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A regular tetrahedron, an example of a solid with full tetrahedral symmetry. A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
Print/export Download as PDF; Printable version; In other projects ... Rotation group: D 4, [2,4] +, (224), order 8 Dual polyhedron: Square antiprism:
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 T d {\displaystyle \mathrm {T} _{\mathrm {d} }} .
It has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by (a,b), where b and a depend on each other such that the tetrahedron defined by the four vertices of a face has volume zero, i.e. is a planar face. (1,1) is the rhombic solution.
It is a special case of a p/q-gonal prismatic compound of antiprisms, where in this case the component p/q = 2 is a digonal antiprism, or tetrahedron. Below are two perspective viewpoints of the uniform compound of four tetrahedra, with each color representing one regular tetrahedron :
Reuleaux tetrahedron. The Reuleaux tetrahedron is the intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s. [1] The spherical surface of the ball centered on each vertex passes through the other three vertices, which also form vertices of the Reuleaux tetrahedron.
Drawing and crystal model of variant with tetrahedral symmetry called hexakis tetrahedron [1] In geometry , a tetrakis hexahedron (also known as a tetrahexahedron , hextetrahedron , tetrakis cube , and kiscube [ 2 ] ) is a Catalan solid .
In geometry, the rectified truncated tetrahedron is a polyhedron, constructed as a rectified, truncated tetrahedron. It has 20 faces: 4 equilateral triangles , 12 isosceles triangles , and 4 regular hexagons .