Editing Tetrahedron (section)

== Regular tetrahedron ==
{{infobox polyhedron
 | name = Regular tetrahedron
 | image = Tetrahedron.svg
 | type = [[Platonic solid]], [[Deltahedron]]
 | faces = 4
 | edges = 6
 | vertices = 4
 | angle = 70.529&deg; (regular)
 | symmetry = [[Tetrahedral symmetry]] <math> T_\mathrm{d} </math>
 | dual = self-dual
 | net = Tetrahedron flat.svg
}}
{{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
 | align = right
 | total_width = 400
}}

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>. Its height 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)}}
The volume is one-third of the base times the height, the general formula for a pyramid;{{sfn|Coxeter|1948|loc=Table I(i)}} this can also be found 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|class=skin-invert-image|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}}