tetrahedron

• 四面体

関

tetrahedral

　 　

WordNet

1. any polyhedron having four plane faces
2. a tetrahedron with four equilateral triangular faces

PrepTutorEJDIC

1. 四面体

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/03/12 19:29:36」(JST)

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三角錐（さんかくすい、triangular pyramid, trigonal pyramid）とは、垂直断面に三角形を持つ錐体のことである。辺6本、頂点4つからなる。さらに、面の数は立体に於ける最小限界の 4 つである。このことからまた、四面体（しめんたい、tetrahedron）とも呼ぶ。三角錐は、最小の頂点数で構成することができる立体であると表現することもできる。

垂直断面が正三角形である場合、特に正三角錐（せいさんかくすい、regular triangular pyramid）という。幾何学に於いて、角錐の側面は全て三角形であるが、この場合は底面も三角形であるから、三角錐は全ての面が三角形である立体である。

正四面体

詳細は「正四面体」を参照

全ての面が正三角形であるような三角錐を、正四面体（せいしめんたい、regular tetrahedron）という。正四面体は、デルタ多面体の一種である。

多面体分割

面積の等しい三角形と四角形は、適当に多角形に有限回分割することによって合同にすることができるが、三角錐と四角錐は、たとえ体積が等しくとも多面体に分割して合同にすることは無理である。これは、ボヤイの定理が 3 次元空間では一般に成立しないことを示す。こうして 3 次元空間に於けるボヤイの定理（それはヒルベルトの第三の問題でもあった）が否定的に解かれた。ただし、分割の仕方を多面体に限らなければ体積が等しくなくても有界な図形は合同にすることができる（バナッハ＝タルスキーのパラドックス）。

一般次元への拡張

一般次元ユークリッド空間 Rⁿ にも、当然、最小の頂点で構成できる立体は存在する。そのような立体を総称して単体あるいは三角錐と言うことがある。一般に、空間の次元が n であるとき、その空間内に存在する三角錐は n+1 個の頂点を持つ。

また、三角錐は、“空間上にある基準点 O を取ったとき、O からの位置ベクトルが互いに一次独立な関係にあるような n+1 個の点 P₁,…,P_n+1 を頂点にもつ多面体。” と定義することもできる。このとき、vecOP_i=(x_1i,…,x_ni) とすれば、この三角錐の表面積 S は、

$S = \frac{1}{n!}{\rm abs} \begin{vmatrix} x_{11} & x_{12} & \cdots & x_{1,n+1} \\ & \cdots & \cdots & \\ x_{n1} & x_{n2} & \cdots & x_{n,n+1} \\ 1 & 1 & 1 & 1 \end{vmatrix}$

と表すことができる。特に、P_n+1=O であるとき、

$S = \frac{1}{n!}{\rm abs} \begin{vmatrix} x_{11} & \cdots & x_{1n} \\ \vdots & \ddots & \vdots \\ x_{n1} & \cdots & x_{nn} \end{vmatrix}$

である。

外部リンク

• 三角錐の体積（三角柱を3つの三角錐に分解することによる証明） - 金沢工業大学

関連項目

• 四角錐
• 五胞体
• 錐体
• 円錐
• 正多角形
• 正多面体
• 正四面体リング
• ピラミンクス

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Not to be confused with tetrahedroid.

For the academic journal, see Tetrahedron (journal).

Regular Tetrahedron
(Click here for rotating model)
Type	Platonic solid
Elements	F = 4, E = 6 V = 4 (χ = 2)
Faces by sides	4{3}
Conway notation	T
Schläfli symbols	{3,3}
	h{4,3}, s{2,4}, sr{2,2}
Wythoff symbol	3 | 2 3 | 2 2 2
Coxeter diagram	=
Symmetry	Td, A3, [3,3], (*332)
Rotation group	T, [3,3]+, (332)
References	U01, C15, W1
Properties	Regular convex deltahedron
Dihedral angle	70.528779° = arccos(1/3)
3.3.3 (Vertex figure)	Self-dual (dual polyhedron)
Net

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, three of which meet at each corner or vertex. It has six edges and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex.

The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".

Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.

For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces.

§Regular tetrahedron

A regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity.

In a regular tetrahedron, not only are all its faces the same size and shape (congruent) but so are all its vertices and edges.

Five tetrahedra are laid flat on a plane, with the highest 3-dimensional points marked as 1, 2, 3, 4, and 5. These points are then attached to each other and a thin volume of empty space is left, where the five edge angles do not quite meet.

Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation.

The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.

§Formulas for a regular tetrahedron

The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin:

(±1, 0, −1/√2)

(0, ±1, 1/√2)

Another set of coordinates are based on an alternated cube with edge length 2. The tetrahedron in this case has edge length . Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube.

Tetrahedon: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1)

Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1)

Regular tetrahedron ABCD and its circumscribed sphere

For a regular tetrahedron of edge length a:

Face area
Surface area
Height of pyramid
Edge to opposite edge distance
Volume
Face-vertex-edge angle	(approx. 54.7356°)
Face-edge-face angle	(approx. 70.5288°)
Edge central angle, known as the tetrahedral angle	(approx. 109.4712°)
Solid angle at a vertex subtended by a face	(approx. 0.55129 steradians)
Radius of circumsphere
Radius of insphere that is tangent to faces
Radius of midsphere that is tangent to edges
Radius of exspheres
Distance to exsphere center from a vertex

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 along the median of a face. In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).

§Isometries of the regular tetrahedron

The proper rotations, (order-3 rotation on a vertex and face, and order-2 on 2 edges) and reflection plane (through 2 faces and one edge) 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 that map the tetrahedra 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, [3,3], (*332), isomorphic to the symmetric group, S₄. They can be categorized as follows:

• T, [3,3]⁺, (332) is 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

§Orthogonal projections of the regular tetrahedron

The regular tetrahedron has two special orthogonal projections, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A₂ Coxeter plane.

Orthogonal projection

Centered by	Face/vertex	Edge
Image
Projective symmetry	[3]	[4]

§Spherical tiling

The tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Orthographic projection	Stereographic projection

§Other special cases

Tetrahedral symmetry subgroup relations

Tetrahedral symmetries shown in tetrahedral diagrams

An isosceles tetrahedron, also called a disphenoid, is a tetrahedron where all four faces are congruent triangles. A space-filling tetrahedron packs with congruent copies of itself to tile space, like the disphenoid tetrahedral honeycomb.

In a trirectangular tetrahedron the three face angles at one vertex are right angles. If all three pairs of opposite edges of a tetrahedron are perpendicular, then it is called an orthocentric tetrahedron. When only one pair of opposite edges are perpendicular, it is called a semi-orthocentric tetrahedron. An isodynamic tetrahedron is one in which the cevians that join the vertices to the incenters of the opposite faces are concurrent, and an isogonic tetrahedron has concurrent cevians that join the vertices to the points of contact of the opposite faces with the inscribed sphere of the tetrahedron.

§Isometries of irregular tetrahedra

The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3-dimensional point group is formed. Two other isometries (C₃, [3]⁺), and (S₄, [2⁺,4⁺]) can exist if the face or edge marking are included. Tetrahedral diagrams are included for each type below, with edges colored by isometric equivalence, and are gray colored for unique edges.

Tetrahedron name				Edge Equivalence diagram	Description
Symmetry
Schön.	Cox.	Orb.	Ord.
Regular Tetrahedron					Four equilateral triangles It forms the symmetry group Td, isomorphic to the symmetric group, S4.
Td T	[3,3] [3,3]+	*332 332	24 12
Triangular pyramid					An equilateral triangle base and three equal isosceles triangle sides It 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 C3v, isomorphic to the symmetric group, S3.
C3v C3	[3] [3]+	*33 33	6 3
Mirrored sphenoid					Two equal scalene triangles with a common base edge 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 Cs, also isomorphic to the cyclic group, Z2.
Cs =C1h =C1v	[ ]	*	2
Irregular tetrahedron (No symmetry)					Four unequal triangles Its only isometry is the identity, and the symmetry group is the trivial group.
C1	[ ]+	1	1
Disphenoids (Four equal triangles)
Tetragonal disphenoid					Four equal isosceles triangles It has 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 D2d.
D2d S4	[2+,4] [2+,4+]	2*2 2×	8 4
Rhombic disphenoid					Four equal scalene triangles It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the Klein four-group V4 or Z22, present as the point group D2.
D2	[2,2]+	222	4
Generalized disphenoids (2 pairs of equal triangles)
Digonal disphenoid					Two pairs of equal isosceles 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 C2v, isomorphic to the Klein four-group V4.
C2v C2	[2] [2]+	*22 22	4 2
Phyllic disphenoid					Two pairs of equal 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 the cyclic group, Z2.
C2	[2]+	22	2

§General properties

§Volume

The volume of a tetrahedron is given by the pyramid volume formula:

where A₀ 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.

