Regular tetrahedron

Properties

Measurement

Let ${\displaystyle a}$ be the edge length of a regular tetrahedron. The height of an equilateral triangle, the segment line is drawn from any vertex to the midpoint of an edge, and by the calculation of Pythagorean theorem, the height of any equilateral triangle is $\frac{\sqrt{3}}{2}a$. The distance between the centroid and a vertex has two different lengths; the longest is two-thirds of the height of an equilateral triangle. By the Pythagorean theorem again, one can find the height of a regular tetrahedron: $${\displaystyle h=\frac{\sqrt{6}}{3}a\approx 0.816a.}$$ Since the shortest distance between the centroid and a vertex is one-third of the height of an equilateral triangle, the dihedral angle (i.e., an angle between two triangular faces) of a regular tetrahedron is $\arccos \left(1/3\right)=\arctan \left(2\sqrt{2}\right)\approx {70.529}^{\circ }$.

The surface area of a regular tetrahedron is four times area of an equilateral triangle:$${\displaystyle A=4\cdot \left(\frac{\sqrt{3}}{4}{a}^2\right)=a^2\sqrt{3}\approx 1.732a^2.}$$

The volume is one-third of the base times the height, the general formula for a pyramid. This can also be found by dissecting a cube into a tetrahedron and four triangular pyramids.$${\displaystyle V=\frac{1}{3}\cdot \left(\frac{\sqrt{3}}{4}{a}^2\right)\cdot \frac{\sqrt{6}}{3}a=\frac{a^3}{6\sqrt{2}}\approx 0.118a^3.}$$

The radii of its circumsphere ${\displaystyle r_{\mathrm{c}\mathrm{i}}}$ (a sphere that contains the regular tetrahedron and touches its vertices), insphere ${\displaystyle r_{\text{in}}}$ (a sphere within a regular tetrahedron and touches to its faces), midsphere ${\displaystyle r_{\mathrm{m}\mathrm{i}}}$ (a sphere that touches its edges), and exsphere (a sphere outside, touches the face of a regular tetrahedron and the planes defined by extending the adjacent faces outwards) ${\displaystyle r_{\mathrm{e}\mathrm{x}}}$ are: $${\displaystyle \begin{array}{cc}{r}_{\mathrm{c}\mathrm{i}}=\frac{a\sqrt{3}}{2\sqrt{2}}\approx 0.612a,& r_{\mathrm{i}\mathrm{n}}=\frac{1}{3}{r}_{\mathrm{c}\mathrm{i}}\approx 0.204a,\\ r_{\mathrm{m}\mathrm{i}}=\sqrt{r_{\mathrm{i}\mathrm{n}}{r}_{\mathrm{c}\mathrm{i}}}\approx 0.354a,& r_{\mathrm{e}\mathrm{x}}=\frac{a}{\sqrt{6}}\approx 0.408a.\end{array}}$$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: $${\displaystyle \begin{array}{cc}\frac{d_1^4+d_2^4+d_3^4+d_4^4}{4}+\frac{16R^4}{9}& ={\left(\frac{d_1^2+d_2^2+d_3^2+d_4^2}{4}+\frac{2R^2}{3}\right)}^2,\\ 4\left(a^4+d_1^4+d_2^4+d_3^4+d_4^4\right)& ={\left(a^2+d_1^2+d_2^2+d_3^2+d_4^2\right)}^2.\end{array}}$$

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 $\arccos \left(\frac{23}{27}\right)=\frac{\pi }{2}-3\arcsin \left(\frac{1}{3}\right)=3\arccos \left(\frac{1}{3}\right)-\pi ,$ or approximately 0.55129 steradians or 1809.8 square degrees.

Symmetry

The regular tetrahedron has a three-dimensional symmetry group known as full tetrahedral symmetry ${\displaystyle {\mathrm{T}}_{\mathrm{d}}}$ or as the symmetric group ${\displaystyle S_4}$. This symmetry group has 24 isometries, containing seven rotational axes and six reflectional planes. The seven rotational axes are the four axes of three-fold rotational symmetry (0°, 120°, and 240°) passing through a vertex to the centroid of an equilateral triangular face, and the three axes of two-fold rotational symmetry (0° and 180°) passing through the midpoint of two edges. This point group has rotational tetrahedral symmetry ${\displaystyle \mathrm{T}}$.

