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Rhombohedron
In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a special case of a parallelepiped in which all six faces are congruent rhombi. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A rhombohedron has two opposite apices at which all face angles are equal; a prolate rhombohedron has this common angle acute, and an oblate rhombohedron has an obtuse angle at these vertices. A cube is a special case of a rhombohedron with all sides square. Special cases The common angle at the two apices is here given as \theta. There are two general forms of the rhombohedron: oblate (flattened) and prolate (stretched). In the oblate case \theta > 90^\circ and in the prolate case \theta < 90^\circ. For the figure is a cube. Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in both prolate and oblate forms. ...
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Rhombohedral
In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a special case of a parallelepiped in which all six faces are congruent rhombus, rhombi. It can be used to define the rhombohedral lattice system, a Honeycomb (geometry), honeycomb with rhombohedral cells. A rhombohedron has two opposite Apex_(geometry), apices at which all face angles are equal; a prolate rhombohedron has this common angle acute, and an oblate rhombohedron has an obtuse angle at these vertices. A cube is a special case of a rhombohedron with all sides square. Special cases The common angle at the two apices is here given as \theta. There are two general forms of the rhombohedron: oblate (flattened) and prolate (stretched). In the oblate case \theta > 90^\circ and in the prolate case \theta < 90^\circ. For the figure is a cube. Certain proportions of the rhombs give rise to some well-known special cases. These typically occur in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the Diamonds (suit), diamonds suit in playing cards which resembles the projection of an Octahedron#Orthogonal projections, octahedral diamond, or a lozenge (shape), lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after calisson, the French sweet—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple polygon, simple (non-self-intersecting), and is a special case of a parallelogram and a Kite (geometry), kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from , meaning something that spins, which derives from the verb , roman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prismatoid Polyhedra
In geometry, a prismatoid is a polyhedron whose vertex (geometry), vertices all lie in two parallel Plane (geometry), 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 = \frac. This formula follows immediately by integral, 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 families Families of prismatoids include: *Pyramid (geometry), Pyramids, in which one plane con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zonohedron
In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional Projection (mathematics), projection of a hypercube. Zonohedra were originally defined and studied by Evgraf Stepanovich Fyodorov, E. S. Fedorove, a Russian Crystallography, 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 (group), lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can Honeycomb (geometry), tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Dodecahedron
In geometry, the rhombic dodecahedron is a Polyhedron#Convex_polyhedra, convex polyhedron with 12 congruence (geometry), congruent rhombus, rhombic face (geometry), faces. It has 24 edge (geometry), edges, and 14 vertex (geometry), vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to Honeycomb (geometry), tesselate its copies in space creating a rhombic dodecahedral honeycomb. There are some variations of the rhombic dodecahedron, one of which is the Bilinski dodecahedron. There are some stellations of the rhombic dodecahedron, one of which is the Escher's solid. The rhombic dodecahedron may also appear in nature (such as in the garnet crystal), the architectural philosophies, practical usages, and toys. As a Catalan solid Metric properties The rhombic dodecahedron is a polyhedron with twelve rhombus, rhombi, each of which long face-diagonal length is exactly \sqrt times the sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombohedral Lattice System
In crystallography, the hexagonal crystal family is one of the six crystal family, crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent (see section hexagonal crystal family#Crystal systems, crystal systems below). In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice (such as α-quartz). The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral. __TOC__ Lattice systems The hexagonal crystal family consists of two lattice systems: hexagonal and rhom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The terms "rhomboid" and "parallelogram" are often erroneously conflated with each other (i.e, when most people refer to a "parallelogram" they almost always mean a rhomboid, a specific subtype of parallelogram); however, while all rhomboids are parallelograms, not all parallelograms are rhomboids. A parallelogram with sides of equal length ( equilateral) is called a ''rhombus'' but not a rhomboid. A parallelogram with right angled corners is a ''rectangle'' but not a rhomboid. A parallelogram is a rhomboid if it is neither a rhombus nor a rectangle. History Euclid introduced the term in his '' Elements'' in Book 1, Definition 22, Euclid never used the definition of rhomboid again and introduced the word parallelogram in Proposition 34 of Book 1; ''"In parallelogrammic areas the opposite sides and angles are equal to on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \frac = \frac = \varphi, where the Greek letter Phi (letter), phi ( or ) denotes the golden ratio. The constant satisfies the quadratic equation and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the Straightedge and compass construction, construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the '' Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthocentric Tetrahedron
In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890.. In an orthocentric tetrahedron the four altitudes are concurrent. This common point is called the tetrahedron orthocenter (a generalization of the orthocenter of a triangle). It has the property that it is the symmetric point of the center of the circumscribed sphere with respect to the centroid. Hence the orthocenter coincides with the Monge point of the tetrahedron. Characterizations All tetrahedra can be inscribed in a parallelepiped. A tetrahedron is orthocentric if and only if its circumscribed parallelepiped is a rhombohedron. Indeed, in any tetrahedron, a pair of opposite edges is perpendicular if and only if the corresponding faces of the circumscrib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
