In

can tessellate

The regular tetrahedron is self-dual, which means that its

_{i}'' from an arbitrary point in 3-space to its four vertices, we have
:$\backslash begin\backslash frac\; +\; \backslash frac\&=\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)^2;\backslash \backslash \; 4\backslash left(a^4\; +\; d\_1^4\; +\; d\_2^4\; +\; d\_3^4\; +\; d\_4^4\backslash right)\; \&=\; \backslash left(a^2\; +\; d\_1^2\; +\; d\_2^2\; +\; d\_3^2\; +\; d\_4^2\backslash right)^2.\backslash end$

_{d}, ,3 (*332), isomorphic to the _{4}. They can be categorized as follows:
* T, ,3sup>+, (332) is isomorphic to _{4} (the identity and 11 proper rotations) with the following

_{2}

_{3}, sup>+), and (S_{4}, +,4+">^{+},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.

_{0} is the area of the

_{''i''} be the area of the face opposite vertex ''P_{i}'' and let ''θ_{ij}'' be the dihedral angle between the two faces of the tetrahedron adjacent to the edge ''P_{i}P_{j}''.
The

_{a}, ''F''_{b}, ''F''_{c}, and ''F''_{d}. Then''Inequalities proposed in “

:$PA\; \backslash cdot\; F\_\backslash mathrm\; +\; PB\; \backslash cdot\; F\_\backslash mathrm\; +\; PC\; \backslash cdot\; F\_\backslash mathrm\; +\; PD\; \backslash cdot\; F\_\backslash mathrm\; \backslash geq\; 9V.$ 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 :$PA+PB+PC+PD\; \backslash geq\; 3(PJ+PK+PL+PM).$

_{''i''} for ''i'' = 1, 2, 3, 4, we have
:$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; \backslash leq\; \backslash frac,$
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
:$r=\backslash frac$.
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 $V\; =\; \backslash frac13A\_1r+\backslash frac13A\_2r+\backslash frac13A\_3r+\backslash frac13A\_4r$.

pyramid
A pyramid (from el, πυραμίς ') is a structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act ...

.
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
Image:CubeAndStel.svg, Stella octangula, 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 Compound of five tetrahedra, five intersecting tetrahedra. This

^{3}-hybridized atoms are surrounded by atoms (or lone pair, lone electron pairs) at the four corners of a tetrahedron. For instance in a methane 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 (journal), Tetrahedron''. The central angle between any two vertices of a perfect tetrahedron is arccos(−), or approximately 109.47°.
Water, , 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 are represented graphically on a two-dimensional plane.

_{4}, and wheel graph, W_{4}. It is one of 5 Platonic graphs, each a skeleton of its

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

geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid
A pyramid (from el, πυραμίς ') is a structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act ...

, is a polyhedron
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

composed of four triangular
A triangle is a polygon
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conce ...

faces
The face is the front of an animal's head that features three of the head's Sense, sense organs, the eyes, nose, and mouth, and through which animals express many of their Emotion, emotions. The face is crucial for human Personal identity, ident ...

, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra
A convex polytope is a special case of a polytope
In elementary geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branc ...

and the only one that has fewer than 5 faces.
The tetrahedron is the three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...

case of the more general concept of a Euclidean
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of:
Geometry
*Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...

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 space.
For e ...

, and may thus also be called a 3-simplex.
The tetrahedron is one kind of pyramid
A pyramid (from el, πυραμίς ') is a structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act ...

, which is a polyhedron with a flat polygon
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

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 convex polytope is a special case of a polytope
In elementary geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branc ...

, 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
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term circumcircle. As in ...

) on which all four vertices lie, and another sphere (the insphere
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

) tangent
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

to the tetrahedron's faces.
Regular tetrahedron

A regular tetrahedron is a tetrahedron in which all four faces areequilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular polygon, equiangular; that is, all three internal angles are also con ...

s. It is one of the five regular Platonic solid
In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...

