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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...
, is a polyhedron composed of four triangular faces, six straight
edges Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
, and four vertex corners. 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 In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...
, and may thus also be called a 3-simplex. The tetrahedron is one kind of
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...
, 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the tetrahedron's faces.


Regular tetrahedron

A regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles. It is one of the five regular
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s, 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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.


Coordinates for a regular tetrahedron

The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, and two level edges: :\left(\pm 1, 0, -\frac\right) \quad \mbox \quad \left(0, \pm 1, \frac\right) Expressed symmetrically as 4 points on the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A u ...
, centroid at the origin, with lower face level, the vertices are: v_1 = \left(\sqrt,0,-\frac\right) v_2 = \left(-\sqrt,\sqrt,-\frac\right) v_3 = \left(-\sqrt,-\sqrt,-\frac\right) v_4 = (0,0,1) with the edge length of \sqrt. Still another set of coordinates are based on an alternated
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...
or demicube with edge length 2. This form has Coxeter diagram and Schläfli symbol h. The tetrahedron in this case has edge length 2. Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube. :Tetrahedron: (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)


Angles and distances

For a regular tetrahedron of edge length ''a'': With respect to the base plane the
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...
of a face (2) 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 Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...
). For a regular tetrahedron with side length ''a'', radius ''R'' of its circumscribing sphere, and distances ''di'' from an arbitrary point in 3-space to its four vertices, we have :\begin\frac + \frac&= \left(\frac + \frac\right)^2;\\ 4\left(a^4 + d_1^4 + d_2^4 + d_3^4 + d_4^4\right) &= \left(a^2 + d_1^2 + d_2^2 + d_3^2 + d_4^2\right)^2.\end


Isometries of the regular tetrahedron

The vertices of a
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...
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 In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
. The regular tetrahedron has 24 isometries, forming the symmetry group Td, ,3 (*332), isomorphic to the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
, ''S''4. They can be categorized as follows: * T, ,3sup>+, (332) is 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 , etc.; ) ** rotation by an angle of 180° such that an edge maps to the opposite edge: , etc.; ) * 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 projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if i ...
s, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A2
Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
.


Cross section of regular tetrahedron

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 In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
. 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 A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by conv ...
. This property also applies for tetragonal disphenoids when applied to the two special edge pairs.


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 Conformal may refer to: * Conformal (software), in ASIC Software * Conformal coating in electronics * Conformal cooling channel, in injection or blow moulding * Conformal field theory in physics, such as: ** Boundary conformal field theory ...
, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.


Helical stacking

Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix. In
four dimensions Four Dimensions may refer to: * ''Four Dimensions'' (Don Patterson album), 1968 * ''Four Dimensions'' (Lollipop F album), 2010 See also *''Four Dimensions of Greta ''Four Dimensions of Greta'' is a 1972 British sex comedy film directed and pro ...
, 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.


Irregular tetrahedra

Tetrahedra 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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
, then it is called an
orthocentric tetrahedron In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, ...
. 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.


Trirectangular 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.


Disphenoid

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.


Orthoschemes

A 3-orthoscheme is a tetrahedron where all four faces are right triangles. An orthoscheme is an irregular
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...
that is the convex hull of a
tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
in which all edges are mutually perpendicular. In a 3-dimensional orthoscheme, the tree consists of three perpendicular edges connecting all four vertices in a linear path that makes two right-angled turns. The 3-orthoscheme is a tetrahedron having two right angles at each of two vertices, so another name for it is birectangular tetrahedron. It is also called a quadrirectangular tetrahedron because it contains four right angles. Coxeter also calls quadrirectangular tetrahedra characteristic tetrahedra, because of their integral relationship to the regular polytopes and their symmetry groups. For example, the special case of a 3-orthoscheme with equal-length perpendicular edges is characteristic of the cube, which means that the cube can be subdivided into instances of this orthoscheme. If its three perpendicular edges are of unit length, its remaining edges are two of length and one of length , so all its edges are edges or diagonals of the cube. The cube can be dissected into six such 3-orthoschemes four different ways, with all six surrounding the same cube diagonal. The cube can also be dissected into 48 ''smaller'' instances of this same characteristic 3-orthoscheme (just one way, by all of its symmetry planes at once). The characteristic tetrahedron of the cube is an example of a Heronian tetrahedron. Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme. There is a 3-orthoscheme which is the characteristic tetrahedron of the regular tetrahedron. The regular tetrahedron is subdivided into 24 instances of its characteristic tetrahedron by its planes of symmetry. If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths \sqrt, 1, \sqrt (the exterior right triangle face, the ''characteristic triangle'' 𝟀, 𝝓, 𝟁), plus \sqrt, \sqrt, \sqrt (edges that are the ''characteristic radii'' of the regular tetrahedron). The 3-edge path along orthogonal edges of the orthoscheme is 1, \sqrt, \sqrt, first from a tetrahedron vertex to an tetrahedron edge center, then turning 90° to an tetrahedron face center, then turning 90° to the tetrahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 60-90-30 triangle which is one-sixth of a tetrahedron face. The three faces interior to the tetrahedron are: a right triangle with edges 1, \sqrt, \sqrt, a right triangle with edges \sqrt, \sqrt, \sqrt, and a right triangle with edges \sqrt, \sqrt, \sqrt.


