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{{Short description|Polyhedron with four faces}}
{{Distinguish|Tetraedron|Tetrahedron (journal)}}
{{Use dmy dates|date=September 2020}}

In [[geometry]], a '''tetrahedron''' ({{plural form}}: '''tetrahedra''' or '''tetrahedrons'''), also known as a ''' triangular pyramid''', is a [[polyhedron]] composed of four triangular [[Face (geometry)|faces]], six straight [[Edge (geometry)|edges]], and four [[vertex (geometry)|vertices]]. The tetrahedron is the simplest of all the ordinary [[convex polytope|convex polyhedra]].<ref name="MW">{{MathWorld |urlname=Tetrahedron |title=Tetrahedron }}</ref>

The tetrahedron is the [[three-dimensional]] case of the more general concept of a [[Euclidean geometry|Euclidean]] [[simplex]], and may thus also be called a '''3-simplex'''.

The tetrahedron is one kind of [[pyramid (geometry)|pyramid]], which is a polyhedron with a flat [[polygon]] base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a [[triangle]] (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".

Like all [[convex polyhedra]], a tetrahedron can be folded from a single sheet of paper. It has two such [[net (polyhedron)|nets]].<ref name="MW" />

For any tetrahedron there exists a sphere (called the [[circumsphere]]) on which all four vertices lie, and another sphere (the [[insphere]]) [[tangent]] to the tetrahedron's faces.<ref>{{citation|title=Plane and Solid Geometry|first1=Walter Burton|last1=Ford|first2=Charles|last2=Ammerman|publisher=Macmillan|year=1913|pages=294–295|url=https://archive.org/stream/planeandsolidge01hedrgoog#page/n315}}</ref>

== Regular tetrahedron ==
{{multiple image
 | image1 = Kepler Tetrahedron Fire.jpg
 | caption1 = Regular tetrahedron, described as the classical element of fire.
 | image2 = Compound of two tetrahedra.png
 | caption2 = The [[stella octangula]]
 | image3 = Tetrahedrons cannot fill space..PNG
 | caption3 = Five tetrahedra are laid flat on a plane, with the highest 3-dimensional points marked as 1, 2, 3, 4, and 5. These points are then attached to each other and [[Angular defect|a thin volume of empty space]] is left, where the five edge angles do not quite meet.
 | perrow = 2
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}}

A '''regular tetrahedron''' is a tetrahedron in which all four faces are [[equilateral triangle]]s. In other words, all of its faces are the same size and shape (congruent) and all edges are the same length. The regular tetrahedron is the simplest [[Convex set|convex]] [[deltahedron]], a polyhedron in which all of its faces are equilateral triangles; there are seven other convex deltahedra.{{sfn|Cundy|1952}}

The regular tetrahedron is also one of the five regular [[Platonic solid]]s, a set of polyhedrons in which all of their faces are [[regular polygons]].{{sfn|Shavinina|2013|p=[https://books.google.com/books?id=JcPd_JRc4FgC&pg=PA333 333]}} Known since antiquity, the Platonic solid is named after the Greek philosopher [[Plato]], who associated those four solids with nature. The regular tetrahedron was considered as the classical element of [[Fire (classical element)|fire]], because of his interpretation of its sharpest corner being most penetrating.{{sfn|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/55 55]}}

The regular tetrahedron is self-dual, meaning its [[Dual polyhedron|dual]] is another regular tetrahedron. The [[Polyhedral compound|compound]] figure comprising two such dual tetrahedra form a ''[[stellated octahedron]]'' or ''stella octangula''. Its interior 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., [[Rectification (geometry)|rectifying]] the tetrahedron). 

The tetrahedron is yet related to another two solids: By [[Truncation (geometry)|truncation]] the tetrahedron becomes a ''[[truncated tetrahedron]]''. The dual of this solid is the [[triakis tetrahedron]], a regular tetrahedron with four triangular pyramids attached to each of its faces. i.e., its [[kleetope]].

Regular tetrahedra alone do not [[Tessellation#Tessellations in higher dimensions|tessellate]] (fill space), but if alternated with [[regular octahedron|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.

=== Measurement ===
[[File:Tetrahedron.stl|thumb|3D model of a regular tetrahedron]]
Consider a regular tetrahedron with edge length <math>a</math>. The height of a regular tetrahedron is <math display="inline"> {\sqrt{\frac{2}{3}}a} </math>.<ref>Köller, Jürgen, [http://www.mathematische-basteleien.de/tetrahedron.htm "Tetrahedron"], Mathematische Basteleien, 2001</ref> Its surface area is four times the area of an equilateral triangle: <math display="inline"> A = 4 \cdot \left(\frac{\sqrt{3}}{4}a^2\right) = a^2 \sqrt{3} \approx 1.732a^2. </math>{{sfn|Coxeter|1948|loc=Table I(i)}}
Obtaining the volume is one-third of the base times the height, the general formula for a pyramid,{{sfn|Coxeter|1948|loc=Table I(i)}} or by dissecting a cube into a tetrahedron and four triangular pyramids.{{sfn|Alsina|Nelsen|2015|p=[https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA68 68]}}.
<math display="block">V = \frac{1}{3} \cdot \left(\frac{\sqrt{3}}{4}a^2\right) \cdot \frac{\sqrt{6}}{3}a = \frac{a^3}{6\sqrt{2}} \approx 0.118a^3.</math>

Its [[dihedral angle]]&mdash;the angle formed by two planes in which adjacent faces lie&mdash;is <math display="inline"> \arccos \left(1/3 \right) = \arctan\left(2\sqrt{2}\right) \approx 70.529^\circ. </math>{{sfn|Coxeter|1948|at=Table I(i)}} {{anchor|Tetrahedral angle}}Its vertex–center–vertex angle—the angle between lines from the tetrahedron center to any two vertices—is <math display="inline"> \arccos \left(-1/3 \right) = 2\arctan\left(\sqrt{2}\right) \approx 109.471^\circ, </math> denoted the '''tetrahedral angle'''.{{sfn|Brittin|1945}} It is the angle between [[Plateau's laws|Plateau borders]] at a vertex. Its value in radians is the length of the circular arc on the unit sphere resulting from centrally projecting one edge of the tetrahedron to the sphere. In chemistry, it is also known as the [[Tetrahedral molecular geometry|tetrahedral bond angle]].

[[File:Вписанный тетраэдр.svg|thumb|right|upright=1.2|Regular tetrahedron ABCD and its circumscribed sphere]]
The radii of its [[circumsphere]] <math> R </math>, [[insphere]] <math> r </math>, [[midsphere]] <math> r_\mathrm{M} </math>, and [[Exsphere (polyhedra)|exsphere]] <math> r_\mathrm{E} </math> are:{{sfn|Coxeter|1948|loc=Table I(i)}}
<math display="block"> \begin{align}
 R = \sqrt{\frac{3}{8}}a, &\qquad r = \frac{1}{3}R = \frac{a}{\sqrt{24}}, \\
 r_\mathrm{M} = \sqrt{rR} = \frac{a}{\sqrt{8}}, &\qquad r_\mathrm{E} = \frac{a}{\sqrt{6}}.
\end{align}
</math>
For a regular tetrahedron with side length <math> a </math> and circumsphere radius <math> R </math>, the distances <math> d_i </math> from an arbitrary point in 3-space to its four vertices satisfy the equations:{{sfn|Park|2016}}
<math display="block"> \begin{align}\frac{d_1^4 + d_2^4 + d_3^4 + d_4^4}{4} + \frac{16R^4}{9}&= \left(\frac{d_1^2 + d_2^2 + d_3^2 + d_4^2}{4} + \frac{2R^2}{3}\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{align}</math>

With respect to the base plane the [[slope]] of a face (2{{sqrt|2}}) is twice that of an edge ({{sqrt|2}}), corresponding to the fact that the ''horizontal'' distance covered from the base to the [[Apex (geometry)|apex]] along an edge is twice that along the [[Median (geometry)|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 [[centroid#Proof that the centroid of a triangle divides each median in the ratio 2:1|proof]]).

