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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.

Hypervolumes

Subfacets

Radii

  • 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

See also

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\} $

Dodecadakon

$ \{3^{10}\} $

Tridecahendon

$ \{3^{11}\} $

Tetradecadokon

$ \{3^{12}\} $

Pentadecatradakon

$ \{3^{13}\} $

Hexadecatedakon

$ \{3^{14}\} $

Heptadecapedakon

$ \{3^{15}\} $

... Omegasimplex
Cross

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

Square

$ \{4\} $

Octahedron

$ \{3, 4\} $

Hexadecachoron

$ \{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\} $

Tetradekacross

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

Pentadekacross

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

Hexadekacross

$ \{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}\} $

Tetradekeract

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

Pentadekeract

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

Hexadekeract

$ \{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} $

Tetradekorb

$ \mathbb B^{14} $

Pentadekorb

$ \mathbb B^{15} $

Hexadekorb

$ \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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