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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.

### Subfacets

• Vertex radius:$\frac{\sqrt{6}{4}}l$
• Edge radius:$\frac{\sqrt{2}{4}}l$
• Face radius:$\frac{\sqrt{6}{12}}l$

### Angles

• Dihedral angle:$\arccos(\frac{1}{3})$

### Vertex coordinates

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

$2\sqrt{2}$ 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:$1^2$

### Related shapes

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

Regular polyhedra
Convex regular polyhedra: tet · cube · oct · doe · ike

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

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
Simplex

$\{3^{n-1}\}$

Null polytope

$)($
$\emptyset$

Point

$()$
$\mathbb{B}^0$

Line segment

$\{\}$
$\mathbb{B}^1$

Triangle

$\{3\}$

Tetrahedron

$\{3^2\}$

Pentachoron

$\{3^3\}$

Hexateron

$\{3^4\}$

Heptapeton

$\{3^5\}$

Octaexon

$\{3^6\}$

Enneazetton

$\{3^7\}$

Decayotton

$\{3^8\}$

Hendecaxennon

$\{3^9\}$

$\{3^{10}\}$

Tridecahendon

$\{3^{11}\}$

$\{3^{12}\}$

$\{3^{13}\}$

$\{3^{14}\}$

$\{3^{15}\}$

... Omegasimplex
Cross

$\{3^{n-2},4\}$

Square

$\{4\}$

Octahedron

$\{3, 4\}$

$\{3^2, 4\}$

Pentacross

$\{3^3, 4\}$

Hexacross

$\{3^4, 4\}$

Heptacross

$\{3^5, 4\}$

Octacross

$\{3^6, 4\}$

Enneacross

$\{3^7, 4\}$

Dekacross

$\{3^8, 4\}$

Hendekacross

$\{3^9, 4\}$

Dodekacross

$\{3^{10}, 4\}$

Tridekacross

$\{3^{11}, 4\}$

$\{3^{12}, 4\}$

$\{3^{13}, 4\}$

$\{3^{14}, 4\}$

... Omegacross
Hydrotopes

$\{3^{n-2}, 5\}$

Pentagon

$\{5\}$

Icosahedron

$\{3, 5\}$

Hexacosichoron

$\{3^2, 5\}$

Order-5 pentachoric tetracomb

$\{3^3, 5\}$

Order-5 hexateric pentacomb

$\{3^4, 5\}$

...
Hypercube

$\{4, 3^{n-2}\}$

Square

$\{4\}$

Cube

$\{4, 3\}$

Tesseract

$\{4, 3^2\}$

Penteract

$\{4, 3^3\}$

Hexeract

$\{4, 3^4\}$

Hepteract

$\{4, 3^5\}$

Octeract

$\{4, 3^6\}$

Enneract

$\{4, 3^7\}$

Dekeract

$\{4, 3^8\}$

Hendekeract

$\{4, 3^9\}$

Dodekeract

$\{4, 3^{10}\}$

Tridekeract

$\{4, 3^{11}\}$

$\{4, 3^{12}\}$

$\{4, 3^{13}\}$

$\{4, 3^{14}\}$

... Omegeract
Cosmotopes

$\{5, 3^{n-2}\}$

Pentagon

$\{5\}$

Dodecahedron

$\{5, 3\}$

Hecatonicosachoron

$\{5, 3^2\}$

Order-3 hecatonicosachoric tetracomb

$\{5, 3^3\}$

Order-3-3 hecatonicosachoric pentacomb

$\{5, 3^4\}$

...
Hyperball

$\mathbb B^n$

Disk

$\mathbb B^2$

Ball

$\mathbb B^3$

Gongol

$\mathbb B^4$

Pentorb

$\mathbb B^5$

Hexorb

$\mathbb B^6$

Heptorb

$\mathbb B^7$

Octorb

$\mathbb B^8$

Enneorb

$\mathbb B^9$

Dekorb

$\mathbb B^{10}$

Hendekorb

$\mathbb B^{11}$

Dodekorb

$\mathbb B^{12}$

Tridekorb

$\mathbb B^{13}$

$\mathbb B^{14}$

$\mathbb B^{15}$

$\mathbb B^{16}$

... Omegaball

$\mathbb B^{\aleph_0}$

$\{2,3\}$ $\{3,3\}$ $\{4,3\}$ $\{5,3\}$ $\{6,3\}$ $\{7,3\}$ $\{8,3\}$ ... $\{\infty,3\}$ $\{\frac{\pi i}{\lambda},3$
Trigonal hosohedron Tetrahedron Cube Dodecahedron Hexagonal tiling Order-3 heptagonal tiling Order-3 octagonal tiling ... Order-3 apeirogonal tiling Order-3 pseudogonal tiling
$\{3,2\}$ $\{3,3\}$ $\{3,4\}$ $\{3,5\}$ $\{3,6\}$ $\{3,7\}$ $\{3,8\}$ ... $\{3,\infty\}$ $\{3, \frac{\pi i}{\lambda}\}$
Trigonal dihedron Tetrahedron Octahedron Icosahedron Triangular tiling Order-7 triangular tiling Order-8 triangular tiling ... Infinite-order triangular tiling Imaginary-order triangular tiling
Regular
$t_0 \{3,3\}$
Rectified
$t_1 \{3,3\}$
Birectified
$t_2 \{3,3\}$
Truncated
$t_{0,1} \{3,3\}$
Bitruncated
$t_{1,2} \{3,3\}$
Cantellated
$t_{0,2} \{3,3\}$
Cantitruncated
$t_{0,1,2} \{3,3\}$
Tetrahedron Octahedron Tetrahedron Truncated tetrahedron Truncated tetrahedron Cuboctahedron Truncated octahedron
$() \wedge \{3\}$ $() \wedge \{4\}$ $() \wedge \{5\}$ $() \wedge \{6\}$
Tetrahedron Square pyramid Pentagonal pyramid Hexagonal pyramid
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