Its [[solid angle]] at a vertex subtended by a face is <math display="inline"> \arccos\left(\frac{23}{27}\right) = \frac{\pi}{2} - 3\arcsin\left(\frac{1}{3}\right) = 3\arccos \left(\frac{1}{3}\right)-\pi, </math> or approximately 0.55129 [[steradian]]s, 1809.8 [[square degree]]s, and 0.04387 [[Spat (angular unit)|spats]].

=== Cartesian coordinates ===
One way to construct a regular tetrahedron is by using the following [[Cartesian coordinates]], defining the four vertices of a tetrahedron with edge length 2, centered at the origin, and two-level edges:
<math display="block"> \left(\pm 1, 0, -\frac{1}{\sqrt{2}}\right) \quad \mbox{and} \quad \left(0, \pm 1, \frac{1}{\sqrt{2}}\right) </math>

Expressed symmetrically as 4 points on the [[unit sphere]], centroid at the origin, with lower face parallel to the <math>xy</math> plane, the vertices are:
<math display="block"> \begin{align}
 \left(\sqrt{\frac{8}{9}},0,-\frac{1}{3}\right), &\quad \left(-\sqrt{\frac{2}{9}},\sqrt{\frac{2}{3}},-\frac{1}{3}\right), \\
 \left(-\sqrt{\frac{2}{9}},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right), &\quad (0,0,1)
\end{align}</math>
with the edge length of <math display="inline">\frac{2\sqrt{6}}{3}</math>.

A regular tetrahedron can be embedded inside a [[cube (geometry)|cube]] in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the [[Cartesian coordinates]] of the vertices are
<math display="block"> \begin{align}
 (1,1,1), &\quad (1,-1,-1), \\
 (-1,1,-1), &\quad (-1,-1,1).
\end{align} </math>
This yields a tetrahedron with edge-length <math> 2 \sqrt{2} </math>, centered at the origin. For the other tetrahedron (which is [[dual polyhedron|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-[[Demihypercube|demicube]], a polyhedron that is by [[Alternation (geometry)|alternating]] a cube. This form has [[Coxeter diagram]] {{CDD|node_h|4|node|3|node}} and [[Schläfli symbol]] <math> \mathrm{h}\{4,3\} </math>.

=== Symmetry ===
[[Image:Tetraeder animation with cube.gif|thumb|The cube and tetrahedron]]
The vertices of a [[cube]] can be grouped into two groups of four, each forming a regular tetrahedron, showing one of the two tetrahedra in the cube. The [[Symmetry in mathematics|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 not mapped to itself by [[point inversion]].

[[Image:Symmetries of the tetrahedron.svg|thumb|upright=2|The proper rotations, (order-3 rotation on a vertex and face, and order-2 on two edges) and reflection plane (through two faces and one edge) in the symmetry group of the regular tetrahedron]]
The regular tetrahedron has 24 isometries, forming the [[symmetry group]] known as [[full tetrahedral symmetry]] <math> \mathrm{T}_\mathrm{d} </math>. This symmetry group is [[Isomorphism|isomorphic]] to the [[symmetric group]] <math> S_4 </math>. They can be categorized as follows:
* It has rotational tetrahedral symmetry <math> \mathrm{T} </math>. This symmetry is isomorphic to [[alternating group]] <math> A_4 </math>&mdash;the identity and 11 proper rotations&mdash;with the following [[conjugacy class]]es (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the [[Quaternions and spatial rotation|unit quaternion representation]]):
** identity (identity; 1)
** 2 conjugacy classes corresponding to positive and negative rotations about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together( {{nowrap|4 (1 2 3)}}, etc., and  {{nowrap|4 (1 3 2)}}, etc.; {{sfrac|1 ± ''i'' ± ''j'' ± ''k''|2}}).
** rotation by an angle of 180° such that an edge maps to the opposite edge: {{nowrap|3 ((1 2)(3 4)}}, etc.; {{nowrap|''i'', ''j'', ''k''}})
* reflections in a plane perpendicular to an edge: 6
* reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion ('''x''' is mapped to −'''x'''): the rotations correspond to those of the cube about face-to-face axes

===Orthogonal projections of the regular tetrahedron===
The regular tetrahedron has two special [[orthogonal projection]]s, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A<sub>2</sub> [[Coxeter plane]].

{| class=wikitable
|+ [[Orthographic projection]]
!Centered by
!Face/vertex
!Edge
|- align=center
!Image
|[[File:3-simplex t0 A2.svg|100px]]
|[[File:3-simplex t0.svg|100px]]
|- align=center
!Projective<br>symmetry
![3]
![4]
|}

===Cross section of regular tetrahedron===
[[File:Regular_tetrahedron_square_cross_section.png|120px|thumb|A central cross section of a ''regular tetrahedron'' is a [[square]].]]
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]].<ref>{{cite web| url = http://www.matematicasvisuales.com/english/html/geometry/space/sectetra.html| title = Sections of a Tetrahedron}}</ref> When the intersecting plane is near one of the edges the rectangle is long and skinny. When halfway between the two edges the intersection is a [[square]]. The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become [[Wedge (geometry)|wedges]].
[[File:Tetragonal disphenoid diagram.png|thumb|100px|left|A tetragonal disphenoid viewed orthogonally to the two green edges.]]
This property also applies for [[tetragonal disphenoid]]s when applied to the two special edge pairs.

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===Spherical tiling===
The tetrahedron can also be represented as a [[spherical tiling]] (of [[spherical triangle]]s), and projected onto the plane via a [[stereographic projection]]. This projection is [[Conformal map|conformal]], preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
{|class=wikitable
|- align=center
|[[File:Uniform tiling 332-t2.svg|160px]]
|[[File:Tetrahedron stereographic projection.svg|160px]]
|-
![[Orthographic projection]]
!colspan=1|[[Stereographic projection]]
|}

=== Helical stacking ===
[[File:600-cell tet ring.png|thumb|A single 30-tetrahedron ring [[Boerdijk–Coxeter helix]] within the [[600-cell]], seen in stereographic projection]]
Regular tetrahedra can be stacked face-to-face in a [[chiral]] aperiodic chain called the [[Boerdijk–Coxeter helix]].

In [[Four-dimensional space|four dimensions]], all the convex [[regular 4-polytope]]s with tetrahedral cells (the [[5-cell#Boerdijk–Coxeter helix|5-cell]], [[16-cell#Helical construction|16-cell]] and [[600-cell#Union of two tori|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.
{{Clear}}

==Irregular tetrahedra==
{| class=wikitable width=440 align=right
|[[File:Tetrahedral subgroup tree.png|240px]]<br>Tetrahedral symmetry subgroup relations
|[[File:Tetrahedron symmetry tree.png|200px]]<br>Tetrahedral symmetries shown in tetrahedral diagrams
|}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]], then it is called an ''[[orthocentric tetrahedron]]''. When only one pair of opposite edges are perpendicular, it is called a '''''semi-orthocentric tetrahedron'''''.
In a ''[[trirectangular tetrahedron]]'' the three face angles at ''one'' vertex are [[right angle]]s, as at the corner of a cube.

An '''isodynamic tetrahedron''' is one in which the [[cevian]]s that join the vertices to the [[Incircle and excircles of a triangle|incenters]] of the opposite faces are [[Concurrent lines|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.
{{Clear}}

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===Disphenoid===
{{Main|Disphenoid}}
[[File:Oblate tetrahedrille cell.png|thumb|A space-filling tetrahedral disphenoid inside a cube. Two edges have [[dihedral angle]]s of 90°, and four edges have dihedral angles of 60°.]]
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.
{{Clear}}

===Orthoschemes===
[[File:Triangulated cube.svg|thumb|400px|A cube dissected into six characteristic orthoschemes.]]
A '''3-orthoscheme''' is a tetrahedron where all four faces are [[Triangle#By internal angles|right triangles]]. A 3-orthoscheme is not a disphenoid, because its opposite edges are not of equal length. It is not possible to construct a [[Disphenoid#Special cases and generalizations|disphenoid]] with right triangle or obtuse triangle faces.

