Tetrahedron

For the academic journal, see Tetrahedron

Regular Tetrahedron
(Click here for rotating model)
Type	Platonic solid
Elements	F=4, E=6, V=4 (χ=2)
Faces by sides	4{3}
Schläfli symbol	{3,3}
Wythoff symbol	3 | 2 3
Symmetry group	Td
Index references	U01, C15, W1
Dual	Tetrahedron
Properties	Regular convex deltahedron
Dihedral angle	70.528779° = arccos(1/3)
Vertex figure 3.3.3

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.

Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper.

Area and volume

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

<math>A=a^2\sqrt{3}</math>

<math>V=\begin{matrix}{1\over12}\end{matrix}a^3\sqrt{2}</math>

The height is <math>h=(a/3) \sqrt{6}</math>, the angle between an edge and a face is arctan <math>\sqrt{2}</math> (ca. 55°), and between two faces arccos (1/3) = arctan <math>2\sqrt{2}</math> (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.

Like for any pyramid, the volume is <math>V = \frac{1}{3} Ah</math> where A is the area of the base and h the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apexes to the opposite faces are inversely proportional to the areas of these faces.

Also, for a tetrahedron ABCT the volume is given by ^{[citation needed]}

<math>V = \frac {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}</math>

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

Any two opposite edges of a tetrahedron lie on two skew lines. If the closest pair of points between these two lines are points in the edges, they define the distance between the edges; otherwise, the distance between the edges equals that between one of the endpoints and the opposite edge.

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. This can be rewritten as a dot and cross product, yielding:

<math>V = \frac { |(\mathbf{d}-\mathbf{a}) \cdot ((\mathbf{d}-\mathbf{b}) \times (\mathbf{d}-\mathbf{c}))| } {6}</math> [1]

Geometric relations

A tetrahedron is a 3-simplex. Unlike in the case of other Platonic solids, all vertices of a regular tetrahedron are equidistant from each other (they are in the only possible arrangement of four equidistant points).

A tetrahedron is a triangular pyramid, and the regular tetrahedron is self-dual.

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. For one such embedding, the Cartesian coordinates of the vertices are

(+1, +1, +1);

(−1, −1, +1);

(−1, +1, −1);

(+1, −1, −1).

For the other tetrahedron (which is dual to the first), reverse all the signs. The volume of this tetrahedron is 1/3 the volume of the cube. Combining both tetrahedra 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).

Inscribing tetrahedra inside the regular compound of five cubes gives two more regular compounds, containing five and ten tetrahedra.

Regular tetrahedra cannot tessellate 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.

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.)

The tetrahedron is unique among the uniform polyhedra in possessing no parallel faces.

Related polyhedra

Truncated tetrahedron

Two tetrahedra in a cube

Five tetrahedra

Ten tetrahedra

Intersecting tetrahedra

An interesting polyhedron can be constructed from five intersecting tetrahedra. This compound of five tetrahedra has been known for hundreds of years. It comes up regularly in the world of origami. Joining the twenty vertices would form a regular dodecahedron. There are both left-handed and right-handed forms which are mirror images of each other.

The isometries of the regular tetrahedron

The proper rotations and reflections in the symmetry group of the regular tetrahedron

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also animation, showing one of the two tetrahedra in the cube). The symmetries of a regular tetrahedron correspond to half of those of a cube: those which map the tetrahedrons to themselves, and not to each other.

The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion.

The regular tetrahedron has 24 isometries, forming the symmetry group T_d, isomorphic to S₄. They can be categorized as follows:

• T, isomorphic to alternating group A₄ (the identity and 11 proper rotations) with the following conjugacy classes (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the unit quaternion representation):
  • identity (identity; 1)
  • rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1±i±j±k)/2)
  • rotation by an angle of 180° such that an edge maps to the opposite edge: 3 ((1 2)(3 4), etc.; i,j,k)
• reflections in a plane perpendicular to an edge: 6
• reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion (x is mapped to −x): the rotations correspond to those of the cube about face-to-face axes

The isometries of irregular tetrahedra

The isometries of an irregular tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3-dimensional point group is formed.

• An equilateral triangle base and isosceles (and non-equilateral) triangle sides gives 6 isometries, corresponding to the 6 isometries of the base. As permutations of the vertices, these 6 isometries are the identity 1, (123), (132), (12), (13) and (23), forming the symmetry group C_3v, isomorphic to S₃.
• Four congruent isosceles (non-equilateral) triangles gives 8 isometries. If edges (1,2) and (3,4) are of different length to the other 4 then the 8 isometries are the identity 1, reflections (12) and (34), and 180° rotations (12)(34), (13)(24), (14)(23) and improper 90° rotations (1234) and (1432) forming the symmetry group D_2d.
• Four congruent scalene triangles gives 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the Klein four-group V₄ ≅ Z₂², present as the point group D₂.
• Two pairs of isomorphic isosceles (non-equilateral) triangles. This gives two opposite edges (1,2) and (3,4) that are perpendicular but different lengths, and then the 4 isometries are 1, reflections (12) and (34) and the 180° rotation (12)(34). The symmetry group is C_2v, isomorphic to V₄.
• Two pairs of isomorphic scalene triangles. This has two pairs of equal edges (1,3), (2,4) and (1,4), (2,3) but otherwise no edges equal. The only two isometries are 1 and the rotation (12)(34), giving the group C₂ isomorphic to Z₂.
• Two unequal isosceles triangles with a common base. This has two pairs of equal edges (1,3), (1,4) and (2,3), (2,4) and otherwise no edges equal. The only two isometries are 1 and the reflection (34), giving the group C_s isomorphic to Z₂.
• No edges equal, so that the only isometry is the identity, and the symmetry group is the trivial group.

Computational uses

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

Euler circuit path

Manipulate the tetrahedron to eulerize a computational path.

Trivia

• The tetrahedron shape is seen in nature in covalent bonds of molecules. For instance in a methane molecule (CH₄) the four hydrogen atoms lie in each corner of a tetrahedron with the carbon atom in the centre. For this reason, one of the leading journals in organic chemistry is called Tetrahedron.

• 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.^[1]

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

• The tetrahedron represents the classical element fire.

• In the Xeelee Sequence of science fiction books by author Stephen Baxter, a blue-green tetrahedron is the symbol of free humanity.

See also

• Spinning tetrahedron
• caltrop
• tetrahedral kite
• triangular dipyramid - constructed by joining two tetrahedra along one face
• tetrahedral number
• tetrahedral molecular geometry
• Tetra-Pak

References

1. ^ Klein, Douglas J. (2002). "Resistance-Distance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. Retrieved on 2006-09-15.

External links

• The Uniform Polyhedra
• A Spinning Tetrahedron
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Paper Models of Polyhedra Many links
• An Amazing, Space Filling, Non-regular Tetrahedron that also includes a description of a "rotating ring of tetrahedra", also known as a kaleidocycle.

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