Tetrahedron

Not to be confused with Tetraedron or Tetrahedron (journal).

In geometry, a tetrahedron (pl.: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.^[1]

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

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

Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.^[1]

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.^[2]

Regular tetrahedron

Regular tetrahedron, described as the classical element of fire.

The stella octangula

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 a thin volume of empty space is left, where the five edge angles do not quite meet.

A regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles. 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 deltahedron, a polyhedron in which all of its faces are equilateral triangles; there are seven other convex deltahedra.^[3]

The regular tetrahedron is also one of the five regular Platonic solids, a set of polyhedrons in which all of their faces are regular polygons.^[4] 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, because of his interpretation of its sharpest corner being most penetrating.^[5]

The regular tetrahedron is self-dual, meaning its dual is another regular tetrahedron. The 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., rectifying the tetrahedron).

The tetrahedron is yet related to another two solids: By 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 tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. Some tetrahedra that are not regular, including the Schläfli orthoscheme and the Hill tetrahedron, can tessellate.

Measurement

3D model of a regular tetrahedron

Consider a regular tetrahedron with edge length ${\displaystyle a}$.

The height of a regular tetrahedron is ${\textstyle {{\sqrt {\frac {2}{3}}}a}}$.^[6]

Its surface area is four times the area of an equilateral triangle: ${\textstyle A=4\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)=a^{2}{\sqrt {3}}\approx 1.732a^{2}.}$^[7]

Its volume can be ascertained similarly as the other pyramids, one-third of the base times height. Because the base is an equilateral, it is:^[7] $${\displaystyle V={\frac {1}{3}}\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)\cdot {\frac {\sqrt {6}}{3}}a={\frac {a^{3}}{6{\sqrt {2}}}}\approx 0.118a^{3}.}$$ Its volume can also be obtained by dissecting a cube into a tetrahedron and four triangular pyramids.^[8]

Its dihedral angle—the angle formed by two planes in which adjacent faces lie—is ${\textstyle \arccos \left({\frac {1}{3}}\right)=\arctan \left(2{\sqrt {2}}\right)\approx 70.529^{\circ }.}$^[7]

Its vertex–center–vertex angle—the angle between lines from the tetrahedron center to any two vertices—is ${\textstyle \arccos \left(-{\frac {1}{3}}\right)=2\arctan \left({\sqrt {2}}\right)\approx 109.471^{\circ },}$ denoted the tetrahedral angle.^[9] It is the angle between 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 bond angle.

Regular tetrahedron ABCD and its circumscribed sphere

The radii of its circumsphere ${\displaystyle R}$, insphere ${\displaystyle r}$, midsphere ${\displaystyle r_{\mathrm {M} }}$, and exsphere ${\displaystyle r_{\mathrm {E} }}$ are:^[7] $${\displaystyle {\begin{aligned}R={\sqrt {\frac {3}{8}}}a,&\qquad r={\frac {1}{3}}R={\frac {a}{\sqrt {24}}},\\r_{\mathrm {M} }={\sqrt {rR}}={\frac {a}{\sqrt {8}}},&\qquad r_{\mathrm {E} }={\frac {a}{\sqrt {6}}}.\end{aligned}}}$$ For a regular tetrahedron with side length ${\displaystyle a}$ and circumsphere radius ${\displaystyle R}$, the distances ${\displaystyle d_{i}}$ from an arbitrary point in 3-space to its four vertices satisfy the equations:^[10] $${\displaystyle {\begin{aligned}{\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+{\frac {16R^{4}}{9}}&=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+{\frac {2R^{2}}{3}}\right)^{2},\\4\left(a^{4}+d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}\right)&=\left(a^{2}+d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}\right)^{2}.\end{aligned}}}$$

With respect to the base plane the slope of a face (2√2) is twice that of an edge (√2), corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that along the median of a face. In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see proof).

Its solid angle at a vertex subtended by a face is ${\textstyle \arccos \left({\frac {23}{27}}\right)={\frac {\pi }{2}}-3\arcsin \left({\frac {1}{3}}\right)=3\arccos \left({\frac {1}{3}}\right)-\pi ,}$ or approximately 0.55129 steradians, 1809.8 square degrees, and 0.04387 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: $${\displaystyle \left(\pm 1,0,-{\frac {1}{\sqrt {2}}}\right)\quad {\mbox{and}}\quad \left(0,\pm 1,{\frac {1}{\sqrt {2}}}\right)}$$

Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face parallel to the ${\displaystyle xy}$ plane, the vertices are: $${\displaystyle {\begin{aligned}\left({\sqrt {\frac {8}{9}}},0,-{\frac {1}{3}}\right),&\quad \left(-{\sqrt {\frac {2}{9}}},{\sqrt {\frac {2}{3}}},-{\frac {1}{3}}\right),\\\left(-{\sqrt {\frac {2}{9}}},-{\sqrt {\frac {2}{3}}},-{\frac {1}{3}}\right),&\quad (0,0,1)\end{aligned}}}$$ with the edge length of ${\textstyle {\frac {2{\sqrt {6}}}{3}}}$.