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.

Spherical tetrahedron

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.

The first few members in the infinite families of regular hosohedra and dihedra are geometrically simpler than the spherical tetrahedron, where the convex regular tetrahedron instead classifies as the simplest regular polyhedron of its class. For example, the hexagonal hosohedron or dihedron and the spherical tetrahedron are regular spherical polyhedra of equal order and total dimensional elements, though they are non-isomorphic (as they have distinct point group symmetries and associated fundamental domains).

Orthogonal projections

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.

Projective spaces

The smallest three-dimensional projective space ${\displaystyle \mathrm{PG}(3,2)}$ shares symmetry with the regular tetrahedron, when extending the two-dimensional Fano plane ${\displaystyle \mathrm{PG}(2,2)}$ in three dimensions. ${\displaystyle \mathrm{PG}(3,2)}$ has four points lying on the four vertices of a regular tetrahedron, six points on the midpoints of its six edges, four on the center (centroids) of its four faces, and a point at the interior core of the tetrahedron. Every single plane is isomorphic to a Fano plane as the smallest projective plane. In the geometry over the field with one element ${\displaystyle {\mathrm{F}}_1}$, vector spaces behave like finite sets, and projective subspaces correspond to ordinary subsets, therefore, ${\displaystyle \mathrm{PG}(3,1)}$ forms a standard 3-simplex, which is abstractly a regular tetrahedron. ${\displaystyle \mathrm{PG}(3,2)}$ serves as the three-dimensional projective realization of the sixteen-dimensional sedenions and their algebraic multiplication.

Related figures

Construction of polyhedra

Many polyhedra are constructed involving regular tetrahedra. By truncating the vertices of a regular tetrahedron, it 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. Some Johnson solid such as elongated triangular pyramid and elongated triangular bipyramid are constructed by attaching one and two regular tetrahedra onto the bases of a triangular prism; the triangular bipyramid is constructed by attaching two regular tetrahedra face-to-face.

Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix.

Polyhedral compounds

The stellated octahedron is constructed by extending its faces to form equilateral triangles on each regular octahedron's face. In the perspective of polyhedral compounds, this figure comprises two such dual tetrahedra, a self-dual because of sharing a common intersphere in the center. Another interesting polyhedral compound involves five intersecting tetrahedra, which has been known for hundreds of years. It comes up regularly in the world of origami. Joining the twenty vertices would form a regular dodecahedron. There are both left-handed and right-handed forms, which are mirror images of each other. Superimposing both forms gives a compound of ten tetrahedra, in which the ten tetrahedra are arranged as five pairs of stellae octangulae.

Dehn invariant and honeycombs

The Dehn invariant of a regular tetrahedron can be defined as a tensor product of the edge length and the dihedral angle of a regular tetrahedron $6a\otimes \arccos \frac{1}{3}$, which is non-zero, where ${\displaystyle a}$ is the edge length of a regular tetrahedron. The Dehn invariant was originally dated from Hilbert's third problem, a set of 23 problems by David Hilbert, asking whether, given any two polyhedra with the same volume, the first polyhedron may be dissected into pieces, which then reassembled into the second polyhedron. His student, Max Dehn, provided the negative answer by showing his invariant.

Every polyhedron with a Dehn invariant of zero can tile a space with its copy by attaching its faces to another, forming a honeycomb. Since the regular tetrahedron has a non-zero Dehn invariant, it cannot alone do such. A related result that two different polyhedra that join to tile the space can have the Dehn invariant of zero. In the case of a regular tetrahedron, it can be alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, forming the alternated cubic honeycomb.

Four-dimensional figures

The pentachoron is a four-dimensional polytope, a generalization of a tetrahedron in four-dimensional space. It is bounded by five regular tetrahedra, known as cells.

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.