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
A tiling or tessellation of a flat surface is the covering of a plane (mathematics), plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to high-dimensional sp ...

(fill space), but if alternated with in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation
A tiling or tessellation of a flat surface is the covering of a plane (mathematics), plane using one or more geometric shapes, call ...

, which is a tessellation. Some tetrahedra that are not regular, including the Schläfli orthoschemeIn geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

and the Hill tetrahedronIn geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

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 ...

is another regular tetrahedron. The compound
Compound may refer to:
Architecture and built environments
* Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall
** Compound (fortification), a version of the above fortified with defensive structu ...

figure comprising two such dual tetrahedra form a stellated octahedron
The stellated octahedron is the only stellation of the octahedron
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldes ...

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: :$\backslash left(\backslash pm\; 1,\; 0,\; -\backslash frac\backslash right)\; \backslash quad\; \backslash mbox\; \backslash quad\; \backslash left(0,\; \backslash pm\; 1,\; \backslash frac\backslash right)$ Expressed symmetrically as 4 points on theunit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...

, centroid at the origin, with lower face level, the vertices are:
$v\_1\; =\; \backslash left(\backslash sqrt,0,-\backslash frac\backslash right)$
$v\_2\; =\; \backslash left(-\backslash sqrt,\backslash sqrt,-\backslash frac\backslash right)$
$v\_3\; =\; \backslash left(-\backslash sqrt,-\backslash sqrt,-\backslash frac\backslash right)$
$v\_4\; =\; (0,0,1)$
with the edge length of $\backslash sqrt$.
Still another set of coordinates are based on an alternated cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

or demicube with edge length 2. This form has Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (February 9, 1907 – March 31, 2003) was a British and later also Canadian geometer
A geometer is a mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
...

and Schläfli symbol
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

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 theslope
In mathematics, the slope or gradient of a line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', ...

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
Apex may refer to:
Arts and media Fictional entities
* Apex (comics), a teenaged super villainess in the Marvel Universe
* Ape-X, a super-intelligent ape in the Squadron Supreme universe
*Apex, a genetically-engineered human population in the TV s ...

along an edge is twice that along the median
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...

of a face. In other words, if ''C'' is the centroid
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a construct in proof theory
* Mathematical proof, a co ...

).
For a regular tetrahedron with side length ''a'', radius ''R'' of its circumscribing sphere, and distances ''dIsometries of the regular tetrahedron

The vertices of acube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also animation
Animation is a method in which figures
Figure may refer to:
General
*A shape, drawing, depiction, or geometric configuration
*Figure (wood), wood appearance
*Figure (music), distinguished from musical motif
*Noise figure, in telecommunication ...

, showing one of the two tetrahedra in the cube). The symmetries
Symmetry (from Ancient Greek, Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" ...

of a regular tetrahedron correspond to half of those of a cube: those that map the tetrahedra to themselves, and not to each other.
The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion.
The regular tetrahedron has 24 isometries, forming the symmetry group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...

Tsymmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

, ''S''alternating group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, ''A''conjugacy class
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

es (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 specialorthogonal projection
In linear algebra and functional analysis
Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common const ...

s, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the ACoxeter plane
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

.
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 arectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

. 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
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

. The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become wedges.
This property also applies for tetragonal disphenoid
In geometry, a disphenoid (from Greek sphenoeides, "wedgelike") is a tetrahedron whose four faces are Congruence (geometry), congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each ...

s when applied to the two special edge pairs.
Spherical tiling

The tetrahedron can also be represented as aspherical tiling
Image:BeachBall.jpg, This beach ball would be a hosohedron with 6 spherical lune faces, if the 2 white caps on the ends were removed.
In mathematics, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the s ...

, and projected onto the plane via a stereographic projection
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

. This projection is , 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 theBoerdijk–Coxeter helix
The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, - ...