For a tetrahedron with vertices a = (a₁, a₂, a₃), b = (b₁, b₂, b₃), c = (c₁, c₂, c₃), and d = (d₁, d₂, d₃), the volume is (1/6)·|det(a − d, b − d, c − d)|, or any other combination of pairs of vertices that form a simply connected graph. This can be rewritten using a dot product and a cross product, yielding

If the origin of the coordinate system is chosen to coincide with vertex d, then d = 0, so

where a, b, and c represent three edges that meet at one vertex, and a · (b × c) is a scalar triple product. Comparing this formula with that used to compute the volume of a parallelepiped, we conclude that the volume of a tetrahedron is equal to 1/6 of the volume of any parallelepiped that shares three converging edges with it.

The triple scalar can be represented by the following determinants:

$6 \cdot V =\begin{vmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{vmatrix}$ or $6 \cdot V =\begin{vmatrix} \mathbf{a} \\ \mathbf{b} \\ \mathbf{c} \end{vmatrix}$ where  is expressed as a row or column vector etc.

Hence

$36 \cdot V^2 =\begin{vmatrix} \mathbf{a^2} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{a} \cdot \mathbf{b} & \mathbf{b^2} & \mathbf{b} \cdot \mathbf{c} \\ \mathbf{a} \cdot \mathbf{c} & \mathbf{b} \cdot \mathbf{c} & \mathbf{c^2} \end{vmatrix}$ where  etc.

which gives

where α, β, γ are the plane angles occurring in vertex d. The angle α, is the angle between the two edges connecting the vertex d to the vertices b and c. The angle β, does so for the vertices a and c, while γ, is defined by the position of the vertices a and b.

Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant:

$288 \cdot V^2 = \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix}$

where the subscripts represent the vertices {a, b, c, d} and is the pairwise distance between them – i.e., the length of the edge connecting the two vertices. A negative value of the determinant means that a tetrahedron cannot be constructed with the given distances. This formula, sometimes called Tartaglia's formula, is essentially due to the painter Piero della Francesca in the 15th century, as a three dimensional analogue of the 1st century Heron's formula for the area of a triangle.

§Heron-type formula for the volume of a tetrahedron

If U, V, W, u, v, w are lengths of edges of the tetrahedron (first three form a triangle; u opposite to U and so on), then

$\text{volume} = \frac{\sqrt {\,( - a + b + c + d)\,(a - b + c + d)\,(a + b - c + d)\,(a + b + c - d)}}{192\,u\,v\,w}$

where

$\begin{align} a & = \sqrt {xYZ} \\ b & = \sqrt {yZX} \\ c & = \sqrt {zXY} \\ d & = \sqrt {xyz} \\ X & = (w - U + v)\,(U + v + w) \\ x & = (U - v + w)\,(v - w + U) \\ Y & = (u - V + w)\,(V + w + u) \\ y & = (V - w + u)\,(w - u + V) \\ Z & = (v - W + u)\,(W + u + v) \\ z & = (W - u + v)\,(u - v + W). \end{align}$

§Volume divider

A plane that divides two opposite edges of a tetrahedron in a given ratio also divides the volume of the tetrahedron in the same ratio. Thus any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron

§Non-Euclidean volume

For tetrahedra in hyperbolic space or in three-dimensional spherical geometry, the dihedral angles of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the Murakami–Yano formula. However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.

§Distance between the edges

Any two opposite edges of a tetrahedron lie on two skew lines, and the distance between the edges is defined as the distance between the two skew lines. Let d be the distance between the skew lines formed by opposite edges a and b − c as calculated here. Then another volume formula is given by

§Properties analogous to those of a triangle

The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. It has respective centers such as incenter, circumcenter, excenters, Spieker center and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes.

Gaspard Monge found a center that exists in every tetrahedron, now known as the Monge point: the point where the six midplanes of a tetrahedron intersect. A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of orthocentric tetrahedron.

An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex.

A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median and a line segment joining the midpoints of two opposite edges is called a bimedian of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all concurrent at a point called the centroid of the tetrahedron. The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle.

The nine-point circle of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the twelve-point sphere and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute Euler points, 1/3 of the way from the Monge point toward each of the four vertices. Finally it passes through the four base points of orthogonal lines dropped from each Euler point to the face not containing the vertex that generated the Euler point.