An [[Schläfli orthoscheme|orthoscheme]] is an irregular [[simplex]] that is the [[convex hull]] of a [[Tree (graph theory)|tree]] 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.<ref>{{Cite journal | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1–3 | pages=59–71 | doi=10.1016/0898-1221(89)90148-X | doi-access=free }}</ref>

Coxeter also calls quadrirectangular tetrahedra "characteristic tetrahedra", because of their integral relationship to the regular polytopes and their symmetry groups.{{Sfn|Coxeter|1973|pp=71-72|loc=§4.7 Characteristic tetrahedra}} For example, the special case of a 3-orthoscheme with equal-length perpendicular edges is [[Polyhedron#Ancient|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 {{radic|2}} and one of length {{radic|3}}, so all its edges are edges or diagonals of the cube. The cube {{CDD|node_1|4|node|3|node}} can be dissected into six such 3-orthoschemes {{CDD|node|4|node|3|node}} four different ways, with all six surrounding the same {{radic|3}} 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 [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]. There is a 3-orthoscheme, which is the "characteristic tetrahedron of the regular tetrahedron". The regular tetrahedron {{Coxeter–Dynkin diagram|node_1|3|node|3|node}} is subdivided into 24 instances of its characteristic tetrahedron {{Coxeter–Dynkin diagram|node|3|node|3|node}} by its planes of symmetry. The 24 characteristic tetrahedra of the regular tetrahedron occur in two mirror-image forms, 12 of each.

{| class="wikitable floatright"
!colspan=6|Characteristics of the regular tetrahedron{{Sfn|Coxeter|1973|pp=292–293|loc=Table I(i); "Tetrahedron, 𝛼<sub>3</sub>"}}
|-
!align=right|
!align=center|edge
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral
|-
!align=right|𝒍
|align=center|<small><math>2</math></small>
|align=center|<small>109°28′16″</small>
|align=center|<small><math>\pi - 2\text{𝜿}</math></small>
|align=center|<small>70°31′44″</small>
|align=center|<small><math>\pi - 2\text{𝟁}</math></small>
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{4}{3}} \approx 1.155</math></small>
|align=center|<small>70°31′44″</small>
|align=center|<small><math>2\text{𝜿}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|<small><math>1</math></small>
|align=center|<small>54°44′8″</small>
|align=center|<small><math>\tfrac{\pi}{2} - \text{𝜿}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small>
|align=center|<small>54°44′8″</small>
|align=center|<small><math>\tfrac{\pi}{2} - \text{𝜿}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
|
|
|
|
|
|-
!align=right|<small><math>_0R/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{3}{2}} \approx 1.225</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_1R/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_2R/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
|
|
|
|
|
|-
!align=right|<small><math>\text{𝜿}</math></small>
|align=center|
|align=center|<small>35°15′52″</small>
|align=center|<small><math>\tfrac{\text{arc sec }3}{2}</math></small>
|align=center|
|align=center|
|}
If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths <small><math>\sqrt{\tfrac{4}{3}}</math></small>, <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (edges that are the ''characteristic radii'' of the regular tetrahedron). The 3-edge path along orthogonal edges of the orthoscheme is <small><math>1</math></small>, <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, 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 [[30-60-90 triangle|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 <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, a right triangle with edges <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, and a right triangle with edges <small><math>\sqrt{\tfrac{4}{3}}</math></small>, <small><math>\sqrt{\tfrac{3}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>.

===Space-filling tetrahedra===
A '''space-filling tetrahedron''' packs with directly congruent or enantiomorphous ([[mirror image]]) copies of itself to tile space.{{Sfn|Coxeter|1973|pp=33–34|loc=§3.1 Congruent transformations}} 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. (The characteristic orthoscheme of the cube is one of the [[Hill tetrahedra]], a family of space-filling tetrahedra. All space-filling tetrahedra are [[scissors-congruent]] to a cube.)

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 (moreover, it is not scissors-congruent to any other polyhedra which can fill the space, see [[Hilbert's third problem]]). The [[tetrahedral-octahedral honeycomb]] fills space with alternating regular tetrahedron cells and regular [[octahedron]] cells in a ratio of 2:1.

===Fundamental domains===
[[File:Coxeter-Dynkin 3-space groups.png|480px|thumb|For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra. They can be seen as points on and within a cube.]]An irregular tetrahedron which is the [[fundamental domain]]{{Sfn|Coxeter|1973|p=63|loc=§4.3 Rotation groups in two dimensions; notion of a ''fundamental region''}} of a [[Coxeter group|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 construction|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. (The [[Coxeter-Dynkin diagram]] of the generated polyhedron contains three ''nodes'' representing the three mirrors. The dihedral angle between each pair of mirrors is encoded in the diagram, as well as the location of a single ''generating point'' which is multiplied by mirror reflections into the vertices of the polyhedron.)

Among the Goursat tetrahedra which generate 3-dimensional [[Honeycomb (geometry)|honeycombs]] we can recognize an orthoscheme (the characteristic tetrahedron of the cube), a double orthoscheme (the characteristic tetrahedron of the cube face-bonded to its mirror image), and the space-filling disphenoid illustrated [[#Disphenoid|above]].{{Sfn|Coxeter|1973|pp=71–72|loc=§4.7 Characteristic tetrahedra}} 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 [[Dissection into orthoschemes|dissected into characteristic tetrahedra of the cube]].
{{Clear}}

===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 [[Point groups in three dimensions|3-dimensional point group]] is formed. Two other isometries (C<sub>3</sub>, [3]<sup>+</sup>), and (S<sub>4</sub>, [2<sup>+</sup>,4<sup>+</sup>]) 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.
{| class=wikitable style="text-align:center; width:800px;"
!colspan=4|Tetrahedron name
!rowspan=3|Edge<br>equivalence<br>diagram
!rowspan=3|Description
|-
!colspan=4|[[List of spherical symmetry groups|Symmetry]]
|-
![[Schönflies notation|Schön.]]
![[Coxeter notation|Cox.]]
![[Orbifold notation|Orb.]]
![[Symmetry order|Ord.]]
|-
!colspan=4|Regular tetrahedron
|rowspan=2|[[File:Regular tetrahedron diagram.png|60px]]
|rowspan=2 align=left|{{center|Four '''equilateral''' triangles}}It forms the symmetry group ''T''<sub>d</sub>, isomorphic to the [[symmetric group]], ''S''<sub>4</sub>. A regular tetrahedron has [[Coxeter diagram]] {{CDD|node_1|3|node|3|node}} and [[Schläfli symbol]] {3,3}.
|-
||''T''<sub>d</sub><br>''T''||[3,3]<br>[3,3]<sup>+</sup>||*332<br>332|| 24<br>12
|-
!colspan=4|Triangular pyramid
|rowspan=2|[[File:Isosceles trigonal pyramid diagram.png|60px]]
|rowspan=2 align=left|{{center|An '''equilateral''' triangle base and three equal '''isosceles''' triangle sides}}It gives 6 isometries, corresponding to the 6 isometries of the base. As permutations of the vertices, these 6 isometries are the identity 1, (123), (132), (12), (13) and (23), forming the symmetry group ''C''<sub>3v</sub>, isomorphic to the [[symmetric group]], ''S''<sub>3</sub>. A triangular pyramid has Schläfli symbol {3}∨(&nbsp;).
|-
||''C''<sub>3v</sub><br>C<sub>3</sub> ||[3]<br>[3]<sup>+</sup> || *33<br>33 ||6<br>3

|-
!colspan=4|Mirrored sphenoid
|rowspan=2|[[File:Sphenoid diagram.png|60px]]
|rowspan=2 align=left|{{center|Two equal '''scalene''' triangles with a common base edge}}This has two pairs of equal edges (1,3), (1,4) and (2,3), (2,4) and otherwise no edges equal. The only two isometries are 1 and the reflection (34), giving the group ''C''<sub>s</sub>, also isomorphic to the [[cyclic group]], '''Z'''<sub>2</sub>.
|-
|''C''<sub>s</sub><br>=''C''<sub>1h</sub><br>=''C''<sub>1v</sub>||[&nbsp;] ||*|| 2

|-
!colspan=4|Irregular tetrahedron<br>(No symmetry)
|rowspan=2|[[File:Scalene tetrahedron diagram.png|60px]]
|rowspan=2 align=left|{{center|Four unequal triangles}}

