 Tetrahedron

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} Schläfli symbol {3,3} and s{2,2} Wythoff symbol 3  2 3
 2 2 2CoxeterDynkin
Symmetry T_{d}, [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)
NetIn geometry, 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. The tetrahedron is the only convex polyhedron that has four 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 nets.^{[1]}
For any tetrahedron there exists a sphere (the circumsphere) such that the tetrahedron's vertices lie on the sphere's surface.
Contents
Formulas for regular tetrahedron
For a regular tetrahedron of edge length a:
Base plane area Surface area^{[2]} Height ^{[3]} Volume^{[2]} Angle between an edge and a face
(approx. 54.7356°)Angle between two faces^{[2]}
(approx. 70.5288°)Angle between the segments joining the center and the vertices,^{[4]} also 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).
Volume
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−b, b−c, 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 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:
 or where is expressed as a row or column vector etc.
 Hence
 where etc.
 which gives
where are the plane angles occurring in vertex d. The angle is the angle between the two edges connecting the vertex d to the vertices b and c. The angle does so for the vertices a and c, while 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 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.^{[5]}
Distance between the edges
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. Let d be the distance between the skew lines formed by opposite edges a and bc as calculated in.^{[6]} Then another volume formula is given by
Properties of a generalized tetrahedron
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 nine point circle, but does not generally pass through the base points of the altitudes of the reference tetrahedron.^{[7]}
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, the Monge point and the orthocenter coincide.
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.
The centroid of a tetrahedron lies at the midpoint of its Monge point and circumcenter, on the Euler line of the tetrahedron.
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.^{[8]}
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 mid point 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.
If OABC forms a generalized tetrahedron with a vertex O as the origin and vectors and represent the positions of the vertices A, B, and C with respect to O, then the radius of the insphere is given by^{[citation needed]}:
and the radius of the circumsphere is given by:
which gives the radius of the twelvepoint sphere:
where:
In the formulas throughout this section, the scalar a² represents the inner vector product a·a; similarly b² and c².
The vector positions of various centers are as follows:
The centroid
The incenter
The circumcenter
The Monge point
The Euler line relationships are:
where is twelvepoint center.
Also:
and:
Geometric relations
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
 (+1, +1, +1);
 (−1, −1, +1);
 (−1, +1, −1);
 (+1, −1, −1).
This yields a tetrahedron with edgelength 2√2, 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, 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 sidenote: these two kinds of tetrahedron have the same volume.)
The tetrahedron is unique among the uniform polyhedra in possessing no parallel faces.
Related 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 tetahedron. The process completes as a birectification, reducing the original faces down to points, and producing the selfdual tetrahedron once again.
Name tetrahedron rectified tetrahedron
(octahedron)truncated tetrahedron cantellated tetrahedron
(cuboctahedron)omnitruncated tetrahedron
(truncated octahedron)Snub tetrahedron
(icosahedron)Schläfli {3,3} t_{1}{3,3} t_{0,1}{3,3} t_{0,2}{3,3} t_{0,1,2}{3,3} s{3,3} CoxeterDynkin Graph
(A_{3})Graph
(A_{2})Solid Tiling
This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,9}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 lefthanded and righthanded forms, which are mirror images of each other.
Isometries
Isometries of regular tetrahedra
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 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 S_{4}. They can be categorized as follows:
 T, isomorphic to alternating group A_{4} (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 facetoface axes
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 3dimensional point group is formed.
 An equilateral triangle base and isosceles (and nonequilateral) 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 S_{3}.
 Four congruent isosceles (nonequilateral) 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 fourgroup V_{4} ≅ Z_{2}^{2}, present as the point group D_{2}. A tetrahedron with this symmetry is called disphenoid.
 Two pairs of isomorphic isosceles (nonequilateral) 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 V_{4}.
 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 C_{2} isomorphic to Z_{2}.
 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 Z_{2}.
 No edges equal, so that the only isometry is the identity, and the symmetry group is the trivial group.
