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The skeleton of the tetrahedron (comprising the vertices and edges) forms a graph, with 4 vertices, and 6 edges. It is a special case of the complete graph, K 4, and wheel graph, W 4. [48] It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.
The 6 edge lengths - associated to the six edges of the tetrahedron. The 12 face angles - there are three of them for each of the four faces of the tetrahedron. The 6 dihedral angles - associated to the six edges of the tetrahedron, since any two faces of the tetrahedron are connected by an edge.
The truncated tetrahedron can be constructed from a regular tetrahedron by cutting all of its vertices off, a process known as truncation. [1] The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices. [2] With edge length 1, the Cartesian coordinates of the 12 vertices are points
In partial truncation, or alternation, half of the vertices and connecting edges are completely removed. The operation applies only to polytopes with even-sided faces. Faces are reduced to half as many sides, and square faces degenerate into edges. For example, the tetrahedron is an alternated cube, h{4,3}.
The smallest polyhedron is the tetrahedron with 4 triangular faces, 6 edges, and 4 vertices. Named polyhedra primarily come from the families of platonic solids , Archimedean solids , Catalan solids , and Johnson solids , as well as dihedral symmetry families including the pyramids , bipyramids , prisms , antiprisms , and trapezohedrons .
Edges Vertices Platonic solid: tetrahedron: 4: 6: 4 Archimedean solid: truncated tetrahedron: 8: 18: 12 Catalan solid: triakis tetrahedron: 12: 18: 8 Near-miss Johnson solid: Truncated triakis tetrahedron: 16 42 28 Tetrated dodecahedron: 28 54 28 Uniform star polyhedron: Tetrahemihexahedron: 7: 12: 6
The chamfered tetrahedron or alternate truncated cube is a convex polyhedron constructed: by chamfering a regular tetrahedron: replacing its 6 edges with congruent flattened hexagons; or by alternately truncating a (regular) cube: replacing 4 of its 8 vertices with congruent equilateral-triangle faces.
In 4-dimensional geometry, the tetrahedral bipyramid is the direct sum of a tetrahedron and a segment, {3,3} + { }. Each face of a central tetrahedron is attached with two tetrahedra, creating 8 tetrahedral cells, 16 triangular faces, 14 edges, and 6 vertices. [1]