Its only isometry is the identity, and the symmetry group is the [[trivial group]]. An irregular tetrahedron has Schläfli symbol (&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;).
|-
|C<sub>1</sub>||[&nbsp;]<sup>+</sup>||1||1

|-
!colspan=6 |[[Disphenoid]]s (Four equal triangles)
|-
!colspan=4|[[Tetragonal disphenoid]]
|rowspan=2|[[File:Tetragonal disphenoid diagram.png|60px]]
|rowspan=2 align=left|{{center|Four equal '''isosceles''' triangles}}

It has 8 isometries. If edges (1,2) and (3,4) are of different length to the other 4 then the 8 isometries are the identity 1, reflections (12) and (34), and 180° rotations (12)(34), (13)(24), (14)(23) and improper 90° rotations (1234) and (1432) forming the symmetry group ''D''<sub>2d</sub>.  A tetragonal disphenoid has Coxeter diagram {{CDD|node_h|2x|node_h|4|node}} and Schläfli symbol s{2,4}.
|-
|''D''<sub>2d</sub><br>S<sub>4</sub> ||[2<sup>+</sup>,4]<br>[2<sup>+</sup>,4<sup>+</sup>] ||2*2<br>2×|| 8<br>4
|-
!colspan=4|[[Rhombic disphenoid]]
|rowspan=2|[[File:Rhombic disphenoid diagram.png|60px]]
|rowspan=2 align=left|{{center|Four equal '''scalene''' triangles}}

It has 4 isometries. The isometries are 1 and the 180° rotations (12)(34), (13)(24), (14)(23). This is the [[Klein four-group]] ''V''<sub>4</sub> or '''Z'''<sub>2</sub><sup>2</sup>, present as the point group ''D''<sub>2</sub>.  A rhombic disphenoid has Coxeter diagram {{CDD|node_h|2x|node_h|2x|node_h}} and Schläfli symbol sr{2,2}.
|-
|''D''<sub>2</sub> ||[2,2]<sup>+</sup> ||222||4

|-
!colspan=6 |Generalized disphenoids (2 pairs of equal triangles)
|-
!colspan=4|[[Digonal disphenoid]]
|rowspan=2|[[File:Digonal disphenoid diagram2.png|80px]]<br>[[File:Digonal disphenoid diagram.png|80px]]
|rowspan=2 align=left|{{center|Two pairs of equal '''isosceles''' triangles}} This gives two opposite edges (1,2) and (3,4) that are perpendicular but different lengths, and then the 4 isometries are 1, reflections (12) and (34) and the 180° rotation (12)(34). The symmetry group is ''C''<sub>2v</sub>, isomorphic to the [[Klein four-group]] ''V''<sub>4</sub>. A digonal disphenoid has Schläfli symbol {&nbsp;}∨{&nbsp;}.
|-
|''C''<sub>2v</sub><br>''C''<sub>2</sub> ||[2]<br>[2]<sup>+</sup> ||*22<br>22||4<br>2

|-
!colspan=4|Phyllic disphenoid
|rowspan=2|[[File:Half-turn tetrahedron diagram.png|80px]]<br> [[File:Half-turn tetrahedron diagram2.png|80px]]
|rowspan=2 align=left|{{center|Two pairs of equal '''scalene''' or '''isosceles''' triangles}}

This has two pairs of equal edges (1,3), (2,4) and (1,4), (2,3) but otherwise no edges equal. The only two isometries are 1 and the rotation (12)(34), giving the group ''C''<sub>2</sub> isomorphic to the [[cyclic group]], '''Z'''<sub>2</sub>.
|-
|''C''<sub>2</sub> ||[2]<sup>+</sup> ||22 ||2
|}

== Subdivision and similarity classes ==
Tetrahedra subdivision is a process used in computational geometry and 3D modeling to divide a tetrahedron into several smaller tetrahedra. This process enhances the complexity and detail of tetrahedral meshes, which is particularly beneficial in numerical simulations, finite element analysis, and computer graphics. One of the commonly used subdivision methods is the '''Longest Edge Bisection (LEB)''', which identifies the longest edge of the tetrahedron and bisects it at its midpoint, generating two new, smaller tetrahedra. When this process is repeated multiple times, bisecting all the tetrahedra generated in each previous iteration, the process is called iterative LEB. 

A '''similarity class''' is the set of tetrahedra with the same geometric shape, regardless of their specific position, orientation, and scale. So, any two tetrahedra belonging to the same similarity class may be transformed to each other by an affine transformation. The outcome of having a limited number of similarity classes in iterative subdivision methods is significant for computational modeling and simulation. It reduces the variability in the shapes and sizes of generated tetrahedra, preventing the formation of highly irregular elements that could compromise simulation results.

The iterative LEB of the regular tetrahedron has been shown to produce only 8 similarity classes. Furthermore, in the case of nearly equilateral tetrahedra where their two longest edges are not connected to each other, and the ratio between their longest and their shortest edge is less than or equal to <math>\sqrt{3/2}</math>, the iterated LEB produces no more than 37 similarity classes.<ref>{{Cite journal |last1=Trujillo-Pino |first1=Agustín |last2=Suárez |first2=Jose Pablo |last3=Padrón |first3=Miguel A. |date=2024 |title=Finite number of similarity classes in Longest Edge Bisection of nearly equilateral tetrahedra |journal=Applied Mathematics and Computation |volume=472 |pages=128631 |doi=10.1016/j.amc.2024.128631 |issn=0096-3003|doi-access=free |hdl=10553/129894 |hdl-access=free }}</ref>

== General properties ==
===Volume===
The volume of a tetrahedron can be obtained in many ways. It can be given by using the formula of the pyramid's volume: 
<math display="block"> V = \frac{1}{3}Ah. </math>
where <math> A </math> is the [[base (geometry)|base]]' area and <math> h </math> 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. Another way is by dissecting a triangular prism into three pieces.{{sfn|Alsina|Nelsen|2015|p=[https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA67 67]}}

Given the vertices of a tetrahedron in the following:
<math display="block"> \begin{align}
 \mathbf{a} &= (a_1, a_2, a_3), \\
 \mathbf{b} &= (b_1, b_2, b_3), \\
 \mathbf{c} &= (c_1, c_2, c_3), \\
 \mathbf{d} &= (d_1, d_2, d_3).
\end{align} </math>
The volume of a tetrahedron can be ascertained in terms of a [[determinant]] <math display="inline"> \frac{1}{6} \det(\mathbf{a} - \mathbf{d}, \mathbf{b} - \mathbf{d}, \mathbf{c} - \mathbf{d}) </math>,{{sfn|Fekete|1985|p=[https://books.google.com/books?id=3_AIXBO11bEC&pg=PA68 68]}} or any other combination of pairs of vertices that form a simply connected [[Graph (discrete mathematics)|graph]]. Comparing this formula with that used to compute the volume of a [[parallelepiped]], we conclude that the volume of a tetrahedron is equal to {{sfrac|1|6}} 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:
:<math>6 \cdot V =\begin{Vmatrix}
\mathbf{a} & \mathbf{b} & \mathbf{c}
\end{Vmatrix}</math>{{pad|2em}}or{{pad|1em}}<math>6 \cdot V =\begin{Vmatrix}
\mathbf{a} \\ \mathbf{b} \\ \mathbf{c}
\end{Vmatrix}</math>{{pad|2em}}where{{pad|1em}}<math>\begin{cases}\mathbf{a} = (a_1,a_2,a_3), \\
 \mathbf{b} = (b_1,b_2,b_3), \\
 \mathbf{c} = (c_1,c_2,c_3), \end{cases}</math>{{pad|1em}}are expressed as row or column vectors.

