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Truncated Trapezohedron
In geometry, an truncated trapezohedron is a polyhedron formed by a trapezohedron with Pyramid (geometry), pyramids Truncation (geometry), truncated from its two polar axis Vertex (geometry), vertices. The vertices exist as 4 in four parallel planes, with alternating orientation in the middle creating the pentagons. The regular dodecahedron is the most common polyhedron in this class, being a Platonic solid, with 12 Congruence (geometry), congruent pentagonal faces. A truncated trapezohedron has all vertices with 3 faces. This means that the dual polyhedra, the set of gyroelongated dipyramids, have all triangular faces. For example, the icosahedron is the dual of the dodecahedron. Forms *Triangular truncated trapezohedron (Dürer's solid) – 6 pentagons, 2 triangles, dual gyroelongated triangular dipyramid *Truncated square trapezohedron – 8 pentagons, 2 squares, dual gyroelongated square dipyramid *''Truncated pentagonal trapezohedron'' or regular dodecahedron – 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Dodecahedron
A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the regular dodecahedron. is a dodecahedron composed of regular polygon, regular pentagonal faces, three meeting at each Vertex (geometry), vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity. The regular dodecahedron is a truncated trapezohedron because it is the result of Truncation (geometry), truncating axial vertices of a pentagonal trapezohedron. It is also a Goldberg polyhedron because it is the initial polyhedron to construct new polyhedrons by the process of chamfering. It has a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Dipyramid
In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid by inserting an antiprism between its congruent halves. Forms Three members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid; the icosahedron, a Platonic solid; and the '' gyroelongated triangular bipyramid'' if it is made with equilateral triangles, but because it has coplanar faces is not strictly convex. With pairs of triangles merged into rhombi, it can be seen as a trigonal trapezohedron. The other members can be constructed with isosceles triangles. Use Pentagonal Gyroelongated Bipyramid * The Pentagonal gyroelongated bipyramid is commonly used in 3D modeling. It appears in various 3D programs such as Autodesk Maya, Blender (software) or Cinema 4D. In most programs, it's referred to as an ico-spere, or Icosahedron. It is also commonly used as a dice in Dungeons and Dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diminished Trapezohedron
In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (Diminishment (geometry), diminishment). It has one regular base face, triangle faces around the base, and kite (geometry), kites meeting on top. The kites can also be replaced by rhombi with specific proportions. Along with the set of pyramid (geometry), pyramids and elongated pyramids, these figures are topologically Self-dual polyhedron, self-dual. It can also be seen as an augmented antiprism, with a pyramid augmented onto one of the faces, and whose height is adjusted so the upper antiprism triangle faces can be made coparallel to the pyramid faces and merged into kite-shaped faces. They're also related to the gyroelongated pyramids, as augmented antiprisms and which are Johnson solids for . This sequence has sets of two triangles instead of kite faces. Examples Special cases ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Hexagonal Dipyramid
In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid by inserting an antiprism between its congruent halves. Forms Three members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid; the icosahedron, a Platonic solid; and the '' gyroelongated triangular bipyramid'' if it is made with equilateral triangles, but because it has coplanar faces is not strictly convex. With pairs of triangles merged into rhombi, it can be seen as a trigonal trapezohedron. The other members can be constructed with isosceles triangles. Use Pentagonal Gyroelongated Bipyramid * The Pentagonal gyroelongated bipyramid is commonly used in 3D modeling. It appears in various 3D programs such as Autodesk Maya, Blender (software) or Cinema 4D. In most programs, it's referred to as an ico-spere, or Icosahedron. It is also commonly used as a dice in Dungeons and Dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Hexagonal Trapezohedron
In geometry, the truncated hexagonal trapezohedron is the fourth in an infinite series of truncated trapezohedra. It has 12 pentagon and 2 hexagon faces. It can be constructed by taking a hexagonal trapezohedron and truncating the polar axis vertices. Weaire–Phelan structure The Weaire–Phelan structure contains another form of this polyhedron that has ''D''''2d'' symmetry and is a part of a space-filling honeycomb along with an irregular dodecahedron. See also * Goldberg polyhedron In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (mathematician), Michael Goldberg (1902–1990 ... External links Wearie-Phelan BubblesTry: "t6dA6". Polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Square Dipyramid
