# Tetrahedron

Tetrahedron | |
---|---|

Rank | 3 |

Type | Regular |

Space | Spherical |

Notation | |

Bowers style acronym | Tet |

Coxeter diagram | x3o3o () |

Schläfli symbol | {3,3} |

Tapertopic notation | 1^{2} |

Stewart notation | Y_{3} |

Elements | |

Faces | 4 triangles |

Edges | 6 |

Vertices | 4 |

Vertex figure | Equilateral triangle, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Edge radius | |

Inradius | |

Volume | |

Dihedral angle | |

Heights | Point atop trig: |

Dyad atop perp dyad: | |

Central density | 1 |

Number of pieces | 4 |

Level of complexity | 1 |

Related polytopes | |

Army | Tet |

Regiment | Tet |

Dual | Tetrahedron |

Petrie dual | Petrial tetrahedron |

Conjugate | None |

Abstract properties | |

Flag count | 24 |

Net count | 2 |

Euler characteristic | 2 |

Topological properties | |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | A_{3}, order 24 |

Convex | Yes |

Nature | Tame |

The **tetrahedron** or **tet**, also sometimes called the **3-simplex**, is the simplest possible non-degenerate polyhedron. The full symmetry version has 4 equilateral triangles as faces, joining 3 to a vertex, and is one of the 5 Platonic solids. It is the 3-dimensional simplex.

It is the uniform digonal antiprism and regular-faced triangular pyramid. Both of these forms are convex segmentohedra.

A regular tetrahedron of edge length √2 can be inscribed in the unit cube. In fact the tetrahedron is the alternated cube, which makes it the 3D demihypercube. The next larger simplex that can be inscribed in a hypercube is the octaexon.

The tetrahedron occurs as cells of three of the six convex regular polychora, namely the pentachoron, hexadecachoron, and hexacosichoron.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a tetrahedron can be given by all even changes of sign of:

These arise from the fact that a tetrahedron can be constructed as the alternation of the cube.

Alternate coordinates can be derived from those of the triangle, by considering the tetrahedron as a triangular pyramid:

These are more complicated, but generalize to simplexes of any dimension.

Simpler coordinates can be given in four dimensions, as all permutations of:

## Representations[edit | edit source]

A regular tetrahedron can be represented by the following Coxeter diagrams:

- x3o3o (full symmetry)
- s2s4o (digonal antiprism, is generally a tetragonal disphenoid)
- s2s2s (alternated cuboid, generally a rhombic disphenoid)
- ox3oo&#x (A2 axial, generally a triangular pyramid)
- xo ox&#x (A1×A1 axial, generally a digonal disphenoid)
- oox&#x (A1 only, generally a sphenoid)
- oooo&#x (no symmetry, fully irregular tetrahedron)

## In vertex figures[edit | edit source]

Name | Picture | Schläfli symbol | Edge length |
---|---|---|---|

Pentachoron | {3,3,3} | ||

Tesseract | {4,3,3} | ||

Hecatonicosachoron | {5,3,3} | ||

Great grand stellated hecatonicosachoron | {5/2,3,3} | ||

Hexagonal tiling honeycomb | {6,3,3} |

## Related polyhedra[edit | edit source]

Two tetrahedra can be attached at a common face to form a triangular tegum, one of the Johnson solids.

A tetrahedron can also be elongated by attaching a triangular prism to one of the faces, forming the elongated triangular pyramid.

A number of uniform polyhedron compounds are composed of tetrahedra:

- Stella octangula (2)
- Chiricosahedron (5)
- Icosicosahedron (10)
- Snubahedron (6)
- Small snubahedron (6, with rotational freedom)
- Disnubahedron (12, with rotational freedom)
- An infinite number of prismatic compounds that are antiprisms of compounds of digons (where the digons degenerate to edges).

Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|

Tetrahedron | tet | {3,3} | x3o3o | |

Truncated tetrahedron | tut | t{3,3} | x3x3o | |

Tetratetrahedron = Octahedron | oct | r{3,3} | o3x3o | |

Truncated tetrahedron | tut | t{3,3} | o3x3x | |

Tetrahedron | tet | {3,3} | o3o3x | |

Small rhombitetratetrahedron = Cuboctahedron | co | rr{3,3} | x3o3x | |

Great rhombitetratetrahedron = Truncated octahedron | toe | tr{3,3} | x3x3x | |

Snub tetrahedron = Icosahedron | ike | sr{3,3} | s3s3s |

## Other kinds of tetrahedra[edit | edit source]

Besides the regular tetrahedron, there are a number of other polyhedra containing four triangular faces. Tetrahedra are generally classified by symmetry. Some of these classes of tetrahedra include:

- Triangular pyramid - one equilateral triangle (base) and three identical isosceles triangles
- Tetragonal disphenoid - four identical isosceles triangles
- Digonal disphenoid - Two pairs of identical isosceles triangles
- Rhombic disphenoid - Four identical scalene triangles
- Phyllic disphenoid - Two pairs of identical scalene triangles
- Sphenoid - Only a single symmetry axis
- Irregular tetrahedron - No symmetry axes at all

## External links[edit | edit source]

- Bowers, Jonathan. "Polyhedron Category 1: Regulars" (#1).

- Klitzing, Richard. "tet".

- Quickfur. "The Tetrahedron".

- Wikipedia Contributors. "Tetrahedron".
- McCooey, David. "Tetrahedron"

- Hi.gher.Space Wiki Contributors. "Tetrahedron".