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Types of tetrahedra

A tetrahedron is a three-dimensional object with four faces, six edges, and four vertices. It can be considered as pyramid whenever one of its faces can be considered as the base. There are many types of tetrahedra. A trirectangular tetrahedron is a tetrahedron whose three face angles at one vertex are right angles, as at the corner of a cube. A disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles, a special case of a regular tetrahedron.

Generally, the tetrahedron can be seen as a wheel graph, meaning it is a triangle in which three vertices connect its center known as the universal vertex in a plane. Unlike other pyramids and other polyhedrons, the tetrahedron is one of the polyhedrons that does not have space diagonal; the other polyhedrons with such property are Császár polyhedron and Schonhardt polyhedron. It is also known as 3-simplex, the generalization of a triangle in multi-dimension. It is self-dual, meaning its dual polyhedron is tetrahedron itself. Many other properties of tetrahedra are explicitly described in the following sections.

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. A convex polyhedron in which all of its faces are equilateral triangles is the deltahedron. There are eight convex deltahedra, one of which is the regular tetrahedron.

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. 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.

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 terahedron. The dual of this solid is the triakis tetrahedron, a regular tetrahedron with four triangular piramids atached 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

Given that the regular tetrahedron with edge length $a$. The surface area of a regular tetrahedron $A$ is four times the area of an equilateral triangle: $$A=4\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)=a^{2}{\sqrt {3}}\approx 1.732a^{2}.$$ The height of a regular tetrahedron is ${\textstyle {\frac {\sqrt {6}}{3}}a}$. The volume of a regular tetrahedron can be ascertained similarly as the other pyramids, one-third of the base and its height. Because the base is an equilateral, it is: $$V={\frac {1}{3}}\cdot \left({\frac {\sqrt {3}}{4}}a^{2}\right)\cdot {\frac {\sqrt {6}}{3}}a={\frac {a^{3}}{6{\sqrt {2}}}}\approx 0.118a^{3}.$$ Its volume can also be obtained by dissecting a cube into three parts.

Its dihedral angle—the angle between two planar—and its angle between lines from the center of a regular tetrahedron between two vertices is respectively: $${\begin{aligned}\arccos \left({\frac {1}{3}}\right)&=\arctan \left(2{\sqrt {2}}\right)\approx 70.529^{\circ },\\\arccos \left(-{\frac {1}{3}}\right)&=2\arctan \left({\sqrt {2}}\right)\approx 109.471^{\circ }.\end{aligned}}$$

Regular tetrahedron ABCD and its circumscribed sphere

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

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 $${\begin{aligned}\arccos \left({\frac {23}{27}}\right)&={\frac {\pi }{2}}-3\arcsin \left({\frac {1}{3}}\right)\\&=3\arccos \left({\frac {1}{3}}\right)-\pi \end{aligned}}$$ This is approximately 0.55129 steradians, 1809.8 square degrees, or 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: $$\left(\pm 1,0,-{\frac {1}{\sqrt {2}}}\right)\quad {\mbox{and}}\quad \left(0,\pm 1,{\frac {1}{\sqrt {2}}}\right)$$

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

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

• It has rotational tetrahedral symmetry $\mathrm {T}$. This symmetry is isomorphic to alternating group $A_{4}$—the identity and 11 proper rotations—with the following conjugacy classes (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the unit quaternion representation):
  • identity (identity; 1)
  • rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together 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. 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, 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.

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