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Elongated Bipyramid
In geometry, the elongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid (by inserting an prism between its congruent halves). There are three ''elongated bipyramids'' that are Johnson solids: * Elongated triangular bipyramid (), * Elongated square bipyramid (), and * Elongated pentagonal bipyramid (). Higher forms can be constructed with isosceles triangles. Forms See also * Gyroelongated bipyramid * Gyroelongated pyramid * Elongated pyramid * 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). It has one regular base face, triangle ... References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Pentagonal Bipyramid
In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism between its congruent halves. Dual polyhedron The dual of the elongated square bipyramid is a pentagonal bifrustum The pentagonal bifrustum or truncated pentagonal bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoid and 2 pentagonal faces. Constructions The pentagonal bifrustum is the dual polyhedron of a Johnson solid, .... See also * Elongated pentagonal pyramid External links * Johnson solids Pyramids and bipyramids {{Polyhedron-stub ... [...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). It has one regular base face, triangle faces around the base, and kites meeting on top. The kites can also be replaced by rhombi with specific proportions. Along with the set of pyramids and elongated pyramids, these figures are topologically 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 There are three special case geometries of the ''diminished trigonal trapezohedron' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Pyramid
In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an pyramid to an prism. Along with the set of pyramids, these figures are topologically self-dual. There are three ''elongated pyramids'' that are Johnson solids: * Elongated triangular pyramid (), *Elongated square pyramid (), and * Elongated pentagonal pyramid (). Higher forms can be constructed with isosceles triangles. Forms See also * Gyroelongated bipyramid * Elongated bipyramid * Gyroelongated pyramid * 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). It has one regular base face, triangle ... References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Pyramid
In geometry, the gyroelongated pyramids (also called ''augmented antiprisms'') are an infinite set of polyhedra, constructed by adjoining an pyramid to an antiprism. There are two ''gyroelongated pyramids'' that are Johnson solids made from regular triangles and square, and pentagons. A triangular and hexagonal form can be constructed with coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ... faces. Others can be constructed allowing for isosceles triangles. Forms See also * Gyroelongated bipyramid * Elongated bipyramid * Elongated pyramid * Diminished trapezohedron References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Bipyramid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ( ); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid () is an example that has a degree-5 vertex. Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of John ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Bifrustum
The hexagonal bifrustum or truncated hexagonal bipyramid is the fourth in an infinite series of bifrustum polyhedra. It has 12 trapezoid and 2 hexagonal faces. This polyhedron can be constructed by taking a hexagonal dipyramid and truncating the polar axis vertices, making it into two end-to-end frustums. Several types of crystal take this shape. It has also been used in the design of 14-sided dice, which may be used to generate randomly chosen playing card A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a f ...s. Patent US 8074986 B1, Douglas A. Gebhart, filed September 30, 2008. It also has app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Bifrustum
The pentagonal bifrustum or truncated pentagonal bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoid and 2 pentagonal faces. Constructions The pentagonal bifrustum is the dual polyhedron of a Johnson solid, the elongated pentagonal bipyramid. This polyhedron can be constructed by taking a pentagonal bipyramid and truncating the polar axis vertices. In Conway's notation for polyhedra, it can be represented as the polyhedron "t5dP5", meaning the truncation of the degree-five vertices of the dual of a pentagonal prism. Alternatively, it can be constructed by gluing together two end-to-end pentagonal frustums, or (if coplanar faces are allowed) by gluing together two pentagonal prisms on their pentagonal faces. Application In the formation of quasicrystals, a 15-site truncated pentagonal bipyramid structure may form the nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Bifrustum
The square bifrustum or ''square truncated bipyramid'' is the second in an infinite series of bifrustum polyhedra. It has 4 trapezoidal and 2 square faces. This polyhedron can be constructed by taking a square bipyramid (octahedron) and truncating the polar axis vertices, making it into two end-to-end frustums. It is dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ... to the elongated square dipyramid. Polyhedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Bifrustum
In geometry, the triangular bifrustum is the second in an infinite series of bifrustum polyhedra. It has 6 trapezoid and 2 triangle faces. It may also be called the truncated triangular bipyramid; however, that term is ambiguous, as it may also refer to polyhedra formed by truncating all five vertices of a triangular bipyramid. This polyhedron can be constructed by taking a triangular bipyramid and truncating the polar axis vertices, making it into two end-to-end frustums. It appears as the form of certain nanocrystals.. A truncated triangular bipyramid can be constructed by connecting two stacked regular octahedra with 3 pairs of tetrahedra around the sides. This represents a portion of the gyrated alternated cubic honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names incl .... : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Hexagonal Dipyramid
In geometry, the elongated hexagonal bipyramid is constructed by elongating a hexagonal bipyramid (by inserting a hexagonal prism between its congruent halves). Related polyhedra This polyhedron is in the family of elongated bipyramids, of which the first three can be Johnson solids: J14, J15 and J16. The hexagonal form can be constructed by all regular faces, but is not a ''Johnson solid'' because 6 equilateral triangles would form six co-planar faces (in a regular hexagon). Uses * A quartz crystal is an example of an elongated hexagonal bipyramid. Because it has 18 faces, it can be called an octadecahedron. Other chemicals also have this shape. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Pentagonal Dipyramid
In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism between its congruent halves. Dual polyhedron The dual of the elongated square bipyramid is a pentagonal bifrustum. See also * Elongated pentagonal pyramid In geometry, the elongated pentagonal pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal pyramid () by attaching a pentagonal prism to its base. Formulae The following formulae ... External links * Johnson solids Pyramids and bipyramids {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