In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid. The dual polyhedron of the gyroelongated square bipyramid is a square truncated trapezohedron with eight pentagons and two squares as its faces. The gyroelongated square pyramid appears in chemistry as the basis for the bicapped square antiprismatic molecular geometry, and in mathematical optimization as a solution to the Thomson problem. Construction Like other gyroelongated bipyramids, the gyroelongated square bipyramid can be constructed by attaching two equilateral square pyramids onto the square faces of a square antiprism; this process is k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Square Trapezohedron
In geometry, the square truncated trapezohedron is the second in an infinite series of truncated trapezohedra. It has 8 pentagon and 2 square faces. This polyhedron can be constructed by taking a tetragonal trapezohedron and truncating the polar axis vertices. The kite faces of the trapezohedron become pentagons. The vertices exist as 4 squares in four parallel planes, with alternating orientation in the middle creating the pentagons. A ''truncated trapezohedron'' has all valence-3 vertices. This means that the dual polyhedron, the gyroelongated square dipyramid, has all triangular faces. It represents the dual polyhedron to the Johnson solid, gyroelongated square dipyramid In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexak ... (), with specific proportions: Polyhedra {{Po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Triangular Dipyramid
In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid by inserting an antiprism between its congruent halves. Forms Three members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid; the icosahedron, a Platonic solid; and the '' gyroelongated triangular bipyramid'' if it is made with equilateral triangles, but because it has coplanar faces is not strictly convex. With pairs of triangles merged into rhombi, it can be seen as a trigonal trapezohedron. The other members can be constructed with isosceles triangles. Use Pentagonal Gyroelongated Bipyramid * The Pentagonal gyroelongated bipyramid is commonly used in 3D modeling. It appears in various 3D programs such as Autodesk Maya, Blender (software) or Cinema 4D. In most programs, it's referred to as an ico-spere, or Icosahedron. It is also commonly used as a dice in Dungeons and Dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dürer's Solid
In geometry, the truncated triangular trapezohedron is the first in an infinite series of truncated trapezohedra. It has 6 pentagon and 2 triangle faces. Geometry This polyhedron can be constructed by truncating two opposite vertices of a cube, of a trigonal trapezohedron (a convex polyhedron with six congruent rhombus sides, formed by stretching or shrinking a cube along one of its long diagonals), or of a rhombohedron or parallelepiped (less symmetric polyhedra that still have the same combinatorial structure as a cube). In the case of a cube, or of a trigonal trapezohedron where the two truncated vertices are the ones on the stretching axes, the resulting shape has three-fold rotational symmetry. Dürer's solid This polyhedron is sometimes called Dürer's solid, from its appearance in Albrecht Dürer's 1514 engraving ''Melencolia I''. The graph formed by its edges and vertices is called the Dürer graph. The shape of the solid depicted by Dürer is a subject of some ac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Truncated Trapezohedron
In geometry, the truncated hexagonal trapezohedron is the fourth in an infinite series of truncated trapezohedra. It has 12 pentagon and 2 hexagon faces. It can be constructed by taking a hexagonal trapezohedron and Truncation (geometry), truncating the polar axis vertices. Weaire–Phelan structure The Weaire–Phelan structure contains another form of this polyhedron that has ''D''''2d'' symmetry and is a part of a honeycomb (geometry), space-filling honeycomb along with an irregular dodecahedron. See also * Goldberg polyhedron External links Wearie-Phelan Bubbles Try: "t6dA6". Polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Truncated Trapezohedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There are numerous other dodecahedra. While the regular dodecahedron shares ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Truncated Trapezohedron
In geometry, the square truncated trapezohedron is the second in an infinite series of truncated trapezohedra. It has 8 pentagon and 2 square faces. This polyhedron can be constructed by taking a tetragonal trapezohedron and truncating the polar axis vertices. The kite faces of the trapezohedron become pentagons. The vertices exist as 4 squares in four parallel planes, with alternating orientation in the middle creating the pentagons. A ''truncated trapezohedron'' has all valence-3 vertices. This means that the dual polyhedron, the gyroelongated square dipyramid, has all triangular faces. It represents the dual polyhedron to the Johnson solid, gyroelongated square dipyramid In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexak ... (), with specific proportions: Polyhedra {{Po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
