Tetrahedron

A **tetrahedron** is a 3-dimensional simplex. Its Bowers acronym is "**tet**". Under the elemental naming scheme it is called a **pyrohedron**.

## Variants

The tetrahedron can be seen as a triangle pyramid where all the sides are equal. It is also a line antiprism.

## Properties

Two tetrahedra can be inscribed in a cube such that no vertex is used twice and the edges are all diagonals of the cube's faces.

The dual of a tetrahedron is another tetrahedron. The dual of a shape is what you get when you turn the faces into vertices, and connect vertices.

The tetrahedron is part of the simplex family of polytopes.

## Symbols

Dynkin symbols for the tetrahedron include:

* x3o3o (regular)
* s4o3o (alternated cube)
* s2s4o (tetragonal disphenoid)
* s2s2s (rhombic disphenoid)
* ox3oo&#x (triangular pyramid)
* xo ox&#x (digonal disphenoid)
* oxo&#x (scalene tet)
* oooo&#x (completely irregular)

## Structure and Sections

### Structure

As a triangular pyramid, the tetrahedron is composed of a point that grows into a triangle. It has 3 triangles around each vertex.

### Sections

When seen vertex first, it is a point that expands into a triangle. When seen edge first, it looks like a line segment expanding into a rectangle, eventually reaching a square as midsection, then reversing in a perpendicular direction, until it becomes a line segment perpendicular to the top.

### Hypervolumes

* vertex count =

4

{\displaystyle 4}
* edge length =

6
l

{\displaystyle 6l}
* surface area =

3

l
2

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

2

12

l
3

{\displaystyle \frac{\sqrt{2}}{12}l^3}

### Subfacets

* 4 points (0D)
* 6 line segments (1D)
* 4 triangles (2D)
* 1 tetrahedron (3D)

### Radii

* Vertex radius:

6

4

l

{\displaystyle \frac{\sqrt{6}{4}}l}
* Edge radius:

2

4

l

{\displaystyle \frac{\sqrt{2}{4}}l}
* Face radius:

6

12

l

{\displaystyle \frac{\sqrt{6}{12}}l}

### Angles

* Dihedral angle:

arccos
⁡
(

1
3

)

{\displaystyle \arccos(\frac{1}{3})}

### Vertex coordinates

The simplest vertex coordinates for a regular tetrahedron can be obtained from alternating a cube. The tetrahedron with side

2

2

{\displaystyle 2\sqrt{2}}

can be given these coordinates:

* (1,1,1)
* (-1,-1,1)
* (-1,1,-1)
* (1,-1,-1)

Alternatively, a tetrahedron can be given as a triangular pyramid with one face parallel to the xy coordinate plane with the coordinates

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

### Notations

* Tapertopic notation:

1
2

{\displaystyle 1^2}

### Related shapes

* Dual: Self-dual
* Vertex figure: Triangle, side length 1

## See also

* Polytope Wiki. "Tetrahedron".
* Bowers, Jonathan. "Polyhedron Category 1: Regulars".

**Self-intersecting regular polyhedra:** gad · sissid · gike · gissid

{
2
,
3
}

{\displaystyle \{2,3\}}

{
3
,
3
}

{\displaystyle \{3, 3\}}

{
4
,
3
}

{\displaystyle \{4,3\}}

{
5
,
3
}

{\displaystyle \{5, 3\}}

{
6
,
3
}

{\displaystyle \{6,3\}}

{
7
,
3
}

{\displaystyle \{7,3\}}

{
8
,
3
}

{\displaystyle \{8,3\}}

{
∞
,
3
}

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

{

π
i

λ

,
3

{\displaystyle \{\frac{\pi i}{\lambda},3}

{
3
,
2
}

{\displaystyle \{3,2\}}

{
3
,
3
}

{\displaystyle \{3, 3\}}

{
3
,
4
}

{\displaystyle \{3, 4\}}

{
3
,
5
}

{\displaystyle \{3, 5\}}

{
3
,
6
}

{\displaystyle \{3,6\}}

{
3
,
7
}

{\displaystyle \{3,7\}}

{
3
,
8
}

{\displaystyle \{3,8\}}

{
3
,
∞
}

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

{
3
,

π
i

λ

}

{\displaystyle \{3,\frac{\pi i}{\lambda}\}}

t
0

{
3
,
3
}

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

t
1

{
3
,
3
}

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

t
2

{
3
,
3
}

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

t

0
,
1

{
3
,
3
}

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

t

1
,
2

{
3
,
3
}

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

t

0
,
2

{
3
,
3
}

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

t

0
,
1
,
2

{
3
,
3
}

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

(
)
∧
{
3
}

{\displaystyle () \wedge \{3\}}

(
)
∧
{
4
}

{\displaystyle () \wedge \{4\}}

(
)
∧
{
5
}

{\displaystyle () \wedge \{5\}}

(
)
∧
{
6
}

{\displaystyle () \wedge \{6\}}