Tetrahedron

Tetrahedron
(3D model) (OFF file)
Rank	3
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Tet
Coxeter diagram	x3o3o ()
Schläfli symbol	{3,3}
Tapertopic notation	12
Stewart notation	Y3
Elements
Faces	4 triangles
Edges	6
Vertices	4
Vertex figure	Equilateral triangle, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt6}{4} ≈ 0.61237$
Edge radius	$\frac{\sqrt2}{4} ≈ 0.35355$
Inradius	$\frac{\sqrt6}{12} ≈ 0.20412$
Volume	$\frac{\sqrt2}{12} ≈ 0.11785$
Dihedral angle	$\arccos\left(\frac13\right) ≈ 70.52878^\circ$
Heights	Point atop trig: $\frac{\sqrt6}{3} ≈ 0.81650$
	Dyad atop perp dyad: $\frac{\sqrt2}{2} ≈ 0.70711$
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	A3, 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

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

• $\left(\frac{\sqrt{2}}{4},\,\frac{\sqrt{2}}{4},\,\frac{\sqrt{2}}{4}\right).$

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:

• $\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12}\right),$
• $\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12}\right),$
• $\left(0,\,0,\,\frac{\sqrt{6}}{4}\right).$

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

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

• $\left(\frac{\sqrt{2}}{2},\,0,\,0,\,0\right).$

Representations

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

Tetrahedra in vertex figures

Name	Picture	Schläfli symbol	Edge length
Pentachoron		{3,3,3}	$1$
Tesseract		{4,3,3}	$\sqrt{2}$
Hecatonicosachoron		{5,3,3}	$\frac{1+\sqrt{5}}{2}$
Great grand stellated hecatonicosachoron		{5/2,3,3}	$\frac{\sqrt{5}-1}{2}$
Hexagonal tiling honeycomb		{6,3,3}

Related polyhedra

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).

o3o3o truncations

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

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

• 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".