The center T of the twelve-point sphere also lies on the Euler line. Unlike its triangular counterpart, this center lies 1/3 of the way from the Monge point M towards the circumcenter. Also, an orthogonal line through T to a chosen face is coplanar with two other orthogonal lines to the same face. The first is an orthogonal line passing through the corresponding Euler point to the chosen face. The second is an orthogonal line passing through the centroid of the chosen face. This orthogonal line through the twelve-point center lies midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelve-point center lies at the midpoint of the corresponding Euler point and the orthocenter for that face.

The radius of the twelve-point sphere is 1/3 of the circumradius of the reference tetrahedron.

There is a relation among the angles made by the faces of a general tetrahedron given by

$\begin{vmatrix} -1 & \cos{(\alpha_{12})} & \cos{(\alpha_{13})} & \cos{(\alpha_{14})}\\ \cos{(\alpha_{12})} & -1 & \cos{(\alpha_{23})} & \cos{(\alpha_{24})} \\ \cos{(\alpha_{13})} & \cos{(\alpha_{23})} & -1 & \cos{(\alpha_{34})} \\ \cos{(\alpha_{14})} & \cos{(\alpha_{24})} & \cos{(\alpha_{34})} & -1 \\ \end{vmatrix} = 0\,$

where is the angle between the faces i and j.

§Geometric relations

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

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).

This yields a tetrahedron with edge-length , centered at the origin. For the other tetrahedron (which is dual to the first), reverse all the signs. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube.

The stella octangula.

The volume of this tetrahedron is 1/3 the volume of the cube. Combining both tetrahedra gives a regular polyhedral compound called the compound of two tetrahedra or stella octangula.

The interior of the stella octangula is an octahedron, and 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).

The above embedding divides the cube into five tetrahedra, one of which is regular. In fact, 5 is the minimum number of tetrahedra required to compose a cube.

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 this result seems likely enough that Aristotle claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron that can tile space.

However, several irregular tetrahedra are known, of which copies can tile space, for instance the disphenoid tetrahedral honeycomb. The complete list remains an open problem.

If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in many different 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.

§A law of sines for tetrahedra and the space of all shapes of tetrahedra

A corollary of the usual law of sines is that in a tetrahedron with vertices O, A, B, C, we have

One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.

Putting any of the four vertices in the role of O yields four such identities, but at most three of them are independent: If the "clockwise" sides of three of them are multiplied and the product is inferred to be equal to the product of the "counterclockwise" sides of the same three identities, and then common factors are cancelled from both sides, the result is the fourth identity.

Three angles are the angles of some triangle if and only if their sum is 180° (π radians). What condition on 12 angles is necessary and sufficient for them to be the 12 angles of some tetrahedron? Clearly the sum of the angles of any side of the tetrahedron must be 180°. Since there are four such triangles, there are four such constraints on sums of angles, and the number of degrees of freedom is thereby reduced from 12 to 8. The four relations given by this sine law further reduce the number of degrees of freedom, from 8 down to not 4 but 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5-dimensional.

§Interior point

Let P be any interior point of a tetrahedron of volume V for which the vertices are A, B, C, and D, and for which the areas of the opposite faces are F_a , F_b , F_c , and F_d. Then

For vertices A, B, C, and D, interior point P, and feet J, K, L, and M of the perpendiculars from P to the faces,

§Inradius

Denoting the inradius of a tetrahedron as r and the inradii of its triangular faces as r_i for i= 1, 2, 3, 4, we have

with equality if and only if the tetrahedron is regular.

§Faces

The sum of the areas of any three faces is greater than the area of the fourth face.

§Integer tetrahedra

There exist tetrahedra having integer-valued edge lengths, face areas and volume. One example has one edge of 896, the opposite edge of 190 and the other four edges of 1073; two faces have areas of 436800 and the other two have areas of 47120, while the volume is 62092800.

§Related polyhedra and compounds

A regular tetrahedron can be seen as a triangular pyramid.

Pyramids

Triangular	Square	Pentagonal	Hexagonal	...
Regular	Equilateral		Isosceles

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform digonal antiprism, where base polygons are reduced digons.

Family of uniform antiprisms

2	3	4	5	6	7	8	9	10	11	12	n
s{2,4} sr{2,2}	s{2,6} sr{2,3}	s{2,8} sr{2,4}	s{2,10} sr{2,5}	s{2,12} sr{2,6}	s{2,14} sr{2,7}	s{2,16} sr{2,8}	s{2,18} sr{2,9}	s{2,20} sr{2,10}	s{2,22} sr{2,11}	s{2,24} sr{2,12}	s{2,2n} sr{2,n}
As spherical polyhedra

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual digonal trapezohedron, containing 6 vertices, in two sets of colinear edges.

