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Zonohedra
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope. Zonohedra that tile space The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron is combinatorially equivalent to one of five types: the rhombohedron (including the cube), hexagonal prism, truncated octahedron, rhombic dodec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long face-diagonal length is exactly times the short face-diagonal length; thus, the acute angles on each face measure arccos(), or approximately 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on its set of faces. In elementary terms, this means that for any two faces A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Square (geometry), squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron. The truncated octahedron was called the "mecon" by Buckminster Fuller. Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths and . Construction A truncated octahedron is constructed from a regular octahedron with side length 3''a'' by the removal of six right square pyramids, one from each point. These pyramids have both base side len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombohedron
In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled. A parallelogram with sides of equal length (equilateral) is a rhombus but not a rhomboi ...) is a three-dimensional figure with six faces which are rhombus, rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a Honeycomb (geometry), honeycomb with rhombohedral cells. A cube is a special case of a rhombohedron with all sides square. In general a ''rhombohedron'' can have up to three types of rhombic faces in congruent opposite pairs, ''C''''i'' symmetry, Order (group theory), order 2. Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, and all orthocentric tetrahedra c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spherical tiling, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Cuboctahedron
In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism. Names There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron. Cartesian coordinates The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of: :(±1, ±(1 + ), ±(1 + 2)). Area and volume The area ''A'' and the volume ''V'' of the truncated cuboctahedron of edge length ''a'' are: :\begin A &= 12\left(2+\sqrt+\sqrt\right) a^2 &&\approx 61.755\,1724~a^2, \\ V &= \left(22+14\sqrt\right) a^3 &&\approx 41. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecagonal Prism
In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps. If faces are all regular, it is a uniform polyhedron. Use It is used in the construction of two prismatic uniform honeycomb In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of face ...s: The new British one pound (£1) coin, which entered circulation in March 2017, is shaped like a dodecagonal prism. Related polyhedra References External links * Prismatoid polyhedra Zonohedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decagonal Prism
In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra. The decagonal prism has 12 faces, 30 edges, and 20 vertices, so, it is a dodecahedron (while the term is usually applied to regular dodecahedron or rhombic dodecahedron.). If faces are all regular, it is a semiregular or prismatic uniform polyhedron. Uses The decagonal prism exists as cells in two four-dimensional uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There ...s: Related polyhedra External links * 3-d model of a Decagonal Prism Prismatoid polyhedra Zonohedra {{Polyhedron-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octagonal Prism
In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by rectangular sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron. Symmetry Images The octagonal prism can also be seen as a tiling on a sphere: : Use In optics, octagonal prisms are used to generate flicker-free images in movie projectors. In uniform honeycombs and 4-polytopes It is an element of three uniform honeycombs: It is also an element of two four-dimensional uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There ...s: Related polyhedra External links * Interactive model of an Octagonal Prism {{Polyhedron-stub Zonohedra Prismatoid polyhedra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombo-hexagonal Dodecahedron
In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongated by a square prism. Parallelohedron Along with the rhombic dodecahedron, it is a space-filling polyhedron, one of the five types of parallelohedron identified by Evgraf Fedorov that tile space face-to-face by translations. It has 5 sets of parallel edges, called zones or belts. : Tessellation * It can tesselate all space by translations. * It is the Wigner–Seitz cell for certain body-centered tetragonal lattices. This is related to the rhombic dodecahedral honeycomb with an elongation of zero. Projected normal to the elongation direction, the honeycomb looks like a square tiling with the rhombi projected into squares. Variat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.. Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used to refer to the ''regular octahedron'', which has eight triangular faces. Because of the ambiguity of the term ''octahedron'' and tilarity of the various eight-sided figures, the term is rarely used without clarification. Before sharpening, many pencils take the shape of a long hexagonal prism. As a semiregular (or uniform) polyhedron If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon Base (geometry), base, a second base which is a Translation (geometry), translated copy (rigidly moved without rotation) of the first, and other Face (geometry), faces, necessarily all parallelograms, joining corresponding sides of the two bases. All Cross section (geometry), cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. Like many basic geometric terms, the word ''prism'' () was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers. Oblique prism An oblique prism is a pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Principal properties Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: *The area is A=\frac a^2, *The perimeter is p=3a\,\! *The radius of the circumscribed circle is R = \frac *The radius of the inscribed circle is r=\frac a or r=\frac *The geometric center of the triangle is the center of the circumscribed and inscribed circles *The altitude (height) from any side is h=\frac a Denoting the radius of the circumscribed circle as ''R'', we can determine using trigonometry that: *The area of the triangle is \mathrm=\fracR^2 Many of these quantities have simple r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
