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Octadecahedron
In geometry, an octadecahedron (or octakaidecahedron) is a polyhedron with 18 face (geometry), faces. No octadecahedron is regular polyhedron, 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 Edge contraction, contracted. It is the shape of the closo-boranate ion [boron, B11hydrogen, 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 (geometry), pyramid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Edge-contracted Icosahedron
In geometry, an edge-contracted icosahedron is a polyhedron with octadecahedron, 18 triangular face (geometry), faces, 27 Edge (geometry), edges, and 11 Vertex (geometry), 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 (geometry), square-pentagonal antiprismatic core (an antiprismatic core with one square base and one pentagonal base); each base is then augmentation (geometry), augmented with a pyramid (geometry), pyramid. Related polytopes The diss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Closo
In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of cluster compound, clusters such as Boranes, 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, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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'' or ''star polygon, star''. 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 polygon also h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Near-miss Johnson Solid
In geometry, a near-miss Johnson solid is a strictly convex set, convex polyhedron whose face (geometry), faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces.. The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons. Some near-misses with high symmetry are also symmetrohedron, symmetrohedra with some truly regular polygon faces. Some near-misses are also zonohedron, zonohedra. Examples Coplanar misses Some failed Johnson solid candidates have coplanar faces. These polyhedra can be perturbed to become convex with faces that are arbitrarily close to regular polygons. These cases use 4.4.4.4 vertex figures of the square tiling, 3.3.3.3.3.3 vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Triangular Hebesphenorotunda
In geometry, the triangular hebesphenorotunda is a Johnson solid with 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon, meaning the total of its faces is 20. Properties The triangular hebesphenorotunda is named by , with the prefix ''hebespheno-'' referring to a blunt wedge-like complex formed by three adjacent ''lunes''—a figure where two equilateral triangles are attached at the opposite sides of a square. The suffix (triangular) ''-rotunda'' refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears a structural resemblance to the pentagonal rotunda. Therefore, the triangular hebesphenorotunda has 20 faces: 13 equilateral triangles, 3 squares, 3 regular pentagons, and 1 regular hexagon. The faces are all regular polygons, categorizing the triangular hebesphenorotunda as a Johnson solid, enumerated the last one J_ . It is an elementary polyhedron, meaning that it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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,Gyr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Elongated Triangular Gyrobicupola
In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in 60^\circ . It is an example of Johnson solid. Construction The elongated triangular gyrobicupola is similarly can be constructed as the elongated triangular orthobicupola, started from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces. This construction process is known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares. The difference between those two polyhedrons is one of two triangular cupolas in the elongated triangular gyrobicupola is rotated in 60^\circ . A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular gyrobicupola is one among them, enumerated as 36th Johnson solid J_ . Properties An elongated triangular gyrobicupola with a given ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Elongated Triangular Orthobicupola
In geometry, the elongated triangular orthobicupola is a polyhedron constructed by attaching two regular triangular cupola into the base of a regular hexagonal prism. It is an example of Johnson solid. Construction The elongated triangular orthobicupola can be constructed from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces. This construction process known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares. A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular orthobicupola is one among them, enumerated as 35th Johnson solid J_ . Properties An elongated triangular orthobicupola with a given edge length a has a surface area, by adding the area of all regular faces: \left(12 + 2\sqrt\right)a^2 \approx 15.464a^2. Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Sphenomegacorona
In geometry, the sphenomegacorona is a Johnson solid with 16 equilateral triangles and 2 squares as its faces. Properties The sphenomegacorona was named by in which he used the prefix ''spheno-'' referring to a wedge-like complex formed by two adjacent ''lunes''—a square with equilateral triangles attached on its opposite sides. The suffix ''-megacorona'' refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona. By joining both complexes, the resulting polyhedron has 16 equilateral triangles and 2 squares, making 18 faces. All of its faces are regular polygons, categorizing the sphenomegacorona as a Johnson solid—a convex polyhedron in which all of the faces are regular polygons—enumerated as the 88th Johnson solid J_ . It is an elementary polyhedron, meaning it cannot be separated by a plane into two small regular-faced polyhedra. The surface area of a sphenomegacorona A is the total of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Elongated Square Cupola
In geometry, the elongated square cupola is a polyhedron constructed from an octagonal prism by attaching square cupola onto its base. It is an example of Johnson solid. Construction The elongated square cupola is constructed from an octagonal prism by attaching a square cupola onto one of its bases, a process known as the elongation. This cupola covers the octagonal face so that the resulting polyhedron has four equilateral triangles, thirteen squares, and one regular octagon. It can also be constructed by removing a square cupola from a rhombicuboctahedron, which would also make it a diminished rhombicuboctahedron. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated square cupola is one of them, enumerated as the nineteenth Johnson solid J_ . Properties The surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Square Orthobicupola
In geometry, the square orthobicupola is a Johnson solid constructed by two square cupolas base-to-base. Construction The square orthobicupola is started by attaching two square cupolae onto their bases. The resulting polyhedron consisted of eight equilateral triangles and ten squares, having eighteen faces in total, as well as thirty-two edges and sixteen vertices. A convex polyhedron in which the faces are all regular polygons is a Johnson solid, and the square orthobicupola is one of them, enumerated as twenty-eighth Johnson solid J_ . This construction is similar to the next one, the square gyrobicupola, which is twisted one of the cupolae around 45°. Properties The square orthobicupola has surface area A of a total sum of its area's faces, eight equilateral triangles and two squares. Its volume V is twice that of the square cupola's volume. With the edge length a , they are: \begin A &= \left(2 \cdot \sqrt + 10\right)a^2 \approx 13.464a^2, \\ V &= \left(2+\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