Family of trapezohedra

2	3	4	5	6	7	8	9	10	11	12	...
As spherical polyhedra

A truncation process applied to the tetrahedron produces a series of uniform polyhedra. Truncating edges down to points produces the octahedron as a rectified tetrahedron. The process completes as a birectification, reducing the original faces down to points, and producing the self-dual tetrahedron once again.

Family of uniform tetrahedral polyhedra

Symmetry: [3,3], (*332)							[3,3]+, (332)
{3,3}	t{3,3}	r{3,3}	t{3,3}	{3,3}	rr{3,3}	tr{3,3}	sr{3,3}
Duals to uniform polyhedra
V3.3.3	V3.6.6	V3.3.3.3	V3.6.6	V3.3.3	V3.4.3.4	V4.6.6	V3.3.3.3.3

This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.

Spherical				Euclid.	Compact hyper.		Paraco.	Noncompact hyperbolic
{3,2}	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	(3,∞}	{3,12i}	{3,9i}	{3,6i}	{3,3i}

The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 vertex figures.

Spherical				Euclidean	Compact hyperb.		Paraco.	Noncompact hyperbolic
{2,3}	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}	{12i,3}	{9i,3}	{6i,3}	{3i,3}

Compounds:

• Two tetrahedra in a cube

• Compound of five tetrahedra

• Compound of ten 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.

§Applications

§Numerical analysis

An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.

In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have wide applications in practical applications in computational fluid dynamics, aerodynamics, electromagnetic fields, civil engineering, chemical engineering, naval architecture and engineering, and related fields.

§Chemistry

The ammonium+ ion is tetrahedral

Main article: Tetrahedral molecular geometry

The tetrahedron shape is seen in nature in covalently bonded molecules. All sp³-hybridized atoms are surrounded by atoms (or lone electron pairs) at the four corners of a tetrahedron. For instance in a methane molecule (CH₄) or an ammonium ion (NH₄⁺), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called Tetrahedron. The central angle between any two vertices of a perfect tetrahedron is , or approximately 109.47°.

Water, H₂O, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O-H bonds.

Quaternary phase diagrams in chemistry are represented graphically as tetrahedra.

However, quaternary phase diagrams in communication engineering are represented graphically on a two-dimensional plane.

§Electricity and electronics

Main articles: Electricity and Electronics

If six equal resistors are soldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.

Since silicon is the most common semiconductor used in solid-state electronics, and silicon has a valence of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how crystals of silicon form and what shapes they assume.

§Games

4-sided die

Main article: Game

The Royal Game of Ur, dating from 2600 BC, was played with a set of tetrahedral dice.

Especially in roleplaying, this solid is known as a 4-sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex. Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix.

The net of a tetrahedron also makes the famous Triforce from Nintendo's The Legend of Zelda franchise.

§Color space

Main article: Color space

Tetrahedra are used in color space conversion algorithms specifically for cases in which the luminance axis diagonally segments the color space (e.g. RGB, CMY).

§Contemporary art

Main article: Contemporary art

The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps. It was shown at the light art biennale Austria 2010.

It is used as album artwork, surrounded by black flames on The End of All Things to Come by Mudvayne.

§Popular culture

Stanley Kubrick originally intended the monolith in 2001: A Space Odyssey to be a tetrahedron, according to Marvin Minsky, a cognitive scientist and expert on artificial intelligence who advised Kubrick on the HAL 9000 computer and other aspects of the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand.

In Season 6, Episode 15 of Futurama, aptly named "Möbius Dick", the Planet Express crew pass through an area in space known as the Bermuda Tetrahedron. Many other ships passing through the area have mysteriously disappeared, including that of the first Planet Express crew.

In the 2013 film Oblivion the large structure in orbit above the Earth is of a tetrahedron design and referred to as the Tet.

§Geology

Main article: Geology

The tetrahedral hypothesis, originally published by William Lowthian Green to explain the formation of the Earth, was popular through the early 20th century.

§Structural engineering

A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as spaceframes.

§Aviation

At some airfields, a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind. It is built big enough to be seen from the air and is sometimes illuminated. Its purpose is to serve as a reference to pilots indicating wind direction.

§Tetrahedral graph

Tetrahedral graph
Vertices	4
Edges	6
Radius	1
Diameter	1
Girth	3
Automorphisms	24
Chromatic number	4
Properties	Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph

The skeleton of the tetrahedron (the vertices and edges) form a graph, with 4 vertices, and 6 edges. It is a special case of the complete graph, K₄, and wheel graph, W₄. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.

