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Space-filling Polyhedron
In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where ''filling'' means that, taken together, all the instances of the polyhedron constitute a partition of three-space. Any periodic tiling or honeycomb of three-space can in fact be generated by translating a primitive cell polyhedron. Any parallelepiped tessellates Eucledian 3-space, and more specifically any of five parallelohedra such as the rhombic dodecahedron, which is one of nine edge-transitive and face-transitive solids. Examples of other space-filling polyhedra include the set of five convex polyhedra with regular faces, which include the triangular prism, hexagonal prism, gyrobifastigium, cube, and 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 octahed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelohedron
In geometry, a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron. Classification Every parallelohedron is a zonohedron, constructed as the Minkowski sum of between three and six line segments. Each of these line segments can have any positive real number as its length, and each edge of a parallelohedron is parallel to one of these generating segments, with the same length. If the length of a segments of a parallelohedron generated from four or more segments is reduced to zero, the result is that the polyhedron degenerates to a simpler form, a parallelohedron formed from one fewer segment. As a zonohedron, these shapes ... [...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] |
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] |
Gyrobifastigium
In geometry, the gyrobifastigium is the 26th Johnson solid (). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space. It is also the vertex figure of the nonuniform duoantiprism (if and are greater than 2). Despite the fact that would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices, except for the case which represents a uniform great duoantiprism. Its dual, the elongated tetragonal disphenoid, can be found as cells of the duals of the duoantiprisms. History and name The name of the gyrobifastigium comes from the Latin ''fastigium'', meaning a sloping roof. In the standard naming convention of the Johnson solids, ''bi-'' means two solids connected at their bases, and ''gyro-'' means the two halves are twisted with respect to each other. The gyro ... [...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] |
Triangular Prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides. Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle. As a semiregular (or uniform) polyhedron A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a triangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge-transitive
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged. Isotoxal polygons An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. Isotoxal 4n-gons are centrally symmetric, so are also zonogons. In general, an isotoxal 2n-gon has \mathrm_n, (^*nn) dihedral symmetry. For example, a rhombus is an isotoxal "2×2-gon" (quadrilateral) with \mathrm_2, (^*22) symmetry. All regular polygons (equilateral triangle, square, etc.) are isotoxal, having double the minimum symmetry order: a regular n-gon has \mathrm_n, (^*nn) dihedral symmetry. An ... [...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] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
