Regular Tetrahedron

The regular tetrahedron, often simply called "the" tetrahedron, is the Platonic solid with four polyhedron vertices, six polyhedron edges, and four equivalent equilateral triangular faces, . It is illustrated above together with a wireframe version and a net that can be used for its construction.

The regular tetrahedron is also the uniform polyhedron with Maeder index 1 (Maeder 1997), Wenninger index 1 (Wenninger 1989), Coxeter index 15 (Coxeter et al. 1954), and Har'El index 6 (Har'El 1993). It is described by the Schläfli symbol and the Wythoff symbol is . It is an isohedron, and a special case of the general tetrahedron and the isosceles tetrahedron.

A number of symmetric projections of the regular tetrahedron are illustrated above.

The regular tetrahedron is implemented in the Wolfram Language as Tetrahedron[] or UniformPolyhedron["Tetrahedron"]. Precomputed properties are available as PolyhedronData["Tetrahedron", prop].

The tetrahedron has 7 axes of symmetry: (axes connecting vertices with the centers of the opposite faces) and (the axes connecting the midpoints of opposite sides).

There are no other convex polyhedra other than the tetrahedron having four faces.

The tetrahedron has two distinct nets (Buekenhout and Parker 1998). Questions of polyhedron coloring of the tetrahedron can be addressed using the Pólya enumeration theorem.

The surface area of the tetrahedron is simply four times the area of a single equilateral triangle face

so

The height of the regular tetrahedron is

and the inradius and circumradius are

where as it must.

Since a tetrahedron is a pyramid with a triangular base, , giving

The dihedral angle is

and the Dehn invariant for a unit regular tetrahedron is

(OEIS A377277), where the first expression uses the basis of Conway et al. (1999).

The solid angle subtended from a vertex by the opposite face of a regular tetrahedron is given by

or approximately 0.55129 steradians.

The midradius of the tetrahedron is

Plugging in for the polyhedron vertices gives

As illustrated above, the dual polyhedron of an tetrahedron with unit edge lengths is another oppositely oriented tetrahedron with unit edge lengths.

The figure above shows an origami tetrahedron constructed from a single sheet of paper (Kasahara and Takahama 1987, pp. 56-57).

It is the prototype of the tetrahedral group . The connectivity of the vertices is given by the tetrahedral graph, equivalent to the circulant graph and the complete graph .

The tetrahedron is its own dual polyhedron, and therefore the centers of the faces of a tetrahedron form another tetrahedron (Steinhaus 1999, p. 201). The tetrahedron is the only simple polyhedron with no polyhedron diagonals, and it cannot be stellated. If a regular tetrahedron is cut by six planes, each passing through an edge and bisecting the opposite edge, it is sliced into 24 pieces (Gardner 1984, pp. 190 and 192; Langman 1951).

Alexander Graham Bell was a proponent of use of the tetrahedron in framework structures, including kites (Bell 1903; Lesage 1956, Gardner 1984, pp. 184-185). The opposite edges of a regular tetrahedron are perpendicular, and so can form a universal coupling if hinged appropriately. Eight regular tetrahedra can be placed in a ring which rotates freely, and the number can be reduced to six for squashed irregular tetrahedra (Wells 1975, 1991).

Let a tetrahedron be length on a side, and let its base lie in the plane with one vertex lying along the positive -axis. The polyhedron vertices of this tetrahedron are then located at (, 0, 0), (, , 0), and (0, 0, ), where

is then

This gives the area of the base as

The height is

The circumradius is found from

Solving gives

The inradius is

which is also

The angle between the bottom plane and center is then given by

Given a tetrahedron of edge length situated with vertical apex and with the origin of coordinate system at the geometric centroid of the vertices, the four polyhedron vertices are located at , , , with, as shown above

The vertices of a tetrahedron of side length can also be given by a particularly simple form when the vertices are taken as corners of a cube (Gardner 1984, pp. 192-194). One such tetrahedron for a cube of side length 1 gives the tetrahedron of side length having vertices (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0), and satisfies the inequalities

The following table gives polyhedra which can be constructed by augmentation of a tetrahedron by pyramids of given heights .

Connecting opposite pairs of edges with equally spaced lines gives a configuration like that shown above which divides the tetrahedron into eight regions: four open and four closed (Steinhaus 1999, p. 246).

Michigan artist David Barr designed his "Four Corners Project" in 1976. It is an Earth-sized regular tetrahedron that spans the planet, with just the tips of its four corners protruding. These visible portions are four-inch tetrahedra, which protrude from the globe at Easter Island, Greenland, New Guinea, and the Kalahari Desert. Barr traveled to these locations and was able to permanently install the four aligned marble tetrahedra between 1981 and 1985 (G. Hart, pers. comm.; Arlinghaus and Nystuen 1986).

By slicing a tetrahedron as shown above, a square can be obtained. This cut divides the tetrahedron into two congruent solids rotated by . The projection of a tetrahedron can be an equilateral triangle or a square (Steinhaus 1999, pp. 191-192).