. In four dimensions, all the convex regular 4-polytope
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s with tetrahedral cells (the 5-cell
In geometry, the 5-cell is a four-dimensional space, four-dimensional object bounded by 5 tetrahedron, tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or #Irregular 5-cell, tetrahedral pyramid. It is the 4-si ...

, 16-cell
In , a 16-cell is a . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid.
It is a part of an infinite family of polytopes ...

and 600-cell
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

) can be constructed as tilings of the 3-sphere
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.
Other special cases

An isosceles tetrahedron, also called adisphenoid
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

, is a tetrahedron where all four faces are congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In modu ...

triangles. A space-filling tetrahedron packs with congruent copies of itself to tile space, like the disphenoid tetrahedral honeycomb.
In a trirectangular tetrahedron
In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one Vertex (geometry), vertex are right angles. That vertex is called the ''right angle'' of the trirectangular tetrahedron and the face opposite it is called ...

the three face angles at one vertex are right angle
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

s. If all three pairs of opposite edges of a tetrahedron are perpendicular
In elementary geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

, then it is called an orthocentric tetrahedronIn geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

. When only one pair of opposite edges are perpendicular, it is called a semi-orthocentric tetrahedron. An isodynamic tetrahedron is one in which the cevian In geometry, a cevian is a Line (geometry), line that intersects both a triangle's Vertex (geometry), vertex, and also the side that is opposite to that vertex. Median (geometry), Medians and angle bisectors are special cases of cevians. The name " ...

s that join the vertices to the incenters of the opposite faces are concurrent
Concurrency, concurrent, or concurrence may refer to:
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion
In law, a concurring opinion is in certain legal systems a written opinion
An opin ...

, and an isogonic tetrahedron has concurrent cevians that join the vertices to the points of contact of the opposite faces with the inscribed sphere
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

of the tetrahedron.
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 (CGeneral properties

Volume

The volume of a tetrahedron is given by the pyramid volume formula: :$V\; =\; \backslash frac13\; A\_0\backslash ,h\; \backslash ,$ where ''A''base
Base or BASE may refer to:
Brands and enterprises
* Base (mobile telephony provider), a Belgian mobile telecommunications operator
*Base CRM
Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...

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 apexes 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 discret ...

. This can be rewritten using a dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

and a cross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

, yielding
:$V\; =\; \backslash frac\; .$
If the origin of the coordinate system is chosen to coincide with vertex d, then d = 0, so
:$V\; =\; \backslash frac\; ,$
where a, b, and c represent three edges that meet at one vertex, and is a scalar triple product
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

. Comparing this formula with that used to compute the volume of a parallelepiped
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

, 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:
:$6\; \backslash cdot\; V\; =\backslash begin\; \backslash mathbf\; \&\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash end$or$6\; \backslash cdot\; V\; =\backslash begin\; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \backslash end$where$\backslash begin\backslash mathbf\; =\; (a\_1,a\_2,a\_3),\; \backslash \backslash \; \backslash mathbf\; =\; (b\_1,b\_2,b\_3),\; \backslash \backslash \; \backslash mathbf\; =\; (c\_1,c\_2,c\_3),\; \backslash end$are expressed as row or column vectors.
Hence
:$36\; \backslash cdot\; V^2\; =\backslash begin\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \&\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash end$where$\backslash begin\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; ab\backslash cos,\; \backslash \backslash \; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; bc\backslash cos,\; \backslash \backslash \; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; ac\backslash cos.\; \backslash end$
which gives
:$V\; =\; \backslash frac\; \backslash sqrt,\; \backslash ,$
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
:$6\; \backslash cdot\; V\; =\; \backslash left,\; \backslash det\; \backslash left(\; \backslash begin\; a\_1\; \&\; b\_1\; \&\; c\_1\; \&\; d\_1\; \backslash \backslash \; a\_2\; \&\; b\_2\; \&\; c\_2\; \&\; d\_2\; \backslash \backslash \; a\_3\; \&\; b\_3\; \&\; c\_3\; \&\; d\_3\; \backslash \backslash \; 1\; \&\; 1\; \&\; 1\; \&\; 1\; \backslash end\; \backslash right)\; \backslash \backslash ,.$
Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinantIn linear algebra, geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properti ...

