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Polyhedron with 4 faces

In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3 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.

A regular tetrahedron is a tetrahedron 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, all faces are the same size and shape (congruent) and all edges are the same length. Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. Some tetrahedra that are not regular, including the Schläfli orthoscheme and the Hill tetrahedron, can tessellate. 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.

The two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is near one of the edges the rectangle is long and skinny. When halfway between the two edges the intersection is a square. The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become wedges. This property also applies for tetragonal disphenoids when applied to the two special edge pairs.

Regular tetrahedra can be stacked face to face in a chiral aperiodic chain called the Boerdijk–Coxeter helix. In four dimensions, all the convex regular 4 polytopes with tetrahedral cells (the 5 cell, 16 cell and 600 cell) can be constructed as tilings of the 3 sphere by these chains, which become periodic in the three dimensional space of the 4 polytope's boundary surface.

Tetrahedral symmetry subgroup relationsTetrahedral symmetries shown in tetrahedral diagramsTetrahedra which do not have four equilateral faces are categorized and named by the symmetries they do possess. 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. 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.

In a trirectangular tetrahedron the three face angles at one vertex are right angles, as at the corner of a cube. Kepler discovered the relationship between the cube, regular tetrahedron and trirectangular tetrahedron.

A disphenoid is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron is a special case of a disphenoid. Other names for the same shape include bisphenoid, isosceles tetrahedron and equifacial tetrahedron.

A space filling tetrahedron packs with directly congruent or enantiomorphous (mirror image) copies of itself to tile space. The cube can be dissected into six 3 orthoschemes, three left handed and three right handed (one of each at each cube face), and cubes can fill space, so the characteristic 3 orthoscheme of the cube is a space filling tetrahedron in this sense. A disphenoid can be a space filling tetrahedron in the directly congruent sense, as in the disphenoid tetrahedral honeycomb. Regular tetrahedra, however, cannot fill space by themselves.

An irregular tetrahedron which is the fundamental domain of a symmetry group is an example of a Goursat tetrahedron. The Goursat tetrahedra generate all the regular polyhedra (and many other uniform polyhedra) by mirror reflections, a process referred to as Wythoff's kaleidoscopic construction. For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a kaleidoscope. Unlike a cylindrical kaleidoscope, Wythoff's mirrors are located at three faces of a Goursat tetrahedron such that all three mirrors intersect at a single point. Among the Goursat tetrahedra which generate 3 dimensional honeycombs we can recognize an orthoscheme (the characteristic tetrahedron of the cube), a double orthoscheme (the characteristic tetrahedron of the cube face bonded to its mirror image), and the space filling disphenoid illustrated above. The disphenoid is the double orthoscheme face bonded to its mirror image (a quadruple orthoscheme). Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can be dissected into characteristic tetrahedra of the cube.

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 nameEdgeequivalencediagramDescriptionSymmetrySchön. Cox. Orb. Ord. Regular tetrahedronFour equilateral trianglesIt forms the symmetry group Td, isomorphic to the symmetric group, S4. A regular tetrahedron has Coxeter diagram and Schläfli symbol {3,3}. TdT[3,3][3,3]+*332332 2412Triangular pyramidAn equilateral triangle base and three equal isosceles triangle sidesIt 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. A triangular pyramid has Schläfli symbol {3}∨( ). C3vC3 [3][3]+ *3333 63Mirrored sphenoidTwo equal scalene triangles with a common base edgeThis 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[ ] * 2Irregular tetrahedron(No symmetry)Four unequal triangles

Its only isometry is the identity, and the symmetry group is the trivial group. An irregular tetrahedron has Schläfli symbol ( )∨( )∨( )∨( ). C1[ ]+11Disphenoids (Four equal triangles)Tetragonal disphenoidFour 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. A tetragonal disphenoid has Coxeter diagram and Schläfli symbol s{2,4}. D2dS4 [2+,4][2+,4+] 2*22× 84Rhombic disphenoidFour 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. A rhombic disphenoid has Coxeter diagram and Schläfli symbol sr{2,2}. D2 [2,2]+ 2224Generalized disphenoids (2 pairs of equal triangles)Digonal disphenoidTwo 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. A digonal disphenoid has Schläfli symbol { }∨{ }. C2vC2 [2][2]+ *222242Phyllic disphenoid Two pairs of equal scalene or isosceles 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

Tetrahedra subdivision is a process used in computational geometry and 3D modeling to divide a tetrahedron into several smaller tetrahedra. This process enhances the complexity and detail of tetrahedral meshes, which is particularly beneficial in numerical simulations, finite element analysis, and computer graphics. One of the commonly used subdivision methods is the Longest Edge Bisection (LEB), which identifies the longest edge of the tetrahedron and bisects it at its midpoint, generating two new, smaller tetrahedra. When this process is repeated multiple times, bisecting all the tetrahedra generated in each previous iteration, the process is called iterative LEB. A similarity class is the set of tetrahedra with the same geometric shape, regardless of their specific position, orientation, and scale. So, any two tetrahedra belonging to the same similarity class may be transformed to each other by an affine transformation. The outcome of having a limited number of similarity classes in iterative subdivision methods is significant for computational modeling and simulation. It reduces the variability in the shapes and sizes of generated tetrahedra, preventing the formation of highly irregular elements that could compromise simulation results. The iterative LEB of the regular tetrahedron has been shown to produce only 8 similarity classes. Furthermore, in the case of nearly equilateral tetrahedra where their two longest edges are not connected to each other, and the ratio between their longest and their shortest edge is less than or equal to , the iterated LEB produces no more than 37 similarity classes.

Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron.

For tetrahedra in hyperbolic space or in three dimensional elliptic 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.

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's center of mass computes as the arithmetic mean of its four vertices, see Centroid.

The sum of the areas of any three faces is greater than the area of the fourth face. A tetrahedron can have integer volume and consecutive integers as edges, an example being the one with edges 6, 7, 8, 9, 10, and 11 and volume 48.

A regular tetrahedron can be seen as a triangular pyramid. A regular tetrahedron can be seen as a degenerate polyhedron, a uniform digonal antiprism, where base polygons are reduced digons. A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual digonal trapezohedron, containing 6 vertices, in two sets of colinear edges. 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. This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane. The tetrahedron is topologically related to a series of regular polyhedra and tilings with order 3 vertex figures. 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. Superimposing both forms gives a compound of ten tetrahedra, in which the ten tetrahedra are arranged as five pairs of stellae octangulae. A stella octangula is a compound of two tetrahedra in dual position and its eight vertices define a cube as their convex hull. The square hosohedron is another polyhedron with four faces, but it does not have triangular faces. The Szilassi polyhedron and the tetrahedron are the only two known polyhedra in which each face shares an edge with each other face. Furthermore, the Császár polyhedron (itself is the dual of Szilassi polyhedron) and the tetrahedron are the only two known polyhedra in which every diagonal lies on the sides.

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.

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. 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.

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 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 arccos(−), or approximately 109.47°. 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.

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).

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

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.