Tetrahedron

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Tetrahedron
Tetrahedron.png
Rank3
TypeRegular
SpaceSpherical
Bowers style acronymTet
Info
Coxeter diagramx3o3o (CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Schläfli symbol{3,3}
Tapertopic notation12
SymmetryA3, order 24
ArmyTet
RegimentTet
Elements
Vertex figureEquilateral triangle, edge length 1
Faces4 triangles
Edges6
Vertices4
Measures (edge length 1)
Circumradius
Edge radius
Inradius
Volume
Dihedral angle
HeightsPoint atop trig:
 Dyad atop perp dyad:
Central density1
Euler characteristic2
Number of pieces4
Level of complexity1
Related polytopes
DualTetrahedron
Petrie dualPetrial tetrahedron
ConjugateTetrahedron
Properties
ConvexYes
OrientableYes
NatureTame

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:

In vertex figures[edit | edit source]

Tetrahedra in vertex figures
Name Picture Schläfli symbol Edge length
Pentachoron
Schlegel wireframe 5-cell.png
{3,3,3}
Tesseract
Schlegel wireframe 8-cell.png
{4,3,3}
Hecatonicosachoron
Schlegel wireframe 120-cell.png
{5,3,3}
Great grand stellated hecatonicosachoron
Gogishi.png
{5/2,3,3}
Hexagonal tiling honeycomb
H3 633 FC boundary.png
{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:

o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Tetrahedron tet {3,3} x3o3o
Uniform polyhedron-33-t0.png
Truncated tetrahedron tut t{3,3} x3x3o
Uniform polyhedron-33-t01.png
Tetratetrahedron = Octahedron oct r{3,3} o3x3o
Uniform polyhedron-33-t1.png
Truncated tetrahedron tut t{3,3} o3x3x
Uniform polyhedron-33-t12.png
Tetrahedron tet {3,3} o3o3x
Uniform polyhedron-33-t2.png
Small rhombitetratetrahedron = Cuboctahedron co rr{3,3} x3o3x
Uniform polyhedron-33-t02.png
Great rhombitetratetrahedron = Truncated octahedron toe tr{3,3} x3x3x
Uniform polyhedron-33-t012.png
Snub tetrahedron = Icosahedron ike sr{3,3} s3s3s
Uniform polyhedron-33-s012.png

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:

External links[edit | edit source]

  • Klitzing, Richard. "Tet".