:
:$288\; \backslash cdot\; V^2\; =\; \backslash begin\; 0\; \&\; 1\; \&\; 1\; \&\; 1\; \&\; 1\; \backslash \backslash \; 1\; \&\; 0\; \&\; d\_^2\; \&\; d\_^2\; \&\; d\_^2\; \backslash \backslash \; 1\; \&\; d\_^2\; \&\; 0\; \&\; d\_^2\; \&\; d\_^2\; \backslash \backslash \; 1\; \&\; d\_^2\; \&\; d\_^2\; \&\; 0\; \&\; d\_^2\; \backslash \backslash \; 1\; \&\; d\_^2\; \&\; d\_^2\; \&\; d\_^2\; \&\; 0\; \backslash end$
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
Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an list of Italian painters, Italian painter of the Italian Renaissance, Early Renaissance. To contemporaries he was also known as a mathematician ...

in the 15th century, as a three dimensional analogue of the 1st century Heron's formula
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

for the area of a triangle.
Denote be three edges that meet at a point, and the opposite edges. Let be the volume of the tetrahedron; then
:$V=\backslash frac$
where
:$\backslash beginX\&=b^2+c^2-x^2,\; \backslash \backslash \; Y\&=a^2+c^2-y^2,\; \backslash \backslash \; Z\&=a^2+b^2-z^2.\; \backslash end$
The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles.
:$V\; =\; \backslash frac\; \backslash sqrt$
Heron-type formula for the volume of a tetrahedron

If , , , , , are lengths of edges of the tetrahedron (first three form a triangle; opposite to and so on), then :$V\; =\; \backslash frac$ where : $\backslash begin\; p\; \&\; =\; \backslash sqrt\; ,\; \&\; q\; \&\; =\; \backslash sqrt\; ,\; \&\; r\; \&\; =\; \backslash sqrt\; ,\; \&\; s\; \&\; =\; \backslash sqrt\; ,\; \backslash end$ : $\backslash begin\; X\; \&\; =\; (w\; -\; U\; +\; v)\backslash ,(U\; +\; v\; +\; w),\; \&\; x\; \&\; =\; (U\; -\; v\; +\; w)\backslash ,(v\; -\; w\; +\; U),\; \backslash \backslash \; Y\; \&\; =\; (u\; -\; V\; +\; w)\backslash ,(V\; +\; w\; +\; u),\; \&\; y\; \&\; =\; (V\; -\; w\; +\; u)\backslash ,(w\; -\; u\; +\; V),\; \backslash \backslash \; Z\; \&\; =\; (v\; -\; W\; +\; u)\backslash ,(W\; +\; u\; +\; v),\; \&\; z\; \&\; =\; (W\; -\; u\; +\; v)\backslash ,(u\; -\; v\; +\; W).\; \backslash end$Volume divider

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

For tetrahedra inhyperbolic space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

or in three-dimensional elliptic geometry
Elliptic geometry is an example of a geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conc ...

, the dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, ele ...

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 twoskew lines
In three-dimensional geometry, skew lines are two Line (geometry), lines that do not Line-line intersection, intersect and are not Parallel (geometry), parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of ...

, 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
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV
Here TV is an America ...

. Then another volume formula is given by
:$V\; =\; \backslash frac\; .$
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
In geometry, the Spieker center is a special point associated with a plane (geometry), plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle ABC is the center of gravity of a homogeneo ...

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
Technical drawing, drafting or drawing, i ...

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 tetrahedronIn geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

.
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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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
Concurrency, concurrent, or concurrence may refer to:
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion
In law, a concurring opinion is in certain legal systems a written opinion
An opin ...

