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The triakis truncated tetrahedron is a polyhedron constructed from a truncated tetrahedron by adding three tetrahedrons onto its triangular faces, as interpreted by the name "triakis". It is classified as plesiohedron , meaning it can tessellate in three-dimensional space known as honeycomb ; an example is triakis truncated tetrahedral honeycomb .
The truncated tetrahedron-first Schlegel diagram projection of the truncated 5-cell into 3-dimensional space has the following structure: The projection envelope is a truncated tetrahedron . One of the truncated tetrahedral cells project onto the entire envelope.
The truncated tetrahedron first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure: The projection envelope is a truncated cube . The nearest truncated tetrahedron to the 4D viewpoint projects to the center of the envelope, with its triangular faces joined to 4 octahedral volumes that connect it to ...
Types of truncation on a square, {4}, showing red original edges, and new truncated edges in cyan. A uniform truncated square is a regular octagon, t{4}={8}. A complete truncated square becomes a new square, with a diagonal orientation. Vertices are sequenced around counterclockwise, 1-4, with truncated pairs of vertices as a and b.
In geometry, a Schlegel diagram is a projection of a polytope from into through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in R d − 1 {\textstyle \mathbb {R} ^{d-1}} that, together with the original facet, is combinatorially equivalent to the original polytope.
It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb. [1] [2] The triakis truncated tetrahedron is the shape of the Voronoi cell of the carbon atoms in diamond, which lie on the diamond cubic crystal structure. [3] [4] As the Voronoi cell of a symmetric space pattern, it is a plesiohedron. [5]
The truncated 600-cell or truncated hexacosichoron is a uniform 4-polytope. It is derived from the 600-cell by truncation. It has 720 cells: 120 icosahedra and 600 truncated tetrahedra. Its vertex figure is a pentagonal pyramid, with one icosahedron on the base, and 5 truncated tetrahedra around the sides.
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 .