Hence
:<math>36 \cdot V^2 =\begin{vmatrix}
\mathbf{a^2} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\
\mathbf{a} \cdot \mathbf{b} & \mathbf{b^2} & \mathbf{b} \cdot \mathbf{c} \\
\mathbf{a} \cdot \mathbf{c} & \mathbf{b} \cdot \mathbf{c} & \mathbf{c^2}
\end{vmatrix}</math>{{pad|1em}}where{{pad|1em}}<math>\begin{cases}\mathbf{a} \cdot \mathbf{b} = ab\cos{\gamma}, \\ \mathbf{b} \cdot \mathbf{c} = bc\cos{\alpha}, \\ \mathbf{a} \cdot \mathbf{c} = ac\cos{\beta}. \end{cases}</math>

where <math>a = \begin{Vmatrix} \mathbf{a} \end{Vmatrix} </math>, <math>b = \begin{Vmatrix} \mathbf{b} \end{Vmatrix} </math>, and <math>c = \begin{Vmatrix} \mathbf{c} \end{Vmatrix} </math>, which gives
:<math>V = \frac {abc} {6} \sqrt{1 + 2\cos{\alpha}\cos{\beta}\cos{\gamma}-\cos^2{\alpha}-\cos^2{\beta}-\cos^2{\gamma}}, \,</math>
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
:<math>6 \cdot V = \left| \det \left( \begin{matrix}
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{matrix} \right) \right|\,.</math>

Given the distances between the vertices of a tetrahedron the volume can be computed using the [[Cayley–Menger determinant]]:
:<math>288 \cdot V^2 =
\begin{vmatrix}
  0 & 1        & 1        & 1        & 1        \\
  1 & 0        & d_{12}^2 & d_{13}^2 & d_{14}^2 \\
  1 & d_{12}^2 & 0        & d_{23}^2 & d_{24}^2 \\
  1 & d_{13}^2 & d_{23}^2 & 0        & d_{34}^2 \\
  1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0
\end{vmatrix}</math>
where the subscripts {{nowrap|''i'', ''j'' ∈ {1, 2, 3, 4}|}} represent the vertices {{nowrap|{'''a''', '''b''', '''c''', '''d'''}|}} and ''d{{sub|ij}}'' 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.<ref>[http://www.mathpages.com/home/kmath664/kmath664.htm "Simplex Volumes and the Cayley-Menger Determinant"], MathPages.com</ref>

Let <math> a </math>, <math> b </math>, and <math> c </math> be the lengths of three edges that meet at a point, and <math> x </math>, <math> y </math>, and <math> z </math> be those of the opposite edges. The volume of the tetrahedron <math> V </math> is:{{sfn|Kahan|2012|p=11}}
<math display="block"> V = \frac{\sqrt{4 a^2 b^2 c^2-a^2 X^2-b^2 Y^2-c^2 Z^2+X Y Z}}{12}</math>
where
<math display="block"> \begin{align}
 X &= b^2+c^2-x^2, \\
 Y &= a^2+c^2-y^2, \\
 Z &= a^2+b^2-z^2.
\end{align}</math>
The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles.{{sfn|Kahan|2012|p=11}}
<math display="block"> V = \frac{abc}{6} \sqrt{1 + 2\cos{\alpha}\cos{\beta}\cos{\gamma}-\cos^2{\alpha}-\cos^2{\beta}-\cos^2{\gamma}}</math>

[[Image:Six edge-lengths of Tetrahedron.png|right|thumb|240px|Six edge-lengths of Tetrahedron]]
The volume of a tetrahedron can be ascertained by using the Heron formula. Suppose <math> U </math>, <math> V </math>, <math> W </math>, <math> u </math>. <math> v </math>, and <math> w </math> are the lengths of the tetrahedron's edges as in the following image. Here, the first three form a triangle, with <math> u </math> opposite <math> U </math>, <math> v </math> opposite <math> V </math>, and <math> w </math> opposite <math> W </math>. Then,
<math display="block"> V = \frac{\sqrt {\,( - p + q + r + s)\,(p - q + r + s)\,(p + q - r + s)\,(p + q + r - s)}}{192\,u\,v\,w}</math>
where
<math display="block"> \begin{align}
 p = \sqrt{xYZ}, &\quad q = \sqrt{yZX}, \\
 r = \sqrt{zXY}, &\quad s = \sqrt {xyz},
\end{align} </math>
and
<math display="block"> \begin{align}
 X = (w - U + v)(U + v + w), &\quad x = (U - v + w)(v - w + U), \\
 Y = (u - V + w)(V + w + u), &\quad y = (V - w + u)(w - u + V), \\
 Z = (v - W + u)(W + u + v), &\quad z = (W - u + v)\,(u - v + W).
\end{align}
</math>

Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron [[bisection|bisects]] the volume of the tetrahedron.{{sfn|Bottema|1969}}

For tetrahedra in [[hyperbolic space]] or in three-dimensional [[elliptic geometry]], the [[dihedral angle]]s of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the [[Murakami–Yano formula]], after Jun Murakami and Masakazu Yano.{{sfn|Murakami|Yano|2005}} However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.

Any two opposite edges of a tetrahedron lie on two [[skew lines]], and the distance between the edges is defined as the distance between the two skew lines. Let <math> d </math> be the distance between the skew lines formed by opposite edges <math> a </math> and <math> \mathbf{b} - \mathbf{c} </math> as calculated [[Skew lines#Distance between two skew lines|here]]. Then another formula for the volume of a tetrahedron <math> V </math> is given by
<math display="block"> V = \frac {d |(\mathbf{a} \times \mathbf{(b-c)})|}{6}. </math>

===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 circle|Spieker center]] and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes.<ref>{{Cite journal |last1=Havlicek |first1=Hans |last2=Weiß |first2=Gunter |title=Altitudes of a tetrahedron and traceless quadratic forms |journal=[[American Mathematical Monthly]] |volume=110 |issue=8 |pages=679–693 |year=2003 |url=http://www.geometrie.tuwien.ac.at/havlicek/pub/hoehen.pdf |doi=10.2307/3647851 |jstor=3647851 |arxiv=1304.0179 }}</ref>

[[Gaspard Monge]] found a center that exists in every tetrahedron, now known as the '''Monge point''': the point where the six midplanes of a tetrahedron intersect. A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of [[orthocentric tetrahedron]].

