# Tetrahedron

Tetrahedron Click on picture for large version.
TypePlatonic
Face polygontriangle
Faces4
Edges6
Vertices4
Faces per vertex3
Vertices per face3
Symmetry grouptetrahedral (Td)
Dual polyhedrontetrahedron (self-dual)
Dihedral Angle70° 32'
Propertiesregular, convex

A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral," and is one of the Platonic solids.

Missing image
Tetrahedron_flat.png
image:tetrahedron flat.png

The area A and the volume V of a regular tetrahedron of edge length a are:

[itex]A=\sqrt{3}a^2[itex]
[itex]V=\begin{matrix}{1\over12}\end{matrix}\sqrt{2}a^3[itex]

A tetrahedron is a 3-simplex.

Tetrahedra are a special type of triangular pyramid and are self-dual. Canonical coordinates of the tetrahedron are (1, 1, 1), (−1, −1, 1), (−1, 1, −1) and (1, −1, −1). A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. The volume of this tetrahedron is 1/3 the volume of the cube. Taking both tetrahedra within a single cube gives a regular polyhedral compound called the stella octangula, whose interior is an octahedron. Inscribing tetrahedra inside the regular compound of five cubes gives two more regular compounds, containing five and ten tetrahedra.

Regular tetrahedra can't tile space by themselves, although it seems likely enough that Aristotle reported it was possible. In fact, octahedra are necessary to fill some of the gaps. This is one of the five Andreini tessellations, and is a limiting case of another, a tiling involving tetrahedra and truncated tetrahedra.

However, irregular tetrahedra can tile space by themselves. Complex shapes are often broken down into a mesh of irregular tetrahedra in preparation for finite element analysis.

The volume of an irregular tetrahedron, given its vertices a, b, c and d, is (1/6)·|det(ab, bc, cd)|, or any other combination of pairs of vertices that form a simply connected graph. (This works for regular tetrahedrons too.)

Especially in roleplaying, this solid is known as a d4, one of the more common Polyhedral dice.

Like all platonic solids, archimedean solids and indeed all convex polyhedra, a tetrahedron can be folded from a single sheet of paper.

If each edge of a tetrahedron were to be replaced by a one ohm resistor, the resistance between any two vertices would be 1/2 ohm.

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