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
Commandino's theorem, named after Federico Commandino (1509–1575), states that the four Median (geometry), medians of a tetrahedron are concurrent at a point ''S'', which divides them in a 3:1 ratio. In a tetrahedron a median is a line segment th ...

). 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 300px, Euler's line (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "ea ...

of a triangle.
The nine-point circle
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

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
:$\backslash begin\; -1\; \&\; \backslash cos\; \&\; \backslash cos\; \&\; \backslash cos\backslash \backslash \; \backslash cos\; \&\; -1\; \&\; \backslash cos\; \&\; \backslash cos\; \backslash \backslash \; \backslash cos\; \&\; \backslash cos\; \&\; -1\; \&\; \backslash cos\; \backslash \backslash \; \backslash cos\; \&\; \backslash cos\; \&\; \backslash cos\; \&\; -1\; \backslash \backslash \; \backslash end\; =\; 0\backslash ,$
where ''α'' is the angle between the faces ''i'' and ''j''.
The geometric median
The geometric median of a discrete set of sample points in a Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there a ...

of the vertex position coordinates of a tetrahedron and its isogonic center are associated, under circumstances analogous to those observed for a triangle. 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
The geometric median of a discrete set of sample points in a Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there a ...

, ''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 space.
For e ...

. 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
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act ...

, and the regular tetrahedron is self-dual.
A regular tetrahedron can be embedded inside a cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

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 ...

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 compoundA polyhedral compound is a figure that is composed of several polyhedra sharing a ''common centre''. They are the three-dimensional analogs of polygonal compounds such as the hexagram
, can be seen as a compound polygon, compound composed of an ...

called the compound of two tetrahedra
In geometry, a Polyhedral compound, compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.
Stellated octahedron
There is only one uniform polyhedral compound, the stellated octahedron, whi ...

or stella octangula
The stellated octahedron is the only stellation
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...

.
The interior of the stella octangula is an octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral tri ...

, 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
A rectifier is an electrical device that converts
Religious conversion is the adoption of a set of beliefs identified with one particular religious denomination
A religious denomination is a subgroup within a religion
Religion is ...

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
The compound of five cube
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concer ...

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
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental questio ...

claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

that can tile space.
However, several irregular tetrahedra are known, of which copies can tile space, for instance the disphenoid tetrahedral honeycomb. The complete list remains an open problem.
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 usuallaw of sines
In trigonometry
Trigonometry (from Greek '' trigōnon'', "triangle" and '' metron'', "measure") is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...

is that in a tetrahedron with vertices ''O'', ''A'', ''B'', ''C'', we have
:$\backslash sin\backslash angle\; OAB\backslash cdot\backslash sin\backslash angle\; OBC\backslash cdot\backslash sin\backslash angle\; OCA\; =\; \backslash sin\backslash angle\; OAC\backslash cdot\backslash sin\backslash angle\; OCB\backslash cdot\backslash sin\backslash angle\; OBA.\backslash ,$
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 system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...

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 Δlaw of cosines
In trigonometry
Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focu ...

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:
: $\backslash Delta\_i^2\; =\; \backslash Delta\_j^2\; +\; \backslash Delta\_k^2\; +\; \backslash Delta\_l^2\; -\; 2(\backslash Delta\_j\backslash Delta\_k\backslash cos\backslash theta\_\; +\; \backslash Delta\_j\backslash Delta\_l\; \backslash cos\backslash theta\_\; +\; \backslash Delta\_k\backslash Delta\_l\; \backslash cos\backslash theta\_)$
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''Crux Mathematicorum
''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students.
The journal was established in 1975, under the name '' ...

