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Rhombic Enneacontahedron
In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 Rhombus, rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron. Construction It can also be seen as a nonuniform truncated icosahedron with pyramids augmented to the pentagonal and hexagonal faces with heights adjusted until the dihedral angles are zero, and the two pyramid type side edges are equal length. This construction is expressed in the Conway polyhedron notation ''jtI'' with join operator ''j''. Without the equal edge constraint, the wide rhombi are kite (geometry), kites if limited only by the icosahedral symmetry. The sixty broad rhombic faces in the rhombic enneacontahedron are identical to those in the rhombic dodecahedron, with diagonals in a ratio of 1 to the square root of 2. The face angles of these rho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Enneacontahedron
In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 Rhombus, rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron. Construction It can also be seen as a nonuniform truncated icosahedron with pyramids augmented to the pentagonal and hexagonal faces with heights adjusted until the dihedral angles are zero, and the two pyramid type side edges are equal length. This construction is expressed in the Conway polyhedron notation ''jtI'' with join operator ''j''. Without the equal edge constraint, the wide rhombi are kite (geometry), kites if limited only by the icosahedral symmetry. The sixty broad rhombic faces in the rhombic enneacontahedron are identical to those in the rhombic dodecahedron, with diagonals in a ratio of 1 to the square root of 2. The face angles of these rho ... [...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] |
Joined Truncated Icosahedron
Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topological spaces ** Join (sigma algebra), a refinement of sigma algebras ** Join (algebraic geometry), a union of lines between two varieties *In computing: ** Join (relational algebra), a binary operation on tuples corresponding to the relation join of SQL *** Join (SQL), relational join, a binary operation on SQL and relational database tables *** join (Unix), a Unix command similar to relational join ** Join-calculus, a process calculus developed at INRIA for the design of distributed programming languages *** Join-pattern, generalization of Join-calculus *** Joins (concurrency library), a concurrent computing API from Microsoft Research * Join Network Studio of NENU, a non-profit organization of Northeast Normal University * Joins.com, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
VRML
VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphics, designed particularly with the World Wide Web in mind. It has been superseded by X3D. WRL file format VRML is a text file format where, e.g., vertices and edges for a 3D polygon can be specified along with the surface color, UV-mapped textures, shininess, transparency, and so on. URLs can be associated with graphical components so that a web browser might fetch a webpage or a new VRML file from the Internet when the user clicks on the specific graphical component. Animations, sounds, lighting, and other aspects of the virtual world can interact with the user or may be triggered by external events such as timers. A special Script Node allows the addition of program code (e.g., written in Java or ECMAScript) to a VRML file. VRML ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombic Dodecahedral Honeycomb
The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal spheres in ordinary space (see Kepler conjecture). Geometry It consists of copies of a single cell, the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1:. Three cells meet at each edge. The honeycomb is thus cell-transitive, face-transitive, and edge-transitive; but it is not vertex-transitive, as it has two kinds of vertex. The vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells. The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, which is the cell of a somewhat similar tessellation, the Voronoi diagram of hexagonal close-packing. Colorings Cells can be given 4 colors in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler Conjecture
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%. In 1998, Thomas Hales, following an approach suggested by , announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inscribed Sphere
In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual to the dual polyhedron's circumsphere. The radius of the sphere inscribed in a polyhedron ''P'' is called the inradius of ''P''. Interpretations All regular polyhedra have inscribed spheres, but most irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes. For such cases, the notion of an insphere does not seem to have been properly defined and various interpretations of an ''insphere'' are to be found: * The sphere tangent to all faces (if one exists). * The sphere tangent to all face planes (if one exists). * The sphere tangent to a given set of faces (if one exists). * The largest sphere that can fit inside the polyhedron. Often ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bravais Lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n_3 \mathbf_3, where the ''ni'' are any integers, and a''i'' are ''primitive translation vectors'', or ''primitive vectors'', which lie in different directions (not necessarily mutually perpendicular) and span the lattice. The choice of primitive vectors for a given Bravais lattice is not unique. A fundamental aspect of any Bravais lattice is that, for any choice of direction, the lattice appears exactly the same from each of the discrete lattice points when looking in that chosen direction. The Bravais lattice concept is used to formally define a ''crystalline arrangement'' and its (finite) frontiers. A crystal is made up of one or more atoms, called the ''basis'' or ''motif'', at each lattice point. The ''basis'' may consist of atoms, mol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Packing Density
A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. In packing problems, the objective is usually to obtain a packing of the greatest possible density. In compact spaces If are measurable subsets of a compact measure space and their interiors pairwise do not intersect, then the collection is a packing in and its packing density is :\eta = \frac. In Euclidean space If the space being packed is infinite in measure, such as Euclidean space, it is customary to define the density as the limit of densities exhibited in balls of larger and larger radii. If is the ball of radius centered at the origin, then the density of a packing is :\eta = \lim_\frac. Since this limit does not always exist, it is also useful to define the upper and lower densities as the limit superior and limit i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lloyd Kahn
Lloyd Kahn (born April 28, 1935) is an American publisher, editor, author, photographer, carpenter, and self-taught architect. He is the founding editor-in-chief of Shelter Publications, Inc., and is the former Shelter editor of the ''Whole Earth Catalog''. He is a pioneer of the green building and green architecture movements. His book ''Shelter'' (1973) about DIY architecture, has sold more than 250,000 copies. He lives and works in Bolinas, Marin County, California. Early life Kahn became interested in construction at age 12 when working on his family's house in Central Valley. He earned a B.A. degree (1957) from Stanford University. During the late 1950s, while serving in the United States Air Force in Germany, Kahn ran the USAF newspaper for two years. He returned to California in 1960 to work as an insurance broker and in 1965 quit his insurance job and began work as a carpenter, eventually building four houses. Career in carpentry and construction Kahn's first project ... [...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 sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( or \phi) denotes the golden ratio. The constant \varphi satisfies the quadratic equation \varphi^2 = \varphi + 1 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, and also goes by several 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 construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of \varphi—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: : History The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