3-fold symmetry

§See also

• Boerdijk–Coxeter helix
• Caltrop
• Demihypercube and simplex – n-dimensional analogues
• Hill tetrahedron
• Pentachoron – 4-dimensional Hyper-Tetrahedron
• Schläfli orthoscheme
• Tetra Pak
• Tetrahedral kite
• Tetrahedral number
• Tetrahedron packing
• Triangular dipyramid – constructed by joining two tetrahedra along one face
• Trirectangular tetrahedron

§References

§External links

• Weisstein, Eric W., "Tetrahedron", MathWorld.
• Weisstein, Eric W., "Monge point", MathWorld.
• Weisstein, Eric W., "Euler points", MathWorld.
• Richard Klitzing, 3D convex uniform polyhedra , x3o3o - tet
• The Uniform Polyhedra
• Editable printable net of a tetrahedron with interactive 3D view
• Tetrahedron: Interactive Polyhedron Model
• Free paper models of a tetrahedron and many other polyhedra
• An Amazing, Space Filling, Non-regular Tetrahedron that also includes a description of a "rotating ring of tetrahedra", also known as a kaleidocycle.
• Tetrahedron Core Network Application of a tetrahedral structure to create resilient partial-mesh data network
• Explicit exact formulas for the inertia tensor of an arbitrary tetrahedron in terms of its vertex coordinates
• Hart, Vi. "ViHart rolls a Tetrahedral Dice Yahtzee". Numberphile. Brady Haran. 

English Journal

• Highly selective and sensitive detection of miRNA based on toehold-mediated strand displacement reaction and DNA tetrahedron substrate.

• Li W1, Jiang W2, Ding Y2, Wang L3.
• Biosensors & bioelectronics.Biosens Bioelectron.2015 Sep 15;71:401-6. doi: 10.1016/j.bios.2015.04.067. Epub 2015 Apr 22.
• MicroRNAs (miRNAs) play important roles in a variety of biological processes and have been regarded as tumor biomarkers in cancer diagnosis and prognosis. In this work, a single-molecule counting method for miRNA analysis was proposed based on toehold-mediated strand displacement reaction (SDR) and
• PMID 25950935

• Electrochemical detection of lung cancer specific microRNAs using 3D DNA origami nanostructures.

• Liu S1, Su W1, Li Z1, Ding X2.
• Biosensors & bioelectronics.Biosens Bioelectron.2015 Sep 15;71:57-61. doi: 10.1016/j.bios.2015.04.006. Epub 2015 Apr 7.
• Recent reports have indicated that aberrant expression of microRNAs is highly correlated with occurrence of lung cancer. Therefore, highly sensitive detection of lung cancer specific microRNAs provides an attractive approach in lung cancer early diagnostics. Herein, we designed 3D DNA origami struct
• PMID 25884735

• Ultrasensitive electrochemical detection of microRNA-21 combining layered nanostructure of oxidized single-walled carbon nanotubes and nanodiamonds by hybridization chain reaction.

• Liu L1, Song C1, Zhang Z1, Yang J1, Zhou L1, Zhang X1, Xie G2.
• Biosensors & bioelectronics.Biosens Bioelectron.2015 Aug 15;70:351-7. doi: 10.1016/j.bios.2015.03.051. Epub 2015 Mar 28.
• Measurement of microRNA (miRNA) levels in body fluids is a crucial tool for the early diagnosis and prognosis of cancers. In this study, we developed an electrochemical assay to detect miRNA-21 by fabricating the electrode with layer-by-layer assembly of oxidized single-walled carbon nanotubes and n
• PMID 25841119

Japanese Journal

• 実験室 最適化テトラヘドロン法を用いた高精度ブリルアン領域積分

• 固体物理 51(1), 25-31, 2016-01
• NAID 40020723284

• Chiral primary amino alcohol organobase catalysts for the asymmetric Diels-Alder reactions of anthrones with maleimides

• Tetrahedron
• NAID 120005713343

• Terpenoids from Ligularia virgaurea collected in China: the first example of two bakkane derivatives with an anhydride-type ring C and nineteen new chemical constituents

• Tetrahedron
• NAID 120005672439

Related Links

• Tetrahedron | Vol 68, Iss 50, Pgs 10257-10390, (16 December, 2012 ...

The online version of Tetrahedron at ScienceDirect.com, the world's leading platform for high quality peer-reviewed full-text journals.

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