Space-filling tetrahedra

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.


Fundamental domains

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 A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of hone ...
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.


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, sup>+), and (S4, +,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.


General properties


Volume

The volume of a tetrahedron is given by the pyramid volume formula: :V = \frac13 A_0\,h \, where ''A''0 is the area of the base and ''h'' is the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apices to the opposite faces are inversely proportional to the areas of these faces. For a tetrahedron with vertices , , , and , the volume is , or any other combination of pairs of vertices that form a simply connected
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
. This can be rewritten using a dot product and a
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
, yielding :V = \frac . If the origin of the coordinate system is chosen to coincide with vertex d, then d = 0, so :V = \frac , 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 of the volume of any parallelepiped that shares three converging edges with it. The absolute value of the scalar triple product can be represented as the following absolute values of determinants: :6 \cdot V =\begin \mathbf & \mathbf & \mathbf \endor6 \cdot V =\begin \mathbf \\ \mathbf \\ \mathbf \endwhere\begin\mathbf = (a_1,a_2,a_3), \\ \mathbf = (b_1,b_2,b_3), \\ \mathbf = (c_1,c_2,c_3), \endare expressed as row or column vectors. Hence :36 \cdot V^2 =\begin \mathbf & \mathbf \cdot \mathbf & \mathbf \cdot \mathbf \\ \mathbf \cdot \mathbf & \mathbf & \mathbf \cdot \mathbf \\ \mathbf \cdot \mathbf & \mathbf \cdot \mathbf & \mathbf \endwhere\begin\mathbf \cdot \mathbf = ab\cos, \\ \mathbf \cdot \mathbf = bc\cos, \\ \mathbf \cdot \mathbf = ac\cos. \end which gives :V = \frac \sqrt, \, 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. If we do not require that d = 0 then :6 \cdot V = \left, \det \left( \begin a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \\ 1 & 1 & 1 & 1 \end \right) \\,. Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant: :288 \cdot V^2 = \begin 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & d_^2 & d_^2 & d_^2 \\ 1 & d_^2 & 0 & d_^2 & d_^2 \\ 1 & d_^2 & d_^2 & 0 & d_^2 \\ 1 & d_^2 & d_^2 & d_^2 & 0 \end where the subscripts represent the vertices and ''d'' 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. Let be three edges that meet at a point, and the opposite edges. Let be the volume of the tetrahedron; then :V=\frac where :\beginX&=b^2+c^2-x^2, \\ Y&=a^2+c^2-y^2, \\ Z&=a^2+b^2-z^2. \end The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles. :V = \frac \sqrt


Heron-type formula for the volume of a tetrahedron

If , , , , , are lengths of edges of the tetrahedron (first three form a triangle; with opposite , opposite , opposite ), then : V = \frac where : \begin p & = \sqrt , & q & = \sqrt , & r & = \sqrt , & s & = \sqrt , \end : \begin 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


Volume divider

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 elliptic geometry, the
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...
s 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 as calculated here. Then another volume formula is given by :V = \frac .