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 lines|concurrent]] at a point called the ''centroid'' of the tetrahedron.<ref>Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53–54</ref> 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.<ref>{{Cite book |last1=Outudee |first1=Somluck |last2=New |first2=Stephen |title=The Various Kinds of Centres of Simplices |publisher=Dept of Mathematics, Chulalongkorn University, Bangkok |url=http://www.math.sc.chula.ac.th/ICAA2002/pages/Somluck_Outudee.pdf |url-status=bot: unknown |archive-url=https://web.archive.org/web/20090227143222/http://www.math.sc.chula.ac.th/ICAA2002/pages/Somluck_Outudee.pdf |archive-date=27 February 2009}}</ref>

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<ref>{{cite web |first=Daniel |last=Audet |title=Déterminants sphérique et hyperbolique de Cayley-Menger |url=http://archimede.mat.ulaval.ca/amq/bulletins/mai11/Chronique_note_math.mai11.pdf |publisher=Bulletin AMQ |date=May 2011 }}</ref>
:<math>\begin{vmatrix}  -1 & \cos{(\alpha_{12})} & \cos{(\alpha_{13})} & \cos{(\alpha_{14})}\\
\cos{(\alpha_{12})} & -1 & \cos{(\alpha_{23})} & \cos{(\alpha_{24})} \\
\cos{(\alpha_{13})} & \cos{(\alpha_{23})} & -1 & \cos{(\alpha_{34})} \\
\cos{(\alpha_{14})} & \cos{(\alpha_{24})} & \cos{(\alpha_{34})} & -1 \\ \end{vmatrix} = 0\,</math>

where ''α{{sub|ij}}'' is the angle between the faces ''i'' and ''j''.

The [[geometric median]] 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 [[Steradian|sr]], and at which the angles subtended by opposite edges are equal.<ref>{{cite journal |last1=Lindelof |first1=L. |title=Sur les maxima et minima d'une fonction des rayons vecteurs menés d'un point mobile à plusieurs centres fixes |journal=Acta Societatis Scientiarum Fennicae |date=1867 |volume=8 |issue=Part 1 |pages=189–203}}</ref>  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]], ''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]]. 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, for an example in [[electromagnetism]] cf. [[Thomson problem]]).

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 [[Polyhedral compound|compound of five cubes]] gives two more regular compounds, containing five and ten tetrahedra.

Regular tetrahedra cannot [[Honeycomb (geometry)|tessellate space]] by themselves, although this result seems likely enough that [[Aristotle]] claimed it was possible. However, two regular tetrahedra can be combined with an octahedron, giving a [[rhombohedron]] that can tile space as the [[tetrahedral-octahedral honeycomb]].

On otherhand, several irregular tetrahedra are known, of which copies can tile space, for instance the [[#Orthoschemes|characteristic orthoscheme of the cube]] and the [[#Disphenoid|disphenoid]] of the [[disphenoid tetrahedral honeycomb]]. The complete list remains an open problem.<ref>{{Cite journal
 | doi = 10.2307/2689983
 | last = Senechal | first = Marjorie | author-link = Marjorie Senechal
 | title = Which tetrahedra fill space?
 | year = 1981
 | journal = [[Mathematics Magazine]]
 | volume = 54
 | issue = 5
 | pages = 227–243
 | publisher = Mathematical Association of America
 | jstor = 2689983
 }}</ref>

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 polyhedron|uniform polyhedra]] in possessing no parallel faces.

===A law of sines for tetrahedra and the space of all shapes of tetrahedra===
[[Image:tetra.png|248px]]

{{main|Trigonometry of a tetrahedron}}

A corollary of the usual [[law of sines]] is that in a tetrahedron with vertices ''O'', ''A'', ''B'', ''C'', we have
:<math>\sin\angle OAB\cdot\sin\angle OBC\cdot\sin\angle OCA = \sin\angle OAC\cdot\sin\angle OCB\cdot\sin\angle OBA.\,</math>

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 (statistics)|degrees of freedom]] is thereby reduced from 12 to 8. The four relations given by this sine law further reduce the number of degrees of freedom, 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.<ref>{{Cite journal |title=Is There a "Most Chiral Tetrahedron"? |first1=André |last1=Rassat |first2=Patrick W. |last2=Fowler |journal=Chemistry: A European Journal |volume=10 |issue=24 |pages=6575–6580 |year=2004 |doi=10.1002/chem.200400869 |pmid=15558830 }}</ref>

=== Law of cosines for tetrahedra ===
{{main|Trigonometry of a tetrahedron}}

Let <math> P_1 </math>, <math> P_2 </math>, <math> P_3 </math>, <math> P_4 </math> be the points of a tetrahedron. Let <math> \Delta_i </math> be the area of the face opposite vertex <math> P_i </math> and let <math> \theta_{ij} </math> be the dihedral angle between the two faces of the tetrahedron adjacent to the edge <math> P_i P_j </math>. The [[law of cosines]] for a tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation:{{sfn|Lee|1997}}
<math display="block"> \Delta_i^2 = \Delta_j^2 + \Delta_k^2 + \Delta_l^2 - 2(\Delta_j\Delta_k\cos\theta_{il} + \Delta_j\Delta_l \cos\theta_{ik} + \Delta_k\Delta_l \cos\theta_{ij})</math>

=== 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''<sub>a</sub>, ''F''<sub>b</sub>, ''F''<sub>c</sub>, and ''F''<sub>d</sub>. Then<ref name="Crux">''Inequalities proposed in “[[Crux Mathematicorum]]”'', [http://www.imomath.com/othercomp/Journ/ineq.pdf].</ref>{{rp|p.62,#1609}}

:<math>PA \cdot F_\mathrm{a} + PB \cdot F_\mathrm{b} + PC \cdot F_\mathrm{c} + PD \cdot F_\mathrm{d} \geq 9V.</math>

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<ref name=Crux/>{{rp|p.226,#215}}

:<math>PA+PB+PC+PD \geq 3(PJ+PK+PL+PM).</math>

===Inradius===

Denoting the inradius of a tetrahedron as ''r'' and the [[inradius|inradii]] of its triangular faces as ''r''<sub>''i''</sub> for ''i'' = 1, 2, 3, 4, we have<ref name=Crux/>{{rp|p.81,#1990}}

:<math>\frac{1}{r_1^2} + \frac{1}{r_2^2} + \frac{1}{r_3^2} + \frac{1}{r_4^2} \leq \frac{2}{r^2},</math>

with equality if and only if the tetrahedron is regular.

If ''A''<sub>''1''</sub>, ''A''<sub>''2''</sub>, ''A''<sub>''3''</sub> and ''A''<sub>''4''</sub> denote the area of each faces, the value of ''r'' is given by

:<math>r=\frac{3V}{A_1+A_2+A_3+A_4}</math>.

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 <math>V = \frac13A_1r+\frac13A_2r+\frac13A_3r+\frac13A_4r</math>.

===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<ref>{{cite book |chapter-url=https://archive.org/stream/sammlungmathemat01crel#page/104 |chapter=Einige Bemerkungen über die dreiseitige Pyramide |title=Sammlung mathematischer Aufsätze u. Bemerkungen 1 |publisher=Maurer |location=Berlin |last=Crelle |first=A. L. |year=1821 |pages=105–132 |language=de |access-date=7 August 2018}}</ref><ref>{{citation|first=I.|last= Todhunter|title=Spherical Trigonometry: For the Use of Colleges and Schools|year=1886|page= 129 |url=//www.gutenberg.org/ebooks/19770}} ( Art. 163 )</ref>

:<math>R=\frac{\sqrt{(aA+bB+cC)(aA+bB-cC)(aA-bB+cC)(-aA+bB+cC)}}{24V}.</math>

===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 {{math|''C''}} of a tetrahedron with vertices {{math|''x''<sub>0</sub>}},{{math|''x''<sub>1</sub>}},{{math|''x''<sub>2</sub>}},{{math|''x''<sub>3</sub>}} can be formulated as matrix-vector product:<ref>{{citation
 | last1 = Lévy | first1 = Bruno
 | last2 = Liu | first2 = Yang
 | doi = 10.1145/1778765.1778856
 | issue = 4
 | journal = ACM Transactions on Graphics
 | pages = 119:1–119:11
 | title = {{math|''L''<sub>p</sub>}} centroidal Voronoi tessellation and its applications
 | volume = 29
 | year = 2010}}</ref>
:<math>\begin{align}
C &= A^{-1}B &
\text{where} & \ &
A = \left(\begin{matrix}\left[x_1 - x_0\right]^T \\ \left[x_2 - x_0\right]^T \\ \left[x_3 - x_0\right]^T \end{matrix}\right) & \ &
\text{and} & \ &
B = \frac{1}{2}\left(\begin{matrix} \|x_1\|^2 - \|x_0\|^2 \\ \|x_2\|^2 - \|x_0\|^2 \\ \|x_3\|^2 - \|x_0\|^2 \end{matrix}\right) \\
\end{align}
</math>
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 can be computed 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.<ref name=Crux/>{{rp|p.225,#159}}

