Verse and Dimensions Wikia
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Verse and Dimensions Wikia

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:
  • Edge radius:
  • Face radius:

Angles[]

  • Dihedral angle:

Vertex coordinates[]

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

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:

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 Seventeen ... Aleph null
Simplex

Null polytope


Point


Line segment


Triangle

Tetrahedron

Pentachoron

Hexateron

Heptapeton

Octaexon

Enneazetton

Decayotton

Hendecaxennon

Dodecadakon

Tridecahendon

Tetradecadokon

Pentadecatradakon

Hexadecatedakon

Heptadecapedakon

Octadecapedakon

... Omegasimplex
Cross

Square

Octahedron

Hexadecachoron

Pentacross

Hexacross

Heptacross

Octacross

Enneacross

Dekacross

Hendekacross

Dodekacross

Tridekacross

Tetradekacross

Pentadekacross

Hexadekacross

Heptadekacross

... Omegacross
Hydrotopes

Pentagon

Icosahedron

Hexacosichoron

Order-5 pentachoric tetracomb

Order-5 hexateric pentacomb

...
Hypercube

Square

Cube

Tesseract

Penteract

Hexeract

Hepteract

Octeract

Enneract

Dekeract

Hendekeract

Dodekeract

Tridekeract

Tetradekeract

Pentadekeract

Hexadekeract

Heptadekeract

... Omegeract
Cosmotopes

Pentagon

Dodecahedron

Hecatonicosachoron

Order-3 hecatonicosachoric tetracomb

Order-3-3 hecatonicosachoric pentacomb

...
Hyperball

Disk

Ball

Gongol

Pentorb

Hexorb

Heptorb

Octorb

Enneorb

Dekorb

Hendekorb

Dodekorb

Tridekorb

Tetradekorb

Pentadekorb

Hexadekorb

Heptadekorb

... Omegaball

...
Trigonal hosohedron Tetrahedron Cube Dodecahedron Hexagonal tiling Order-3 heptagonal tiling Order-3 octagonal tiling ... Order-3 apeirogonal tiling Order-3 pseudogonal tiling
...
Trigonal dihedron Tetrahedron Octahedron Icosahedron Triangular tiling Order-7 triangular tiling Order-8 triangular tiling ... Infinite-order triangular tiling Imaginary-order triangular tiling
Regular
Rectified
Birectified
Truncated
Bitruncated
Cantellated
Cantitruncated
Tetrahedron Octahedron Tetrahedron Truncated tetrahedron Truncated tetrahedron Cuboctahedron Truncated octahedron
Tetrahedron Square pyramid Pentagonal pyramid Hexagonal pyramid
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