The regular tetrahedron was considered as the classical element of fire, because of his interpretation of its sharpest corner being most penetrating.

The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.

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).

Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face parallel to the

The two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes.

For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly.

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.

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.

around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),[a] plus

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

This process enhances the complexity and detail of tetrahedral meshes, which is particularly beneficial in numerical simulations, finite element analysis, and computer graphics.

A similarity class is the set of tetrahedra with the same geometric shape, regardless of their specific position, orientation, and scale.

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.

[22] For tetrahedra in hyperbolic space or in three-dimensional elliptic geometry, the dihedral angles of the tetrahedron determine its shape and hence its volume.

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.

However, two regular tetrahedra can be combined with an octahedron, giving a rhombohedron that can tile space as the tetrahedral-octahedral honeycomb.

A corollary of the usual law of sines is that in a tetrahedron with vertices O, A, B, C, we have One may view the two sides of this identity as corresponding to clockwise and counterclockwise orientations of the surface.

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.

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 Fa, Fb, Fc, and Fd.

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

The process completes as a birectification, reducing the original faces down to points, and producing the self-dual tetrahedron once again.

This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.

A stella octangula is a compound of two tetrahedra in dual position and its eight vertices define a cube as their convex hull.

At some airfields, 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.

There are molecules with the shape based on four nearby atoms whose bonds form the sides of a tetrahedral structure, such as white phosphorus allotrope[40] and tetra-t-butyltetrahedrane, known derivative of the hypothetical tetrahedrane.

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.

The tetrahedral hypothesis, originally published by William Lowthian Green to explain the formation of the Earth,[43] was popular through the early 20th century.

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.

[46] The tetrahedron with regular faces is a solution to an old puzzle asking to form four equilateral triangles using six unbroken matchsticks.