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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.
It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes. It contains the 75 nonprismatic uniform polyhedra , as well as 44 stellated forms of the convex regular and quasiregular polyhedra.
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} }} .
The regular tetrahedron is the simplest deltahedron. A polyhedron is said to be convex if a line between any two of its vertices lies either within its interior or on its boundary, and additionally, if no two faces are coplanar (lying in the same plane) and no two edges are collinear (segments of the same line).
A regular tetrahedron is invariant under twelve distinct rotations (if the identity transformation is included as a trivial rotation and reflections are excluded). These are illustrated here in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (pink and orange arrows) rotations that permute the tetrahedron through the positions.
(rotating and 3D model) Type: Catalan: Conway notation: oC or deC Coxeter diagram: Face polygon: Kite with 3 equal acute angles & 1 obtuse angle Faces: 24, congruent: Edges: 24 short + 24 long = 48 Vertices: 8 (connecting 3 short edges) + 6 (connecting 4 long edges) + 12 (connecting 4 alternate short & long edges) = 26 Face configuration: V3.4. ...
Print/export Download as PDF; Printable version; In other projects ... Rotation group: D 4, [2,4] +, (224), order 8 Dual polyhedron: Square antiprism:
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 .