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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] |
Plesiohedron
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy. The plesiohedra include such well-known shapes as the cube, hexagonal prism, rhombic dodecahedron, and truncated octahedron. The largest number of faces that a plesiohedron can have is 38. Definition A set S of points in Euclidean space is a Delone set if there exists a number \varepsilon>0 such that every two points of S are at least at distance \varepsilon apart from each other and such that every point of space is within distance 1/\varepsilon of at least one point in S. So S fills space, but its points never come too close to each other. For this to be true, S must be infinite. Additionally, the set S is symmetric (in the sense needed ... [...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] |
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 truncated tetrahedron. The triakis tetrahedron can be seen as a tetrahedron with a triangular pyramid added to each face; that is, it is the Kleetope of the tetrahedron. It is very similar to the net for the 5-cell, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell a tetrahedron with pyramids attached to each face. This interpretation is expressed in the name. The length of the shorter edges is that of the longer edges. If the triakis tetrahedron has shorter edge length 1, it has area and volume . Cartesian coordinates Cartesian coordinates for the 8 vertices of a triakis tetrahedron centered at the origin, are the points (±5/3, ±5/3, ±5/3) with an even number of minus signs, along with the points (±1, ±1, ±1) with an odd numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length. A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron. A ''truncated tetrahedron'' is the Goldberg polyhedron containing triangular and hexagonal faces. A ''truncated tetrahedron'' can be called a cantic cube, with Coxeter diagram, , having half of the vertices of the cantellated cube (rhombicuboctahedron), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra. Area and volume The area ''A'' and the volume ''V'' of a truncated tetrahedron of edge length ''a'' are: :\begin A &= 7\sqrta^2 &&\appro ... [...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] |
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] |
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] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plesiohedron
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy. The plesiohedra include such well-known shapes as the cube, hexagonal prism, rhombic dodecahedron, and truncated octahedron. The largest number of faces that a plesiohedron can have is 38. Definition A set S of points in Euclidean space is a Delone set if there exists a number \varepsilon>0 such that every two points of S are at least at distance \varepsilon apart from each other and such that every point of space is within distance 1/\varepsilon of at least one point in S. So S fills space, but its points never come too close to each other. For this to be true, S must be infinite. Additionally, the set S is symmetric (in the sense needed ... [...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] |
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] |
Hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular polygon, regular hexagon'' has Schläfli symbol and can also be constructed as a Truncation (geometry), truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral polygon, equilateral and equiangular polygon, equiangular. It is bicentric polygon, bicentric, meaning that it is both cyclic polygon, cyclic (has a circumscribed circle) and tangential polygon, tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed figure, inscribed circle). All internal angles are 120 degree (angle), degrees. A regular hexago ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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