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 in a sense 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. One reason to be interested in this "independence" relation is this: It is widely known that three angles are the angles of some triangle if and only if their sum is a halfcircle. 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 a halfcircle. 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, not from 8 down to 4, but only from 8 down to 5, since the fourth constraint is not independent of the first three. Thus the space of all shapes of tetrahedra is 5dimensional.
Applications
Numerical analysis
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.
Chemistry
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.
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.^{[9]}^{[10]}
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.
Games
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.
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).^{[11]}
Contemporary Art
The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps. It was shown at the light art biennale Austria 2010.^{[12]}
It is used as album artwork, surrounded by black flames on The End of All Things to Come by Mudvayne.
Geology
The tetrahedral hypothesis, originally published by William Lowthian Green to explain the formation of the Earth,^{[13]} was popular through the early 20th century.^{[14]}^{[15]}
See also
 Boerdijk–Coxeter helix
 Caltrop
 Demihypercube
 Disphenoid — a tetrahedron with mirror symmetry
 Hill tetrahedron
 Simplex
 TetraPak
 Tetrahedral kite
 Tetrahedral number
 Triangular dipyramid — constructed by joining two tetrahedra along one face
 Tetrahedron packing
References
 ^ ^{a} ^{b} Weisstein, Eric W., "Tetrahedron" from MathWorld.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} Coxeter, H. S. M.: Regular Polytopes (Methuen and Co., 1948). Table I(i).
 ^ http://www.mathematischebasteleien.de/tetrahedron.htm
 ^ "Angle Between 2 Legs of a Tetrahedron" – Maze5.net
 ^ http://www.mathpages.com/home/kmath424.htm
 ^ http://en.wikipedia.org/wiki/Skew_lines#Distance_between_two_skew_lines</a>
 ^ Havlicek, H.; Weiß, G. (2003). "Altitudes of a tetrahedron and traceless quadratic forms". American Mathematical Monthly 110 (8): 679–693.. doi:10.2307/3647851. JSTOR 3647851. http://www.geometrie.tuwien.ac.at/havlicek/pub/hoehen.pdf.
 ^ Outudee, Somluck; New, Stephen. The Various Kinds of Centres of Simplices. Dept of Maths., Chulalongkorn University, Bangkok. http://www.math.sc.chula.ac.th/ICAA2002/pages/Somluck_Outudee.pdf.^{[dead link]}
 ^ Klein, Douglas J. (2002). "ResistanceDistance Sum Rules" (PDF). Croatica Chemica Acta 75 (2): 633–649. http://jagor.srce.hr/ccacaa/CCAPDF/cca2002/v75n2/CCA_75_2002_633_649_KLEIN.pdf. Retrieved 20060915.
 ^ Tomáš Záležák (18 October 2007). Resistance of a regular tetrahedron (PDF). Retrieved 25 Jan 2011.
 ^ Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). HP Technical Report HPL9895: 1–32. http://www.hpl.hp.com/techreports/98/HPL9895.pdf.
 ^ LightartBiennale Austria 2010
 ^ William Lowthian Green (1875). Vestiges of the Molten Globe, as exhibited in the figure of the earth, volcanic action and physiography. Part I. London: E. Stanford. OCLC 3571917. http://books.google.com/books?id=9DkDAAAAQAAJ.
 ^ Arthur Holmes (1965). Principles of physical geology. Nelson. p. 32. http://books.google.com/books?id=XUJRAAAAMAAJ.
 ^ Charles Henry Hitchcock (January 1900). "William Lowthian Green and his Theory of the Evolution of the Earth's Features". The American geologist (Geological Publishing Company) XXV: pp. 1–10. http://books.google.com/books?id=_Ty8AAAAIAAJ&pg=PA1.
External links
 F. M. Jackson and Weisstein, Eric W., "Tetrahedron" from MathWorld.
 Weisstein, Eric W., "Monge point" from MathWorld.
 Weisstein, Eric W., "Euler points" from 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
 Piero della Francesca's formula for tetrahedron volume at MathPages
 Free paper models of a tetrahedron and many other polyhedra
 An Amazing, Space Filling, Nonregular 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 partialmesh data network
 Explicit exact formulas for the inertia tensor of an arbitrary tetrahedron in terms of its vertex coordinates
 The inertia tensor of a tetrahedron
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