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Octadecahedron
In geometry, an octadecahedron (or octakaidecahedron) is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron. In chemistry, "''the'' octadecahedron" commonly refers to a specific structure with C2v symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [ B11 H11]2−. Convex There are 107,854,282,197,058 topologically distinct ''convex'' octadecahedra, excluding mirror images, having at least 11 vertices. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) Examples The most familiar octadecahedra are the heptadecagonal pyramid, hexadecagonal prism, and the octagonal antiprism. The hexadecagonal prism and the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge-contracted Icosahedron
In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices. Construction It can be constructed from the regular icosahedron, with one edge contraction, removing one vertex, 3 edges, and 2 faces. This contraction distorts the circumscribed sphere original vertices. With all equilateral triangle faces, it has 2 sets of 3 coplanar equilateral triangles (each forming a half-hexagon), and thus is not a Johnson solid. If the sets of three coplanar triangles are considered a single face (called a triamond), it has 10 vertices, 22 edges, and 14 faces, 12 triangles and 2 triamonds . It may also be described as having a hybrid square-pentagonal antiprismatic core (an antiprismatic core with one square base and one pentagonal base); each base is then augmented with a pyramid. Related polytopes The dissected regular icosahedron is a variant topologically equivalent to the sphenocorona with the two sets of 3 coplanar faces as tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closo
In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by Kenneth Wade, and were further developed by others including Michael Mingos; they are sometimes known as Wade's rules or the Wade–Mingos rules. The rules are based on a molecular orbital treatment of the bonding. These notes contained original material that served as the basis of the sections on the 4''n'', 5''n'', and 6''n'' rules. These rules have been extended and unified in the form of the Jemmis ''mno'' rules. Predicting structures of cluster compounds Different rules (4''n'', 5''n'', or 6''n'') are invoked depending on the number of electrons per vertex. The 4''n'' rules are reasonably accurate in predicting the structures of clusters having about 4 electrons per vertex, as is the case for many boranes and carboranes. For suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex polygon, convex, star polygon, star or Skew polygon, skew. In the limit (mathematics), limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a Line (geometry), straight line), if the edge length is fixed. General properties ''These properties apply to all regular polygons, whether convex or star polygon, star.'' A regular ''n''-sided polygon has rotational symmetry of order ''n''. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octagonal Antiprism
In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. In the case of a regular 8-sided base, one usually considers the case where its copy is twisted by an angle 180°/''n''. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two ''n''-gonal bases and, connecting those bases, 2''n'' isosceles triangles. If faces are all regular, it is a semiregular polyhedron. See also External links * Octagonal Antiprism -- Interactive Polyhedron ModelmodelpolyhedronismeA8 {{Polyhedron-stub Prismatoid polyhedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Hebesphenorotunda
In geometry, the triangular hebesphenorotunda is one of the Johnson solids (). . It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid. If these faces are aligned with a congruent patch of faces on the icosidodecahedron, then the hexagonal face will lie in the plane midway between two opposing triangular faces of the icosidodecahedron. The triangular hebesphenorotunda also has clusters of faces that can be aligned with corresponding faces of the rhombicosidodecahedron: the three ''lunes'', each ''lune'' consisting of a square and two antipodal triangles adjacent to the square. The faces around each vertex can also be aligned with the corresponding faces of various diminished icosahedra. Johnson uses the prefix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Triangular Bicupola
In geometry, the gyroelongated triangular bicupola is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a triangular bicupola (either triangular orthobicupola, , or the cuboctahedron) by inserting a hexagonal antiprism between its congruent halves. The gyroelongated triangular bicupola is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each square face on the bottom half of the figure is connected by a path of two triangular faces to a square face above it and to the right. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom square would be connected to a square face above it and to the left. The two chiral forms of are not considered different Johnson solids. Formulae The following formulae for volume and surface area can be used if all faces are regular, with edge length ''a'':Stephen Wolfram,Gy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Triangular Gyrobicupola
In geometry, the elongated triangular gyrobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a "triangular gyrobicupola," or cuboctahedron, by inserting a hexagonal prism between its two halves, which are congruent triangular cupolae (). Rotating one of the cupolae through 60 degrees before the elongation yields the triangular orthobicupola (). Formulae The following formulae for volume and surface area can be used if all faces are regular, with edge length ''a'': :V=\left(\frac+\frac\right)a^3\approx4.9551...a^3 :A=2\left(6+\sqrt\right)a^2\approx15.4641...a^2 Related polyhedra and honeycombs The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Triangular Orthobicupola
In geometry, the elongated triangular orthobicupola or cantellated triangular prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a triangular orthobicupola () by inserting a hexagonal prism between its two halves. The resulting solid is superficially similar to the rhombicuboctahedron (one of the Archimedean solids), with the difference that it has threefold rotational symmetry about its axis instead of fourfold symmetry. Volume The volume of ''J''35 can be calculated as follows: ''J''35 consists of 2 cupolae and hexagonal prism. The two cupolae makes 1 cuboctahedron = 8 tetrahedra + 6 half-octahedra. 1 octahedron = 4 tetrahedra, so total we have 20 tetrahedra. What is the volume of a tetrahedron? Construct a tetrahedron having vertices in common with alternate vertices of a cube (of side \textstyle\frac, if tetrahedron has unit edges). The 4 triangular pyramids left if the tetrahedron is removed from the cube form half an oc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphenomegacorona
In geometry, the sphenomegacorona is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. . Johnson uses the prefix ''spheno-'' to refer to a wedge-like complex formed by two adjacent ''lunes'', a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix ''-megacorona'' refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona. Joining both complexes together results in the sphenomegacorona. Cartesian coordinates Let ''k'' ≈ 0.59463 be the smallest positive root of the polynomial :\begin &1680 x^- 4800 x^ - 3712 x^ + 17216 x^+ 1568 x^ - 24576 x^ + 2464 x^ + 17248 x^9 \\ &\quad -3384 x^8 - 5584 x^7 + 2000 x^6+ 240 x^5- 776 x^4+ 304 x^3 + 200 x^2 - 56 x -23. \end Then, Cartesian coordinates of a sphenomegacorona with edge length 2 are given by the union of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Square Cupola
In geometry, the elongated square cupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square cupola () by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" (another square cupola) removed. Formulae The following formulae for volume, surface area and circumscribed sphere, circumradius can be used if all face (geometry), faces are regular polygon, regular, with edge length ''a'': :V=\left(3+\frac\right)a^3\approx6.77124...a^3 :A=\left(15+2\sqrt+\sqrt\right)a^2\approx19.5605...a^2 :C=\left(\frac\sqrt\right)a\approx1.39897...a Dual polyhedron The dual of the elongated square cupola has 20 faces: 8 isosceles triangles, 4 kites, 8 quadrilaterals. Related polyhedra and honeycombs The elongated square cupola forms space-filling Honeycomb (geometry), honeycombs with Tetrahedron, tetrahedra and cubes; with cubes and Cuboctahedron, cuboctahedra; and with tetrahedra, elongated squ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Orthobicupola
In geometry, the square orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by joining two square cupolae () along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola (). The ''square orthobicupola'' is the second in an infinite set of orthobicupolae. The square orthobicupola can be elongated by the insertion of an octagonal prism between its two cupolae to yield a rhombicuboctahedron, or collapsed by the removal of an irregular hexagonal prism to yield an elongated square dipyramid (), which itself is merely an elongated octahedron. It can be constructed from the disphenocingulum () by replacing the band of up-and-down triangles by a band of rectangles, while fixing two opposite sphenos. Related polyhedra and honeycombs The square orthobicupola forms space-filling honeycombs with tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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