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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  T_{d}, A_{3}, [3,3], (*332) 
Rotation group  T, [3,3]^{+}, (332) 
References  U_{01}, C_{15}, W_{1} 
Properties  Regular convex deltahedron 
Dihedral angle  70.528779° = arccos(1/3) 
3.3.3 (Vertex figure)  Selfdual (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.^{[1]}
The tetrahedron is the threedimensional 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.^{[1]}
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.
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.
Regular tetrahedra alone cannot fill space, but if alternated with regular octahedra they form the alternated cubic honeycomb which does fill space.
The regular tetrahedron is selfdual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.
The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin:
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.
For a regular tetrahedron of edge length a:
Face area  
Surface area^{[2]}  
Height of pyramid^{[3]}  
Volume^{[2]}  
Facevertexedge angle  (approx. 54.7356°) 
Faceedgeface angle^{[2]}  (approx. 70.5288°) 
Edge central angle,^{[4]} ^{[5]} known as the tetrahedral angle  (approx. 109.4712°) 
Solid angle at a vertex subtended by a face  (approx. 0.55129 steradians) 
Radius of circumsphere^{[2]}  
Radius of insphere that is tangent to faces^{[2]}  
Radius of midsphere that is tangent to edges^{[2]}  
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).
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, S_{4}. They can be categorized as follows:
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 A_{2} Coxeter plane.
Centered by  Face/vertex  Edge 

Image  
Projective symmetry  [3]  [4] 
The tetrahedron 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 

An isosceles tetrahedron, also called a disphenoid, is a tetrahedron where all four faces are congruent triangles. A spacefilling 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 semiorthocentric 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.
The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3dimensional point group is formed. Two other isometries (C_{3}, [3]^{+}), and (S_{4}, [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  It forms the symmetry group T_{d}, isomorphic to the symmetric group, S_{4}.  
T_{d} T  [3,3] [3,3]^{+}  *332 332  24 12  
Triangular pyramid  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 C_{3v}, isomorphic to the symmetric group, S_{3}.  
C_{3v} C_{3}  [3] [3]^{+}  *33 33  6 3  
Mirrored sphenoid  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}, also isomorphic to the cyclic group, Z_{2}.  
C_{s} =C_{1h} =C_{1v}  [ ]  *  2  
Irregular tetrahedron (No symmetry)  Its only isometry is the identity, and the symmetry group is the trivial group.  
C_{1}  [ ]^{+}  1  1  
Disphenoids (Four equal triangles)  
Tetragonal disphenoid  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 D_{2d}.  
D_{2d} S_{4}  [2^{+},4] [2^{+},4^{+}]  2*2 2×  8 4  
Rhombic disphenoid  It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the Klein fourgroup V_{4} or Z_{2}^{2}, present as the point group D_{2}.  
D_{2}  [2,2]^{+}  222  4  
Generalized disphenoids (2 pairs of equal triangles)  
Digonal disphenoid  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 the Klein fourgroup V_{4}.  
C_{2v} =D_{1h}  [2]  *22  4  
Phyllic disphenoid  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 C_{2} isomorphic to the cyclic group, Z_{2}.  
C_{2} =D_{1}  [2]^{+}  22  2 
The volume of a tetrahedron is given by the pyramid volume formula:
where A_{0} 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 = (a_{1}, a_{2}, a_{3}), b = (b_{1}, b_{2}, b_{3}), c = (c_{1}, c_{2}, c_{3}), and d = (d_{1}, d_{2}, d_{3}), 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:
Hence
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:
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.^{[6]}
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^{[7]}
where
For tetrahedra in hyperbolic space or in threedimensional 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.^{[8]} However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.
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
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. The circumsphere of the medial tetrahedron is analogous to the triangle's ninepoint circle, but does not generally pass through the base points of the altitudes of the reference tetrahedron.^{[9]}
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.^{[10]} 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 ninepoint circle of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the twelvepoint 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.^{[11]}
The center T of the twelvepoint 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 twelvepoint center lies midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelvepoint center lies at the midpoint of the corresponding Euler point and the orthocenter for that face.
The radius of the twelvepoint 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 ^{[12]}
where is the angle between the faces i and j.
A tetrahedron is a 3simplex. 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 3dimensional space).
A tetrahedron is a triangular pyramid, and the regular tetrahedron is selfdual.
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
This yields a tetrahedron with edgelength , 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 3demicube.
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.^{[13]}
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 sidenote: these two kinds of tetrahedron have the same volume.)
The tetrahedron is unique among the uniform polyhedra in possessing no parallel faces.
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 5dimensional.^{[14]}
A regular tetrahedron can be seen as a triangular pyramid.
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.
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.
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 selfdual tetrahedron once again.
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.
Finite  Euclidean  Compact hyperbolic  Paracompact  

{3,2}  {3,3}  {3,4}  {3,5}  {3,6}  {3,7}  {3,8}  {3,9}  ...  (3,∞} 
The tetrahedron is topologically related to a series of regular polyhedra and tilings with order3 vertex figures.
Spherical Polyhedra  Polyhedra  Euclidean  Hyperbolic tilings  

{2,3}  {3,3}  {4,3}  {5,3}  {6,3}  {7,3}  {8,3}  ...  (∞,3} 
Compounds:
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 lefthanded and righthanded forms, which are mirror images of each other.
In numerical analysis, complicated threedimensional 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.
The tetrahedron shape is seen in nature in covalent bonds of molecules. All sp^{3}hybridized atoms are surrounded by atoms lying in each corner of a tetrahedron. For instance in a methane molecule (CH_{4}) or an ammonium ion (NH_{4}^{+}), 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. See also tetrahedral molecular geometry. The central angle between any two vertices of a perfect tetrahedron is , or approximately 109.47°.
Water, H_{2}O, 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 OH bonds.
Quaternary phase diagrams in chemistry are represented graphically as tetrahedra.
However, quaternary phase diagrams in communication engineering are represented graphically on a twodimensional plane.
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.^{[15]}^{[16]}
Since silicon is the most common semiconductor used in solidstate 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.
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 4sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex. Some Rubik's Cubelike 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.
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).^{[17]}
The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps. It was shown at the light art biennale Austria 2010.^{[18]}
It is used as album artwork, surrounded by black flames on The End of All Things to Come by Mudvayne.
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.^{[19]}
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. Where 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.
The tetrahedral hypothesis, originally published by William Lowthian Green to explain the formation of the Earth,^{[20]} was popular through the early 20th century.^{[21]}^{[22]}
A tetrahedron having stiff edges is inherently rigid. For this reason it is often used to stiffen frame structures such as spaceframes.
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. Its construction to be 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.^{[23]}
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