4-D demihypercube, meaning it can be constructed by taking half the vertices of a tesseract. Because of this it can be called a "**demitesseract**". It is also a bipyramid of an octahedron, or a square duopyramid. Eight tetrahedra join at each vertex.

## Symbols

Dynkin symbols for this shape include:

* o4o3o3x (regular)
* x3o3o \*b3o (demitesseract)
* s4o3o3o, s2s4o3o s4o2s4o, s2s2s4o, s2s2s2s (alternated tesseract)
* xo3oo3ox&#xt (tetrahedral antiprism)
* oxo3ooo4ooo&#xt (octahedral bipyramid)
* xox oxo4ooo&#xt (edge first)
* xoo3oox oqo&#xt (triangle first)
* xo4oo ox4oo&#zx (square duopyramid)
* qo ox3oo4oo&#xt (octahedarl bipyramid)
* xoo4ooo oqo ooq&#zx (square bipyramidal bipyramid)
* qooo oqoo ooqo oooq&#zx (rhombic bipyramidal bipyramid)

## Structure and Sections

### Structure

When seen from one of its cells, the hexadecachoron first has one cell, then four attached to its faces. The next six share an edge with the top cell and are perpendicular to it. The remaining four side cells share only a vertex with the top cell, and are themselves joined to the final cell, in dual orientation to the top cell.

In vertex-first orientation, it is just two octahedral pyramids joined together.

###  Hypervolumes

* vertex count =

8

{\displaystyle 8}
* edge length =

24
l

{\displaystyle 24 l}
* surface area =

8

3

l
2

{\displaystyle 8\sqrt{3}l^2}
* surcell volume =

4

2

3

l
3

{\displaystyle 4\frac{\sqrt{2}}{3}l^3}
* surteron bulk =

l
4

6

{\displaystyle \frac{l^4}{6}}

### Subfacets

* 8 points (0D)
* 24 line segments (1D)
* 32 triangles (2D)
* 16 tetrahedra (3D)

### Radii

* Vertex radius:

2

2

l

{\displaystyle \frac{\sqrt{2}}{2}l}
* Edge radius:

1
2

l

{\displaystyle \frac{1}{2}l}
* Face radius:

6

6

l

{\displaystyle \frac{\sqrt{6}}{6}l}
* Cell radius:

2

4

l

{\displaystyle \frac{\sqrt{2}}{4}l}

### Equations

The surface of a hexadecachoron can be given by

x

y

z

w

=
1

### Vertex coordinates

The vertices of a hexadecachoron of side 2 are all permutations of (±√2,0,0,0).

### Related shapes

* Dual: Tesseract
* Vertex figure: Octahedron, side length 1
* Diminishings/caps: Octahedral pyramid (single), Square pyramidal pyramid (two adjacent)
* Regiment members: 2 (other uniform member: Tesseractihemioctachoron)

## See also

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

{
2
,
3
,
4
}

{\displaystyle \{2,3,4\}}

{
3
,
3
,
4
}

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

{
4
,
3
,
4
}

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

{
5
,
3
,
4
}

{\displaystyle \{5,3,4\}}

{
6
,
3
,
4
}

{\displaystyle \{6,3,4\}}

{
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

{
4
,
3
,
3
}

{\displaystyle t\_0 \{4,3,3\}}

t
1

{
4
,
3
,
3
}

{\displaystyle t\_1 \{4,3,3\}}

t
2

{
4
,
3
,
3
}

{\displaystyle t\_2 \{4,3,3\}}

t
3

{
4
,
3
,
3
}

{\displaystyle t\_3 \{4,3,3\}}

t

0
,
1

{
4
,
3
,
3
}

{\displaystyle t\_{0,1} \{4,3,3\}}

t

1
,
2

{
4
,
3
,
3
}

{\displaystyle t\_{1,2} \{4,3,3\}}

t

2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{2,3} \{4,3,3\}}

t

0
,
2

{
4
,
3
,
3
}

{\displaystyle t\_{0,2} \{4,3,3\}}

t

1
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{1,3} \{4,3,3\}}

t

0
,
1
,
2

{
4
,
3
,
3
}

{\displaystyle t\_{0,1,2} \{4,3,3\}}

t

1
,
2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{1,2,3} \{4,3,3\}}

t

0
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{0,3} \{4,3,3\}}

t

0
,
2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{0,2,3} \{4,3,3\}}

t

0
,
1
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{0,1,3} \{4,3,3\}}

t

0
,
1
,
2
,
3

{
4
,
3
,
3
}

{\displaystyle t\_{0,1,2,3} \{4,3,3\}}

t
0

{
3
,
3
,
b
3
}

{\displaystyle t\_0 \{3,3,b3\}}

t
1

{
3
,
3
,
b
3
}

{\displaystyle t\_1 \{3,3,b3\}}

t
2

{
3
,
3
,
b
3
}

{\displaystyle t\_2 \{3,3,b3\}}

t
3

{
3
,
3
,
b
3
}

{\displaystyle t\_3 \{3,3,b3\}}

t

0
,
1

{
3
,
3
,
b
3
}

{\displaystyle t\_{0,1} \{3,3,b3\}}

t

1
,
2

{
3
,
3
,
b
3
}

{\displaystyle t\_{1,2} \{3,3,b3\}}

t

2
,
3

{
3
,
3
,
b
3
}

{\displaystyle t\_{2,3} \{3,3,b3\}}

t

0
,
2

{
3
,
3
,
b
3
}

{\displaystyle t\_{0,2} \{3,3,b3\}}

t

1
,
3

{
3
,
3
,
b
3
}

{\displaystyle t\_{1,3} \{3,3,b3\}}

t

0
,
1
,
2

{
3
,
3
,
b
3
}

{\displaystyle t\_{0,1,2} \{3,3,b3\}}

t

1
,
2
,
3

{
3
,
3
,
b
3
}

{\displaystyle t\_{1,2,3} \{3,3,b3\}}

t

0
,
3

{
3
,
3
,
b
3
}

{\displaystyle t\_{0,3} \{3,3,b3\}}

t

0
,
2
,
3

{
3
,
3
,
b
3
}

{\displaystyle t\_{0,2,3} \{3,3,b3\}}

t

0
,
1
,
3

{
3
,
3
,
b
3
}

{\displaystyle t\_{0,1,3} \{3,3,b3\}}

t

0
,
1
,
2
,
3

{
3
,
3
,
b
3
}

{\displaystyle t\_{0,1,2,3} \{3,3,b3\}}