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 Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. Duri ...
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 In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, ...
. 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. In addition the four medians are divided in a 3:1 ratio by the centroid (see Commandino's theorem). 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'', one third 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 one third 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 one third 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 -1 & \cos & \cos & \cos\\ \cos & -1 & \cos & \cos \\ \cos & \cos & -1 & \cos \\ \cos & \cos & \cos & -1 \\ \end = 0\, where ''α'' is the angle between the faces ''i'' and ''j''. The
geometric median In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distance ...
of the vertex position coordinates of a tetrahedron and its isogonic center are associated, under circumstances analogous to those observed for a triangle. Lorenz Lindelöf found that, corresponding to any given tetrahedron is a point now known as an isogonic center, ''O'', at which the solid angles subtended by the faces are equal, having a common value of π sr, and at which the angles subtended by opposite edges are equal. A solid angle of π sr is one quarter of that subtended by all of space. When all the solid angles at the vertices of a tetrahedron are smaller than π sr, ''O'' lies inside the tetrahedron, and because the sum of distances from ''O'' to the vertices is a minimum, ''O'' coincides with the
geometric median In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distance ...
, ''M'', of the vertices. In the event that the solid angle at one of the vertices, ''v'', measures exactly π sr, then ''O'' and ''M'' coincide with ''v''. If however, a tetrahedron has a vertex, ''v'', with solid angle greater than π sr, ''M'' still corresponds to ''v'', but ''O'' lies outside the tetrahedron.


Geometric relations

A tetrahedron is a 3-
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...
. 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 A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...
, and the regular tetrahedron is self-dual. A regular tetrahedron can be embedded inside a
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...
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 2, centered at the origin. For the other tetrahedron (which is
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
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 volume of this tetrahedron is one-third 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, five is the minimum number of tetrahedra required to compose a cube. To see this, starting from a base tetrahedron with 4 vertices, each added tetrahedra adds at most 1 new vertex, so at least 4 more must be added to make a cube, which has 8 vertices. 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 Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron that can tile space as the tetrahedral-octahedral honeycomb. However, several irregular tetrahedra are known, of which copies can tile space, for instance the characteristic orthoscheme of the cube and the disphenoid of 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 :\sin\angle OAB\cdot\sin\angle OBC\cdot\sin\angle OCA = \sin\angle OAC\cdot\sin\angle OCB\cdot\sin\angle OBA.\, 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 Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
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.


Law of cosines for tetrahedra

Let be the points of a tetrahedron. Let Δ''i'' be the area of the face opposite vertex ''Pi'' and let ''θij'' be the dihedral angle between the two faces of the tetrahedron adjacent to the edge ''PiPj''. The
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines stat ...
for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation: : \Delta_i^2 = \Delta_j^2 + \Delta_k^2 + \Delta_l^2 - 2(\Delta_j\Delta_k\cos\theta_ + \Delta_j\Delta_l \cos\theta_ + \Delta_k\Delta_l \cos\theta_)


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''Inequalities proposed in “ Crux Mathematicorum”''

:PA \cdot F_\mathrm + PB \cdot F_\mathrm + PC \cdot F_\mathrm + PD \cdot F_\mathrm \geq 9V. 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, and suppose the faces have equal areas, then :PA+PB+PC+PD \geq 3(PJ+PK+PL+PM).


Inradius

Denoting the inradius of a tetrahedron as ''r'' and the inradius, inradii of its triangular faces as ''r''''i'' for ''i'' = 1, 2, 3, 4, we have :\frac + \frac + \frac + \frac \leq \frac, with equality if and only if the tetrahedron is regular. If ''A''''1'', ''A''''2'', ''A''''3'' and ''A''''4'' denote the area of each faces, the value of ''r'' is given by :r=\frac. This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the original faces and the incenter. Since the four subtetrahedra fill the volume, we have V = \frac13A_1r+\frac13A_2r+\frac13A_3r+\frac13A_4r.


Circumradius

Denote the circumradius of a tetrahedron as ''R''. Let ''a'', ''b'', ''c'' be the lengths of the three edges that meet at a vertex, and ''A'', ''B'', ''C'' the length of the opposite edges. Let ''V'' be the volume of the tetrahedron. Then :R=\frac.


Circumcenter

The circumcenter of a tetrahedron can be found as intersection of three bisector planes. A bisector plane is defined as the plane centered on, and orthogonal to an edge of the tetrahedron. With this definition, the circumcenter of a tetrahedron with vertices ,,, can be formulated as matrix-vector product: :\begin C &= A^B & \text & \ & A = \left(\begin\left _1 - x_0\rightT \\ \left _2 - x_0\rightT \\ \left _3 - x_0\rightT \end\right) & \ & \text & \ & B = \frac\left(\begin \, x_1\, ^2 - \, x_0\, ^2 \\ \, x_2\, ^2 - \, x_0\, ^2 \\ \, x_3\, ^2 - \, x_0\, ^2 \end\right) \\ \end In contrast to the centroid, the circumcenter may not always lay on the inside of a tetrahedron. Analogously to an obtuse triangle, the circumcenter is outside of the object for an obtuse tetrahedron.