==Integer tetrahedra==
{{main|Heronian tetrahedron}}
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 triangle]]s with areas of {{val|436800}} and the other two are isosceles with areas of {{val|47120}}, while the volume is {{val|124185600}}.<ref>{{citation
 | date = May 1985
 | department = Solutions
 | issue = 5
 | journal = Crux Mathematicorum
 | pages = 162–166
 | title = Problem 930
 | url = https://cms.math.ca/wp-content/uploads/crux-pdfs/Crux_v11n05_May.pdf
 | volume = 11}}</ref>

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.<ref name=Sierpinski>[[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.</ref>

==Related polyhedra and compounds==
A regular tetrahedron can be seen as a triangular [[Pyramid (geometry)|pyramid]].
{{Pyramids}}

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform ''digonal [[antiprism]]'', where base polygons are reduced [[digon]]s.
{{UniformAntiprisms}}

A regular tetrahedron can be seen as a degenerate polyhedron, a uniform dual ''digonal [[trapezohedron]]'', containing 6 vertices, in two sets of colinear edges.
{{Trapezohedra}}

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.
{{Tetrahedron family}}

This polyhedron is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,''n''}, continuing into the [[Hyperbolic space|hyperbolic plane]].
{{Triangular regular tiling}}

The tetrahedron is topologically related to a series of regular polyhedra and tilings with order-3 [[vertex figure]]s.
{{Order-3 tiling table}}

<gallery mode="packed" caption="Compounds of tetrahedra">
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]]
</gallery>

{{clear}}

An interesting polyhedron can be constructed from [[Compound of five tetrahedra|five intersecting tetrahedra]]. This [[Polyhedral compound|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 image]]s 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. 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===
[[File:Malla irregular de triángulos modelizando una superficie convexa.png|thumb|An irregular volume in space can be approximated by an irregular triangulated surface, and irregular tetrahedral volume elements.]]
In [[numerical analysis]], complicated three-dimensional shapes are commonly broken down into, or [[approximate]]d 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 field]]s, [[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 [[spaceframe]]s.

===Fortification===
Tetrahedrons are used in [[caltrop]]s to provide an [[area denial weapon]]. This is due to their nature of having a sharp corner that always points upwards.

Large concrete tetrahedrons have been used as [[anti-tank]] measures, or as [[Tetrapod (structure)|Tetrapods]] to break down waves at coastlines.

===Aviation===
At some [[airfield]]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.<ref>{{citation|title=Pilot's Handbook of Aeronautical Knowledge|author=Federal Aviation Administration|publisher=U. S. Government Printing Office|year=2009|isbn=9780160876110|page=13{{hyphen}}10<!--hyphenated page-->|url=https://books.google.com/books?id=0l8WO6Drz50C&pg=SA13-PA10}}.</ref>

===Chemistry===
[[Image:Ammonium-3D-balls.png|100px|thumb|The [[ammonium]] ion is tetrahedral]]
[[Image:Tetrahedral_angle_calculation.svg|thumb|216px|<!-- specify width as minus sign vanishes at most sizes --> Calculation of the central angle with a [[dot product]] ]]
{{main|Tetrahedral molecular geometry}}
The tetrahedron shape is seen in nature in [[covalent bond|covalently bonded]] molecules. All [[Orbital hybridisation|sp<sup>3</sup>-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 ({{chem|CH|4}}) or an [[ammonium]] ion ({{chem|NH|4|+}}), four hydrogen atoms surround a central carbon or nitrogen atom with tetrahedral symmetry. For this reason, one of the leading journals in organic chemistry is called ''[[Tetrahedron (journal)|Tetrahedron]]''. The [[central angle]] between any two vertices of a perfect tetrahedron is arccos(−{{sfrac|1|3}}), or approximately 109.47°.<ref name="pubs.acs.org">{{cite journal|doi=10.1021/ed022p145 | volume=22 | issue=3 | title=Valence angle of the tetrahedral carbon atom | year=1945 | journal=Journal of Chemical Education | page=145 | last1 = Brittin | first1 = W. E.| bibcode=1945JChEd..22..145B }}</ref>

[[Water]], {{chem|H|2|O}}, also has a tetrahedral structure, with two hydrogen atoms and two lone pairs of electrons around the central oxygen atoms. Its tetrahedral symmetry is not perfect, however, because the lone pairs repel more than the single O–H bonds.

Quaternary [[phase diagram]]s 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.

There are molecules with the shape based on four nearby atoms whose bonds form the sides of a tetrahedral structure, such as [[Allotropes of phosphorus|white phosphorus]] allotrope<ref>{{Cite web |title=White phosphorus |url=https://www.acs.org/molecule-of-the-week/archive/w/white-phosphorus.html |access-date=2024-05-26 |website=American Chemical Society |language=en}}</ref> and tetra-''t''-butyltetrahedrane, known derivative of the hypothetical [[tetrahedrane]].

===Electricity and electronics===
{{main|Electricity|Electronics}}
If six equal [[resistor]]s are [[solder]]ed together to form a tetrahedron, then the resistance measured between any two vertices is half that of one resistor.<ref>{{cite journal |last=Klein |first=Douglas J. |year=2002 |title=Resistance-Distance Sum Rules |journal=Croatica Chemica Acta |volume=75 |issue=2 |pages=633–649 |url=http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |access-date=2006-09-15 |url-status=dead |archive-url=https://web.archive.org/web/20070610165115/http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |archive-date=10 June 2007}}</ref>

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 [[crystal]]s of silicon form and what shapes they assume.

===Color space===
{{main|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).<ref>{{cite journal |last=Vondran |first=Gary L. |date=April 1998 |title=Radial and Pruned Tetrahedral Interpolation Techniques |journal=HP Technical Report |volume=HPL-98-95 |pages=1–32 |url=http://www.hpl.hp.com/techreports/98/HPL-98-95.pdf |access-date=11 November 2009 |archive-date=7 June 2011 |archive-url=https://web.archive.org/web/20110607102757/http://www.hpl.hp.com/techreports/98/HPL-98-95.pdf |url-status=dead }}</ref>

===Games===
[[Image:4-sided dice 250.jpg|100px|thumb|[[4-sided dice]]]]
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===
{{main|tetrahedral hypothesis}}
The [[tetrahedral hypothesis]], originally published by [[William Lowthian Green]] to explain the formation of the Earth,<ref>{{cite book |title=Vestiges of the Molten Globe, as exhibited in the figure of the earth, volcanic action and physiography |volume=Part I |author-link=William Lowthian Green |first=William Lowthian |last=Green |publisher=E. Stanford |place=London |year=1875 |bibcode=1875vmge.book.....G |oclc=3571917 |url=https://books.google.com/books?id=9DkDAAAAQAAJ }}</ref> was popular through the early 20th century.<ref>{{cite book |author-link=Arthur Holmes |first=Arthur |last=Holmes |title=Principles of physical geology |url=https://archive.org/details/principlesofphys0000holm |url-access=registration |year=1965 |publisher=Nelson |page=[https://archive.org/details/principlesofphys0000holm/page/32 32] |isbn=9780177612992 }}</ref><ref>{{cite news |author-link=Charles Henry Hitchcock |first=Charles Henry |last=Hitchcock |editor-first=Newton Horace |editor-last=Winchell |title=William Lowthian Green and his Theory of the Evolution of the Earth's Features |work=The American Geologist |url=https://books.google.com/books?id=_Ty8AAAAIAAJ&pg=PA1 |date=January 1900 |volume=XXV |publisher=Geological Publishing Company |pages=1–10 }}</ref>

