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 2 
CoxeterDynkin  
Symmetry  T_{d} or (*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) 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 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.
The following Cartesian coordinates define the vertices of a tetrahedron with edgelength 2√2, centered at the origin:
The compound of two tetrahedra shows the regular tetrahedron exists as two alternate sets of vertices of the cube.
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 [2], [3]  (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 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 which shares with it three converging edges.
The triple scalar can be represented by the following determinants:
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.^{[4]}
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 ^{[5]}. 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 nine point circle, but does not generally pass through the base points of the altitudes of the reference tetrahedron.^{[6]}
To resolve these inconsistencies, a substitute center known as the Monge point that always exists for a generalized tetrahedron is introduced. This point was first identified by Gaspard Monge. For tetrahedra where the altitudes do intersect, the Monge point and the orthocenter coincide. The Monge point is defined as 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.
An orthogonal line dropped from the Monge point to any face is coplanar with two other orthogonal lines to the same face. The first is an altitude dropped from a corresponding vertex to the chosen face. The second is an orthogonal line to the chosen face that passes through the orthocenter of that face. This orthogonal line through the Monge point lies mid way between the altitude and the orthocentric orthogonal line.
The Monge point, centroid and circumcenter of a tetrahedron are colinear and form the Euler line of the tetrahedron. However, unlike the triangle, the centroid of a tetrahedron lies at the midpoint of its Monge point and circumcenter.
There is an equivalent sphere to the triangular ninepoint circle for the generalized tetrahedron. It is the circumsphere of its medial tetrahedron. It is a twelve point sphere centered at the circumcenter of the medial tetrahedron. By definition it passes through the centroids of the four faces of the reference tetrahedron. It passes through four substitute Euler points that are located at a distance of 1/3 of the way from M, 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.^{[7]}
If T represents the center of the twelvepoint sphere, then it also lies on the Euler line. However, unlike its triangular counterpart, the 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:
and the radius of the circumsphere is given by:
which gives the radius of the twelvepoint sphere:
where:
The vector position of various centers are given as follows:
The centroid
The circumcenter
The Monge point
The Euler line relationships are:
where is twelvepoint center.
Also:
and:
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
For the other tetrahedron (which is dual to the first), reverse all the signs. The volume of this tetrahedron is 1/3 the volume of the cube. Combining both tetrahedra gives a regular polyhedral compound called the stella octangula, whose interior is an octahedron. Correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron). 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 which can tile space.
However, there is at least one irregular tetrahedron of which copies can tile space. If one relaxes the requirement that the tetrahedra be all the same shape, one can tile space using only tetrahedra in various ways. For example, one can divide an octahedron into four identical tetrahedra and combine them again with two regular ones. (As a sidenote: these two kinds of tetrahedron have the same volume.)
The tetrahedron is unique among the uniform polyhedra in possessing no parallel faces.
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  Truncated tetrahedron 
Rectified tetrahedron (Octahedron) 
Bitruncated tetrahedron (Truncated tetrahedron) 
Birectified tetrahedron (Tetrahedron) 

Picture  
CoxeterDynkin diagram 
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.
The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also animation, showing one of the two tetrahedra in the cube). The symmetries of a regular tetrahedron correspond to half of those of a cube: those which map the tetrahedrons to themselves, and not to each other.
The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion.
The regular tetrahedron has 24 isometries, forming the symmetry group T_{d}, isomorphic to S_{4}. They can be categorized as follows:
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.
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.
Color Translation




Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
TETRAHEDRON (Gr. TErpa, four, Spa, face or base), in geometry, a solid bounded by four triangular faces. It consequently has four vertices and six edges. If the faces be all equal equilateral triangles the solid is termed the "regular" tetrahedron. This is one of the Platonic solids, and is treated in the article Polyhedron, as is also the derived Archimedean solid named the "truncated tetrahedron"; in addition, the regular tetrahedron has important crystallographic relations, being the hemihedral form of the regular octahedron and consequently a form of the cubic system. The bisphenoids (the hemihedral forms of the tetragonal and rhombic bipyramids)., and the trigonal pyramid of the hexagonal system, are examples of nonregular tetrahedra (see Crystallography). "Tetrahedral coordinates" are a system of quadriplanar coordinates, the fundamental planes being the faces of a tetrahedron, and the coordinates the perpendicular distances of the point from the faces, a positive sign being given if the point be between the face and the opposite vertex, and a negative sign if not. If (u, v, w, t) be the coordinates of any point, then the relation u+vFwfit=R, where R is a constant, invariably holds. This system is of much service in following out mathematical, physical and chemical problems in which it is necessary to represent four variables.
Related to the tetrahedron are two spheres which have received much attention. The "twelvepoint sphere," discovered by P. M. E. Prouhet (18171867) in 1863, is somewhat analogous to the ninepoint circle of a triangle. If the perpendiculars from the vertices to the opposite faces of a tetrahedron be concurrent, then a sphere passes through the four feet of the perpendiculars, and consequently through the centre of gravity of each of the four faces, and through the midpoints of the segments of the perpendiculars between the vertices and their common point of intersection. This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the midpoints of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.
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Categories: TENTHE  Mathematics
A tetrahedron (triangular pyramid) is a three dimensional shape. It has four corners. It looks like a pyramid. It has six equally long edges, four corners and four equilateral triangular faces. Every two edges meet on one of those corners forming a sixtydegree angle.
