The regular tetrahedron has 24 isometries, forming the symmetry group T d,, (*332), isomorphic to the symmetric group, S 4. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion. 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 vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also animation, showing one of the two tetrahedra in the cube). 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 ( ± 1, 0, − 1 2 ) and ( 0, ± 1, 1 2 ) Isometries of the regular tetrahedron The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, and two level edges: The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula.Ĭoordinates for a regular tetrahedron The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. Some tetrahedra that are not regular, including the Schläfli orthoscheme and the Hill tetrahedron, can tessellate. 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. 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.

In a regular tetrahedron, all faces are the same size and shape (congruent) and all edges are the same length.įive tetrahedra are laid flat on a plane, with the highest 3-dimensional points marked as 1, 2, 3, 4, and 5. It is one of the five regular Platonic solids, which have been known since antiquity. Regular tetrahedron Ī regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles. įor 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. Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. 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'. 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. 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 the simplest of all the ordinary convex polyhedra. In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.

Tetrahedral molecular geometry