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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 Applications Elongated bipyramids are sometimes used as dice, especially to make dice with atypical side count, such as 5 or 7. Such a die has numbers written on the square faces, which are usually heightened into rectangles for convenience in rolling. Whichever number comes face-up when the die is rolled is the side that is to be read. 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 pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Hexagonal Bipyramid
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. * The ''edge-first orthogonal projection'' of a 24-cell is an elongated hexagonal bipyramid. * Used as the shape of Fruit Gushers candy. * Used as a physical manifestation for assisting various branches of three-dimensional graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathema ... [...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] |
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 References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics The ''Canadian Journal of Mathematics'' () is a bimonthly mathematics journal published by the Canadian Mathematical Society. It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief of the journal ar ..., 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the co ... [...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 faces. Others can be constructed allowing for isosceles triangles. Forms See also * Gyroelongated bipyramid * Elongated bipyramid * 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 (geometry), diminishment). It has one regul ... 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Bipyramid
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] |
Dice
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing games, and games of chance. A traditional die is a cube with each of its six faces marked with a different number of dots ( pips) from one to six. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. Dice may also have other polyhedral or irregular shapes, may have faces marked with numerals or symbols instead of pips and may have their numbers carved out from the material of the dice instead of marked on it. Loaded dice are specifically designed or modified to favor some results over others, for cheating or entertainment purposes. History Dice have been used since before recorded history, and their origin is uncertain. It is hypoth ... [...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. ...
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Pentagonal Bifrustum
In geometry, the pentagonal bifrustum or truncated pentagonal bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoidal 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 polyhedron notation, it can be represented as the polyhedron "", 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 nanoparticles, a 15-site truncated pentagonal bipyramid structure may form the nucleus of larger twinned structures with five-fold or icosahedral In geometry, an icosahedron ( or ) is a polyhedron with 20 face ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Bifrustum
In geometry, an -agonal bifrustum is a polyhedron composed of three parallel planes of polygon, -agons, with the middle plane largest and usually the top and bottom congruent. It can be constructed as two congruent frustum, frusta combined across a plane of symmetry, and also as a bipyramid with the two polar vertices truncated. They are dual polyhedron, duals to the family of elongated bipyramids. Formulae For a regular -gonal bifrustum with the equatorial polygon sides , bases sides and semi-height (half the distance between the planes of bases) , the lateral surface area , total area and volume are: and \begin A_l &= n (a+b) \sqrt \\[4pt] A &= A_l + n \frac \\[4pt] V &= n \frach \end Note that the volume V is twice the volume of a frustum, frusta. Forms Three bifrusta are dual polyhedron, duals to three Johnson solids, . In general, a -agonal bifrustum has trapezoids, 2 -agons, and is dual to the elongated dipyramids. References [...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 Truncation (geometry), 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. : References {{reflist External linksConway Notation for Polyhedra Try: t3dP3 Polyhedra ... [...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. * The ''edge-first orthogonal projection'' of a 24-cell is an elongated hexagonal bipyramid. * Used as the shape of Fruit Gushers candy. * Used as a physical manifestation for assisting various branches of three-dimensional graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
