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Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—''parallelepiped'' and ''cube'' in three dimensions, ''parallelogram'' and ''square'' in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only ''parallelograms'' and ''parallelepipeds'' exist. Three equivalent definitions of ''parallelepiped'' are *a polyhedron with six faces (hexahedron), each of which is a parallelogram, *a hexahedron with three pairs of parallel faces, and *a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. "Parallelepiped" is now usually pronounced or ; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelepiped 2013-11-29
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—''parallelepiped'' and ''cube'' in three dimensions, ''parallelogram'' and ''square'' in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only ''parallelograms'' and ''parallelepipeds'' exist. Three equivalent definitions of ''parallelepiped'' are *a polyhedron with six faces (hexahedron), each of which is a parallelogram, *a hexahedron with three pairs of parallel faces, and *a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. "Parallelepiped" is now usually pronounced or ; trad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prismatoid
In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid. Volume If the areas of the two parallel faces are and , the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is , and the height (the distance between the two parallel faces) is , then the volume of the prismatoid is given by V = \fracB. E. Meserve, R. E. Pingry: ''Some Notes on the Prismoidal Formula''. The Mathematics Teacher, Vol. 45, No. 4 (April 1952), pp. 257-263 (This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic function in the height.) Prismatoid famil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zonohedron
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] |
Hexahedron
A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. There are seven topologically distinct ''convex'' hexahedra, one of which exists in two mirror image forms. There are three topologically distinct concave hexahedra. Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces. Convex, Cuboid Convex, Others Concave There are three further topologically distinct hexahedra that can only be realised as ''concave'' figures: A digonal antiprism can be considered a degenerate form of hexahedron, having two opposing digonal faces and four triangular faces. However, digons are usually disregarded in the defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-dimensional Figure
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), including pyramids, prisms and other polyhedrons; cylinders; cones; truncated cones; and balls bounded by spheres. History The Pythagoreans dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its radius.Paraphrased and taken in part from the '' 1911 Encyclopædia Britannica''. Topics Basic topics in solid geometry and stereometry include: * incidence of planes and lines * dihedral angle and solid ang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuboid
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cube. In mathematical language a cuboid is a convex polyhedron, whose polyhedral graph is the same as that of a cube. Special cases are a cube, with 6 squares as faces, a rectangular prism, rectangular cuboid or rectangular box, with 6 rectangles as faces, for both, cube and rectangular prism, adjacent faces meet in a right angle. General cuboids By Euler's formula the numbers of faces ''F'', of vertices ''V'', and of edges ''E'' of any convex polyhedron are related by the formula ''F'' + ''V'' = ''E'' + 2. In the case of a cuboid this gives 6 + 8 = 12 + 2; that is, like a cube, a cuboid has 6 faces, 8 vertices, and 12 edges. Along with the rectangular cuboids, any parallelepiped ... [...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] |
Honeycomb (geometry)
In geometry, a honeycomb is a ''space filling'' or ''close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dimensions. Its dimension can be clarified as ''n''-honeycomb for a honeycomb of ''n''-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. Classification There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or ''slabs'' of prisms based on some tessellations of the plane. In particula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chirality (mathematics)
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be ''achiral''. A chiral object and its mirror image are said to be enantiomorphs. The word ''chirality'' is derived from the Greek (cheir), the hand, the most familiar chiral object; the word ''enantiomorph'' stems from the Greek (enantios) 'opposite' + (morphe) 'form'. Examples Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out. The J, L, S and Z-shaped ''tetrominoes'' of the popul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have the ... [...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] |
Point Symmetry
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center. Terminology The term ''reflection'' is loose, and considered by some an abuse of language, with ''inversion'' preferred; however, ''point reflection'' is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