A regular tetrahedron can be embedded inside a cube in two ways such that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces. For one such embedding, the Cartesian coordinates of the vertices are $${\displaystyle {\begin{aligned}(1,1,1),&\quad (1,-1,-1),\\(-1,1,-1),&\quad (-1,-1,1).\end{aligned}}}$$ This yields a tetrahedron with edge-length ${\displaystyle 2{\sqrt {2}}}$, centered at the origin. For the other tetrahedron (which is 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, a polyhedron that is by alternating a cube. This form has Coxeter diagram and Schläfli symbol ${\displaystyle \mathrm {h} \{4,3\}}$.

Symmetry

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 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.

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 ${\displaystyle \mathrm {T} _{\mathrm {d} }}$. This symmetry group is isomorphic to the symmetric group ${\displaystyle S_{4}}$. They can be categorized as follows:

• It has rotational tetrahedral symmetry ${\displaystyle \mathrm {T} }$. This symmetry is isomorphic to alternating group ${\displaystyle A_{4}}$—the identity and 11 proper rotations—with the following conjugacy classes (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the unit quaternion representation):
  • identity (identity; 1)
  • rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; ⁠1 ± i ± j ± k/2⁠)
  • rotation by an angle of 180° such that an edge maps to the opposite edge: 3 ((1 2)(3 4), etc.; 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 projections, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A₂ Coxeter plane.

Orthographic projection

Centered by	Face/vertex	Edge
Image
Projective symmetry	[3]	[4]

Cross section of regular tetrahedron

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.^[11] 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 wedges.

A tetragonal disphenoid viewed orthogonally to the two green edges.

This property also applies for tetragonal disphenoids when applied to the two special edge pairs.

Spherical tiling

The tetrahedron can also be represented as a spherical tiling (of spherical triangles), and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Orthographic projection	Stereographic projection

Helical stacking

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 dimensions, all the convex regular 4-polytopes with tetrahedral cells (the 5-cell, 16-cell and 600-cell) can be constructed as tilings of the 3-sphere by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.

Irregular tetrahedra

Tetrahedral symmetry subgroup relations

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 angles, as at the corner of a cube.

An isodynamic tetrahedron is one in which the cevians that join the vertices to the incenters of the opposite faces are concurrent.

An isogonic tetrahedron has concurrent cevians that join the vertices to the points of contact of the opposite faces with the inscribed sphere of the tetrahedron.

Disphenoid

Main article: Disphenoid

A space-filling tetrahedral disphenoid inside a cube. Two edges have dihedral angles 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.

Orthoschemes

A cube dissected into six characteristic orthoschemes.

A 3-orthoscheme is a tetrahedron where all four faces are 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 with right triangle or obtuse triangle faces.

An orthoscheme is an irregular simplex that is the convex hull of a 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.^[12]

Coxeter also calls quadrirectangular tetrahedra "characteristic tetrahedra", because of their integral relationship to the regular polytopes and their symmetry groups.^[13] For example, the special case of a 3-orthoscheme with equal-length perpendicular edges is characteristic of the cube, which means that the cube can be subdivided into instances of this orthoscheme. If its three perpendicular edges are of unit length, its remaining edges are two of length √2 and one of length √3, so all its edges are edges or diagonals of the cube. The cube can be dissected into six such 3-orthoschemes four different ways, with all six surrounding the same √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 characteristic orthoscheme. There is a 3-orthoscheme, which is the "characteristic tetrahedron of the regular tetrahedron". The regular tetrahedron is subdivided into 24 instances of its characteristic tetrahedron by its planes of symmetry. The 24 characteristic tetrahedra of the regular tetrahedron occur in two mirror-image forms, 12 of each.

Characteristics of the regular tetrahedron[14]
	edge	arc		dihedral
𝒍	${\displaystyle 2}$	109°28′16″	${\displaystyle \pi -2{\text{𝜿}}}$	70°31′44″	${\displaystyle \pi -2{\text{𝟁}}}$
𝟀	${\displaystyle {\sqrt {\tfrac {4}{3}}}\approx 1.155}$	70°31′44″	${\displaystyle 2{\text{𝜿}}}$	60°	${\displaystyle {\tfrac {\pi }{3}}}$
𝝉[a]	${\displaystyle 1}$	54°44′8″	${\displaystyle {\tfrac {\pi }{2}}-{\text{𝜿}}}$	60°	${\displaystyle {\tfrac {\pi }{3}}}$
𝟁	${\displaystyle {\sqrt {\tfrac {1}{3}}}\approx 0.577}$	54°44′8″	${\displaystyle {\tfrac {\pi }{2}}-{\text{𝜿}}}$	60°	${\displaystyle {\tfrac {\pi }{3}}}$
${\displaystyle _{0}R/l}$	${\displaystyle {\sqrt {\tfrac {3}{2}}}\approx 1.225}$
${\displaystyle _{1}R/l}$	${\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}$
${\displaystyle _{2}R/l}$	${\displaystyle {\sqrt {\tfrac {1}{6}}}\approx 0.408}$
${\displaystyle {\text{𝜿}}}$		35°15′52″	${\displaystyle {\tfrac {{\text{arc sec }}3}{2}}}$