”'':$PA\; \backslash cdot\; F\_\backslash mathrm\; +\; PB\; \backslash cdot\; F\_\backslash mathrm\; +\; PC\; \backslash cdot\; F\_\backslash mathrm\; +\; PD\; \backslash cdot\; F\_\backslash mathrm\; \backslash geq\; 9V.$ 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 :$PA+PB+PC+PD\; \backslash geq\; 3(PJ+PK+PL+PM).$

Inradius

Denoting the inradius of a tetrahedron as ''r'' and the inradius, inradii of its triangular faces as ''r''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 :$R=\backslash frac.$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: :$\backslash begin\; C\; \&=\; A^B\; \&\; \backslash text\; \&\; \backslash \; \&\; A\; =\; \backslash left(\backslash begin\backslash left;\; href="/html/ALL/s/\_1\_-\_x\_0\backslash right.html"\; ;"title="\_1\; -\; x\_0\backslash right">\_1\; -\; x\_0\backslash right$ 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 of its four vertices, see Centroid#Of a tetrahedron and n-dimensional simplex, 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 tetrahedron, 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 triangles 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'', 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 triangularoctahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral tri ...

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 symbol
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

s , continuing into the Hyperbolic space, hyperbolic plane.
The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 vertex figures.
compound
Compound may refer to:
Architecture and built environments
* Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall
** Compound (fortification), a version of the above fortified with defensive structu ...

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 stella octangula, stellae octangulae. A stella octangula is a compound of two tetrahedra in dual position and its eight vertices define a cube as their convex hull.
The square hosohedron is another polyhedron with four faces, but it does not have triangular faces.
The Szilassi polyhedron and the tetrahedron are the only two known polyhedra in which each face shares an edge with each other face.
Applications

Numerical analysis

In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygon mesh, 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, 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 airfields, a large frame in the shape of a tetrahedron with two sides covered with a thin material is mounted on a rotating pivot and always points into the wind. It is built big enough to be seen from the air and is sometimes illuminated. Its purpose is to serve as a reference to pilots indicating wind direction.Chemistry

The tetrahedron shape is seen in nature in covalent bond, covalently bonded molecules. All Orbital hybridisation, spElectricity 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. Since silicon is the most common semiconductor used in solid-state electronics, and silicon has a valence (chemistry), 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.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, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top vertex. Some Rubik's Cube-like puzzles are tetrahedral, such as the Pyraminx and Pyramorphix.Geology

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

Some caltrops are based on tetrahedra as one spike points upwards regardless of how they land and can be easily made by welding two bent nails together.Contemporary art

The Austrian artist Martina Schettina created a tetrahedron using fluorescent lamps. It was shown at the light art biennale Austria 2010. It is used as album artwork, surrounded by black flames on ''The End of All Things to Come'' by Mudvayne.Popular culture

Stanley Kubrick originally intended the Monolith (Space Odyssey), monolith in ''2001: A Space Odyssey (film), 2001: A Space Odyssey'' to be a tetrahedron, according to Marvin Minsky, a cognitive scientist and expert on artificial intelligence who advised Kubrick on the HAL 9000 computer and other aspects of the movie. Kubrick scrapped the idea of using the tetrahedron as a visitor who saw footage of it did not recognize what it was and he did not want anything in the movie regular people did not understand. In Season 6, Episode 15 of ''Futurama'', named "Möbius Dick", the Planet Express crew pass through an area in space known as the Bermuda Tetrahedron. Many other ships passing through the area have mysteriously disappeared, including that of the first Planet Express crew. In the 2013 film ''Oblivion (2013 film), Oblivion'' the large structure in orbit above the Earth is of a tetrahedron design and referred to as the Tet.Tetrahedral graph

The n-skeleton, skeleton of the tetrahedron (comprising the vertices and edges) forms a Graph (discrete mathematics), graph, with 4 vertices, and 6 edges. It is a special case of the complete graph, KPlatonic solid
In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...

.
See also

*Boerdijk–Coxeter helix
The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, - ...

*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 space.
For e ...

– ''n''-dimensional analogues
* Pentachoron – 4-dimensional analogue
* Tetra Pak
* Tetrahedral kite
* Tetrahedral number
* Tetrahedron packing
* Triangular dipyramid – constructed by joining two tetrahedra along one face
* Trirectangular tetrahedron
References

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