Tetrahedron

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.

Related Topics:
Polyhedron - Platonic solid

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:A=sqrt{3}a^2

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:V=egin{matrix}{1over12}end{matrix}sqrt{2}a^3

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The height is h=(a/3) sqrt{6}, the angle between an edge and a face is arctan sqrt{2} (ca. 55°), and between two faces arccos (1/3) = arctan 2sqrt{2} (ca. 71°). Note that with respect to the base plane the slope of a face is twice that of an edge, corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that in a face, from the midpoint at the base.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A tetrahedron is a 3-simplex.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Unlike in the case of other Platonic solids, all vertices are equidistant from each other (they are in the only possible arrangement of four equidistant points).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron).

Related Topics:
Pyramid - Dual - Cube - Polyhedral compound - Octahedron - Rectifying

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron which can tile space. This is one of the five Andreini tessellations, and is a limiting case of another, a tiling involving tetrahedra and truncated tetrahedra.

Related Topics:
Tile space - Aristotle - Rhombohedron - Andreini tessellation - Truncated tetrahedra

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

However, there is at least one irregular tetrahedron of which copies can tile space. If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in various ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a side-note: these two kinds of tetrahedron have the same volume.)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Complex shapes are often broken down into a mesh of irregular tetrahedra in preparation for finite element analysis and computational fluid dynamics studies.

Related Topics:
Mesh - Finite element analysis - Computational fluid dynamics

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The volume of any tetrahedron, given its vertices a, b, c and d, is (1/6)·|det(a−b, b−c, c−d)|, or any other combination of pairs of vertices that form a simply connected graph.

Related Topics:
Det - Graph

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Also, for a tetrahedron ABCT the volume is given by

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

V = rac {AT cdot BT cdot CT}{6} cdot sqrt {1 + 2 cdot cos a cdot cos b cdot cos c - cos^2 a - cos^2 b - cos^2 c}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where a is angle ATB, b angle BTC, and c angle CTB.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For the distance between edges, see skew line.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

Related Topics:
Roleplaying - D4 - 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.

Related Topics:
Ohm - Resistor

~ ~ ~ ~ ~ ~ ~ ~ ~ ~