==={{anchor|Popular Culture}}Popular culture===
{{multiple image
 | align = right | perrow = 2 | total_width = 250
 | image1 = Tétraèdres en béton.jpg
 | image2 = Coffee cream TetraPak.jpg
 | image3 = M tic.jpg
 | image4 = Master Pyramorphix cubemeister com.jpg
 | footer = Tetrahedral objects
}}
[[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.<ref>{{cite web |title=Marvin Minsky: Stanley Kubrick Scraps the Tetrahedron |url=http://www.webofstories.com/play/53140?o=R |publisher=Web of Stories |access-date=20 February 2012 }}</ref>

The tetrahedron with regular faces is a solution to an old puzzle asking to form four equilateral triangles using six unbroken matchsticks. The solution places the matchsticks along the edges of a tetrahedron.<ref>{{cite journal
 | last = Bell | first = Alexander Graham | author-link = Alexander Graham Bell
 | date = June 1903
 | doi = 10.1038/scientificamerican06131903-22947supp
 | issue = 1432supp
 | journal = Scientific American
 | pages = s2294–22950
 | title = The tetrahedral principle in kite structure
 | url = https://scholar.archive.org/work/jstv7c4lvbcenod5l4cpldrqsi
 | volume = 55}}</ref>

==Tetrahedral graph==
{{Infobox graph
 | name = Tetrahedral graph
 | image = [[File:3-simplex t0.svg|160px]]
 | image_caption =
 | namesake =
 | vertices = 4
 | edges = 6
 | automorphisms = 24
 | radius = 1
 | diameter = 1
 | girth = 3
 | chromatic_number = 4
 | chromatic_index =
 | fractional_chromatic_index =
 | properties = [[Hamiltonian graph|Hamiltonian]], [[regular graph|regular]], [[symmetric graph|symmetric]], [[distance-regular graph|distance-regular]], [[distance-transitive graph|distance-transitive]], [[K-vertex-connected graph|3-vertex-connected]], [[planar 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]], K<sub>4</sub>, and [[wheel graph]], W<sub>4</sub>.<ref>{{MathWorld |urlname=TetrahedralGraph |title=Tetrahedral graph}}</ref> It is one of 5 [[Platonic graph]]s, each a skeleton of its [[Platonic solid]].

{| class="wikitable skin-invert-image"
|- align=center
|[[File:3-simplex t0 A2.svg|160px]]<br>3-fold symmetry
|}
{{Clear}}

==See also==
* [[Boerdijk–Coxeter helix]]
* [[Möbius configuration]]
* [[Caltrop]]
* [[Demihypercube]] and [[simplex]] – ''n''-dimensional analogues
* [[Pentachoron]] – 4-dimensional analogue
* [[Synergetics (Fuller)]]
* [[Tetrahedral kite]]
* [[Tetrahedral number]]
* [[Tetrahedroid]]
* [[Tetrahedron packing]]
* [[Triangular dipyramid]] – constructed by joining two tetrahedra along one face
* [[Trirectangular tetrahedron]]
* [[Orthoscheme]]

==Notes==
{{Notelist}}

==References==
{{Reflist}}

==Bibliography==
{{refbegin|25em}}
* {{cite book
 | last1 = Alsina | first1 = C.
 | last2 = Nelsen | first2 = R. B.
 | year = 2015
 | title = A Mathematical Space Odyssey: Solid Geometry in the 21st Century
 | publisher = [[Mathematical Association of America]]
 | isbn = 978-1-61444-216-5
}}
* {{cite journal
 | last = Bottema | first = O.
 | title = A Theorem of Bobillier on the Tetrahedron
 | journal = [[Elemente der Mathematik]]
 | volume = 24
 | year = 1969
 | page = 6–10
}}
* {{cite book 
 | last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter
 | title = Regular Polytopes | title-link = Regular Polytopes (book)
 | publisher = Methuen and Co.
 | year = 1948
}}
* {{cite book
 | last = Coxeter | first = H.S.M. | author-link = Harold Scott MacDonald Coxeter
 | year = 1973
 | title = Regular Polytopes
 | publisher = [[Dover Publications]]
 | place = New York
 | edition = 3rd
 | title-link = Regular Polytopes (book)
}}
* {{cite book
 | last = Cromwell
 | first = Peter R.
 | title = Polyhedra
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 | url = https://archive.org/details/polyhedra0000crom
 | publisher = [[Cambridge University Press]]
 | isbn = 978-0-521-55432-9
 }}
* {{cite journal
 | last = Cundy | first = H. Martyn
 | year = 1952
 | title = Deltahedra
 | journal = The Mathematical Gazette
 | volume = 36
 | issue = 318
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 | doi = 10.2307/3608204
 | jstor = 3608204
 | s2cid = 250435684
}}
* {{cite book
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 | publisher = Marcel Dekker Inc.
 | isbn = 978-0-8247-7238-3
 }}
* {{cite thesis
 | last = Kahan
 | first = W. M.
 | year = 2012
 | title = What has the Volume of a Tetrahedron to do with Computer Programming Languages?
 | pages = 16–17
 | url = http://www.cs.berkeley.edu/~wkahan/VtetLang.pdf
 }}
* {{cite book
 | last = Kepler | first = Johannes | author-link = Johannes Kepler
 | title = Harmonices Mundi (The Harmony of the World) | title-link = Harmonices Mundi
 | publisher = Johann Planck
 | year = 1619
}}
* {{cite journal
 | last = Lee | first = Jung Rye
 | year = 1997
 | title = The Law of Cosines in a Tetrahedron
 | journal = J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math.
 | volume = 4 | issue = 1 | pages = 1&ndash;6
}}
* {{cite journal
 | last1 = Murakami | first1 = Jun
 | last2 = Yano | first2 = Masakazu
 | title = On the volume of a hyperbolic and spherical tetrahedron
 | mr = 2154824
 | year = 2005
 | journal = Communications in Analysis and Geometry
 | issn = 1019-8385
 | volume = 13 | issue = 2 | pages = 379–400
 | doi = 10.4310/cag.2005.v13.n2.a5
 | doi-access = free
}}
* {{cite journal
 | last = Park
 | first = Poo-Sung
 | title = Regular polytope distances
 | journal = [[Forum Geometricorum]]
 | volume = 16
 | year = 2016
 | pages = 227–232
 | url = http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
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 | access-date = 24 May 2016
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 | url-status = dead
 }}
* {{cite book
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 | year = 2013
 | title = The Routledge International Handbook of Innovation Education
 | publisher = Routledge
 | isbn = 978-0-203-38714-6
}}
{{refend}}

==External links==
{{Commons category |Tetrahedron }}
* {{mathworld |urlname=Tetrahedron |title=Tetrahedron }}
* [http://www.korthalsaltes.com/model.php?name_en=tetrahedron Free paper models of a tetrahedron and many other polyhedra]
* [http://mathforum.org/pcmi/hstp/resources/dodeca/paper.html ''An Amazing, Space Filling, Non-regular Tetrahedron''] that also includes a description of a "rotating ring of tetrahedra", also known as a [[kaleidocycle]].

{{Polyhedra}}
{{Polyhedron navigator}}
{{Polytopes}}
{{Authority control}}

[[Category:Deltahedra]]
[[Category:Platonic solids]]
[[Category:Individual graphs]]
[[Category:Self-dual polyhedra]]
[[Category:Prismatoid polyhedra]]
[[Category:Pyramids (geometry)]]
[[Category:Tetrahedra| ]]