Centroid

The tetrahedron's center of mass computes as the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...
of its four vertices, see Centroid.


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. These are called Heronian tetrahedra. One example has one edge of 896, the opposite edge of 990 and the other four edges of 1073; two faces are
isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s with areas of and the other two are isosceles with areas of , while the volume is . 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. Wacław Sierpiński, ''
Pythagorean Triangles ''Pythagorean Triangles'' is a book on right triangles, the Pythagorean theorem, and Pythagorean triples. It was originally written in the Polish language by Wacław Sierpiński (titled ''Trójkąty pitagorejskie''), and published in Warsaw i ...
'', Dover Publications, 2003 (orig. ed. 1962), p. 107. Note however that Sierpiński repeats an erroneous calculation of the volume of the Heronian tetrahedron example above.


Related polyhedra and compounds

A regular tetrahedron can be seen as a triangular
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrila ...
. 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 , continuing into the hyperbolic plane. The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 vertex figures. Image:CubeAndStel.svg, Two tetrahedra in a cube Image:Compound of five tetrahedra.png, Compound of five tetrahedra Image:Compound of ten tetrahedra.png, 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. 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 In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. Coloring and symmetry The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surf ...
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.


Applications


Numerical analysis

In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
, complicated three-dimensional shapes are commonly broken down into, or
approximate An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
d 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 In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
. These methods have wide applications in practical applications in
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate t ...
, aerodynamics,
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classica ...
s,
civil engineering Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...
,
chemical engineering Chemical engineering is an engineering field which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials in ...
, naval architecture and engineering, and related fields.


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
airfield An aerodrome (Commonwealth English) or airdrome (American English) is a location from which aircraft flight operations take place, regardless of whether they involve air cargo, passengers, or neither, and regardless of whether it is for publ ...
s, 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.


Chemistry

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 Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Ear ...
molecule () or an ammonium ion (), 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°.
Water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
, , 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 of mixtures of chemical substances are represented graphically as tetrahedra. However, quaternary phase diagrams in
communication engineering Telecommunications Engineering is a subfield of electrical engineering which seeks to design and devise systems of communication at a distance. The work ranges from basic circuit design to strategic mass developments. A telecommunication engin ...
are represented graphically on a two-dimensional plane.


Electricity and electronics

If six equal
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias activ ...
s are soldered together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor. Since
silicon Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic ...
is the most common
semiconductor A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
used in solid-state electronics, and silicon has a
valence Valence or valency may refer to: Science * Valence (chemistry), a measure of an element's combining power with other atoms * Degree (graph theory), also called the valency of a vertex in graph theory * Valency (linguistics), aspect of verbs rel ...
of four, the tetrahedral shape of the four chemical bonds in silicon is a strong influence on how
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macr ...
s of silicon form and what shapes they assume.


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).


Games

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 Four-sided dice, abbreviated d4, are often used in tabletop role-playing games to obtain random integers in the range 1–4. Three forms exist of this die: a tetrahedron (pyramid shape) with four equilateral triangle-shaped faces, a rectangular pr ...
, one of the more common
polyhedral dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing ga ...
, 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.


Geology

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


Popular culture

Stanley Kubrick Stanley Kubrick (; July 26, 1928 – March 7, 1999) was an American film director, producer, screenwriter, and photographer. Widely considered one of the greatest filmmakers of all time, his films, almost all of which are adaptations of nove ...
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 Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machine A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, moveme ...
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.


Tetrahedral graph

The skeleton of the tetrahedron (comprising the vertices and edges) forms a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
, with 4 vertices, and 6 edges. It is a special case of the
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices ...
, K4, and wheel graph, W4. It is one of 5 Platonic graphs, each a skeleton of its
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
.


See also

* Boerdijk–Coxeter helix * Möbius configuration * Caltrop * Demihypercube and
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...
– ''n''-dimensional analogues * Pentachoron – 4-dimensional analogue * Synergetics (Fuller) * Tetrahedral kite * Tetrahedral number * Tetrahedron packing * Triangular dipyramid – constructed by joining two tetrahedra along one face * Trirectangular tetrahedron * Orthoscheme


Notes


References


Bibliography

* *


External links

*
Free paper models of a tetrahedron and many other polyhedra


that also includes a description of a "rotating ring of tetrahedra", also known as a kaleidocycle. {{Authority control Deltahedra Platonic solids Individual graphs Self-dual polyhedra Prismatoid polyhedra Pyramids and bipyramids