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Triakis Truncated Tetrahedral Honeycomb
The triakis truncated tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of triakis truncated tetrahedra. It was discovered in 1914. Voronoi tessellation It is the Voronoi tessellation of the carbon atoms in diamond, which lie in the diamond cubic crystal structure. Being composed entirely of triakis truncated tetrahedra, it is cell-transitive. Relation to quarter cubic honeycomb It can be seen as the uniform quarter cubic honeycomb where its tetrahedral cells are subdivided by the center point into 4 shorter tetrahedra, and each adjoined to the adjacent truncated tetrahedral cells. : See also *Disphenoid tetrahedral honeycomb The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Hor ... References Honeycombs (geometry) Truncated tilin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triakis Truncated Tetrahedral Honeycomb
The triakis truncated tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of triakis truncated tetrahedra. It was discovered in 1914. Voronoi tessellation It is the Voronoi tessellation of the carbon atoms in diamond, which lie in the diamond cubic crystal structure. Being composed entirely of triakis truncated tetrahedra, it is cell-transitive. Relation to quarter cubic honeycomb It can be seen as the uniform quarter cubic honeycomb where its tetrahedral cells are subdivided by the center point into 4 shorter tetrahedra, and each adjoined to the adjacent truncated tetrahedral cells. : See also *Disphenoid tetrahedral honeycomb The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Hor ... References Honeycombs (geometry) Truncated tilin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Tessellation
Voronoi or Voronoy is a Slavic masculine surname; its feminine counterpart is Voronaya. It may refer to *Georgy Voronoy (1868–1908), Russian and Ukrainian mathematician **Voronoi diagram **Weighted Voronoi diagram ** Voronoi deformation density **Voronoi formula **Voronoi pole In geometry, the positive and negative Voronoi poles of a cell in a Voronoi diagram are certain vertices of the diagram. Definition Let V be the Voronoi diagram for a set of sites P, and let V_p be the Voronoi cell of V corresponding to a site ... ** Centroidal Voronoi tessellation {{Disambiguation, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disphenoid Tetrahedral Honeycomb
The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton Conway calls it an ''oblate tetrahedrille'' or shortened to ''obtetrahedrille''. Symmetry of Things, Table 21.1. Prime Architectonic and Catopric tilings of space, p. 293, 295. A cell can be seen as 1/12 of a translational cube, with its vertices centered on two faces and two edges. Four of its edges belong to 6 cells, and two edges belong to 4 cells. : The tetrahedral disphenoid honeycomb is the dual of the uniform bitruncated cubic honeycomb. Its vertices form the A / D lattice, which is also known as the body-centered cubic lattice. Geometry This honeycomb's vertex figure is a tetrakis cube: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms a rhombic dodecahedron. Each edge of the tessellation is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
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. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. 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 one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. 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". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quarter Cubic Honeycomb
The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb. It is vertex-transitive with 6 truncated tetrahedra and 2 tetrahedra around each vertex. It is one of the 28 convex uniform honeycombs. The faces of this honeycomb's cells form four families of parallel planes, each with a 3.6.3.6 tiling. Its vertex figure is an isosceles antiprism: two equilateral triangles joined by six isosceles triangles. John Horton Conway calls this honeycomb a truncated tetrahedrille, and its dual oblate cubille. The vertices and edges represent a Kagome lattice in three dimensions, which is the pyrochlore lattice. Construction The quarter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not merely Congruence (geometry), congruent but must be ''transitive'', i.e. must lie within the same ''symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by Translation (geometry), translations, Rotation (mathematics), rotations, and/or Reflection (mathematics), reflections that maps onto . For this reason, Convex polytope, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an Parity (mathematics), even number of faces. The Dual polyhedron, dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diamond Cubic
The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, including α-tin, the semiconductors silicon and germanium, and silicon–germanium alloys in any proportion. There are also crystals, such as the high-temperature form of cristobalite, which have a similar structure, with one kind of atom (such as silicon in cristobalite) at the positions of carbon atoms in diamond but with another kind of atom (such as oxygen) halfway between those (see :Minerals in space group 227). Although often called the diamond lattice, this structure is not a lattice in the technical sense of this word used in mathematics. Crystallographic structure Diamond's cubic structure is in the Fdm space group (space group 227), which follows the face-centered cubic Bravais lattice. The lattice describes the repeat pattern; for diamond cubic cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diamond
Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the Chemical stability, chemically stable form of carbon at Standard conditions for temperature and pressure, room temperature and pressure, but diamond is metastable and converts to it at a negligible rate under those conditions. Diamond has the highest Scratch hardness, hardness and thermal conductivity of any natural material, properties that are used in major industrial applications such as cutting and polishing tools. They are also the reason that diamond anvil cells can subject materials to pressures found deep in the Earth. Because the arrangement of atoms in diamond is extremely rigid, few types of impurity can contaminate it (two exceptions are boron and nitrogen). Small numbers of lattice defect, defects or impurities (about one per million of lattice atoms) color diamond blue (bor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carbon
Carbon () is a chemical element with the symbol C and atomic number 6. It is nonmetallic and tetravalent In chemistry, the valence (US spelling) or valency (British spelling) of an element is the measure of its combining capacity with other atoms when it forms chemical compounds or molecules. Description The combining capacity, or affinity of an ...—its atom making four electrons available to form covalent bond, covalent chemical bonds. It belongs to group 14 of the periodic table. Carbon makes up only about 0.025 percent of Earth's crust. Three Isotopes of carbon, isotopes occur naturally, Carbon-12, C and Carbon-13, C being stable, while Carbon-14, C is a radionuclide, decaying with a half-life of about 5,730 years. Carbon is one of the Timeline of chemical element discoveries#Ancient discoveries, few elements known since antiquity. Carbon is the 15th Abundance of elements in Earth's crust, most abundant element in the Earth's crust, and the Abundance of the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triakis Truncated Tetrahedron
In geometry, the triakis truncated tetrahedron is a convex polyhedron made from 4 hexagons and 12 isosceles triangles. It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb. 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. As the Voronoi cell of a symmetric space pattern, it is a plesiohedron.. Construction For space-filling, the triakis truncated tetrahedron can be constructed as follows: # Truncate a regular tetrahedron such that the big faces are regular hexagons. # Add an extra vertex at the center of each of the four smaller tetrahedra that were removed. See also *Quarter cubic honeycomb *Truncated tetrahedron *Triakis tetrahedron In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the trun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triakis Truncated Tetrahedron
In geometry, the triakis truncated tetrahedron is a convex polyhedron made from 4 hexagons and 12 isosceles triangles. It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb. 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. As the Voronoi cell of a symmetric space pattern, it is a plesiohedron.. Construction For space-filling, the triakis truncated tetrahedron can be constructed as follows: # Truncate a regular tetrahedron such that the big faces are regular hexagons. # Add an extra vertex at the center of each of the four smaller tetrahedra that were removed. See also *Quarter cubic honeycomb *Truncated tetrahedron *Triakis tetrahedron In geometry, a triakis tetrahedron (or kistetrahedron) is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the trun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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