If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths ${\displaystyle {\sqrt {\tfrac {4}{3}}}}$, ${\displaystyle 1}$, ${\displaystyle {\sqrt {\tfrac {1}{3}}}}$ around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),^[a] plus ${\displaystyle {\sqrt {\tfrac {3}{2}}}}$, ${\displaystyle {\sqrt {\tfrac {1}{2}}}}$, ${\displaystyle {\sqrt {\tfrac {1}{6}}}}$ (edges that are the characteristic radii of the regular tetrahedron). The 3-edge path along orthogonal edges of the orthoscheme is ${\displaystyle 1}$, ${\displaystyle {\sqrt {\tfrac {1}{3}}}}$, ${\displaystyle {\sqrt {\tfrac {1}{6}}}}$, first from a tetrahedron vertex to an tetrahedron edge center, then turning 90° to an tetrahedron face center, then turning 90° to the tetrahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a 60-90-30 triangle which is one-sixth of a tetrahedron face. The three faces interior to the tetrahedron are: a right triangle with edges ${\displaystyle 1}$, ${\displaystyle {\sqrt {\tfrac {3}{2}}}}$, ${\displaystyle {\sqrt {\tfrac {1}{2}}}}$, a right triangle with edges ${\displaystyle {\sqrt {\tfrac {1}{3}}}}$, ${\displaystyle {\sqrt {\tfrac {1}{2}}}}$, ${\displaystyle {\sqrt {\tfrac {1}{6}}}}$, and a right triangle with edges ${\displaystyle {\sqrt {\tfrac {4}{3}}}}$, ${\displaystyle {\sqrt {\tfrac {3}{2}}}}$, ${\displaystyle {\sqrt {\tfrac {1}{6}}}}$.

Space-filling tetrahedra

A space-filling tetrahedron packs with directly congruent or enantiomorphous (mirror image) copies of itself to tile space.^[15] 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

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^[16] of a symmetry group is an example of a Goursat tetrahedron. The Goursat tetrahedra generate all the regular polyhedra (and many other uniform polyhedra) by mirror reflections, a process referred to as Wythoff's kaleidoscopic construction.

For polyhedra, Wythoff's construction arranges three mirrors at angles to each other, as in a kaleidoscope. Unlike a cylindrical kaleidoscope, Wythoff's mirrors are located at three faces of a Goursat tetrahedron such that all three mirrors intersect at a single point. (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 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 above.^[13] The disphenoid is the double orthoscheme face-bonded to its mirror image (a quadruple orthoscheme). Thus all three of these Goursat tetrahedra, and all the polyhedra they generate by reflections, can be dissected into characteristic tetrahedra of the cube.

Isometries of irregular tetrahedra

The isometries of an irregular (unmarked) tetrahedron depend on the geometry of the tetrahedron, with 7 cases possible. In each case a 3-dimensional point group is formed. Two other isometries (C₃, [3]⁺), and (S₄, [2⁺,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.

Tetrahedron name				Edge equivalence diagram	Description
Symmetry
Schön.	Cox.	Orb.	Ord.
Regular tetrahedron					Four equilateral triangles It forms the symmetry group Td, isomorphic to the symmetric group, S4. A regular tetrahedron has Coxeter diagram and Schläfli symbol {3,3}.
Td T	[3,3] [3,3]+	*332 332	24 12
Triangular pyramid					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 C3v, isomorphic to the symmetric group, S3. A triangular pyramid has Schläfli symbol {3}∨( ).
C3v C3	[3] [3]+	*33 33	6 3
Mirrored sphenoid					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 Cs, also isomorphic to the cyclic group, Z2.
Cs =C1h =C1v	[ ]	*	2
Irregular tetrahedron (No symmetry)					Four unequal triangles Its only isometry is the identity, and the symmetry group is the trivial group. An irregular tetrahedron has Schläfli symbol ( )∨( )∨( )∨( ).
C1	[ ]+	1	1
Disphenoids (Four equal triangles)
Tetragonal disphenoid					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 D2d. A tetragonal disphenoid has Coxeter diagram and Schläfli symbol s{2,4}.
D2d S4	[2+,4] [2+,4+]	2*2 2×	8 4
Rhombic disphenoid					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 V4 or Z22, present as the point group D2. A rhombic disphenoid has Coxeter diagram and Schläfli symbol sr{2,2}.
D2	[2,2]+	222	4
Generalized disphenoids (2 pairs of equal triangles)
Digonal disphenoid