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Quarter 5-cubic Honeycomb
In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation (or honeycomb). It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb.Coxeter, Regular and Semi-Regular Polytopes III, (1988), p318 Its facets are 5-demicubes and runcinated 5-demicubes. Related honeycombs See also Regular and uniform honeycombs in 5-space: * 5-cube honeycomb * 5-demicube honeycomb * 5-simplex honeycomb * Truncated 5-simplex honeycomb * Omnitruncated 5-simplex honeycomb Notes References * Kaleidoscopes: Selected Writings of H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ..., edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 5-honeycomb
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes: the 6-simplex , the 6-cube (hexeract) , and the 6-orthoplex (hexacross) . History of discovery * Regular polytopes: (convex faces) ** 1852: Ludwig Schläfli proved in his manuscript ''Theorie der vielfachen Kontinuität'' that there are exactly 3 regular polytopes in 5 or more dimensions. * Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) ** 190 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that mathematics and music were intimately related, outlining his i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omnitruncated 5-simplex Honeycomb
In Five-dimensional space, five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb (geometry), honeycomb). It is composed entirely of omnitruncated 5-simplex facets. The facets of all omnitruncated simplectic honeycombs are called permutahedron, permutahedra and can be positioned in ''n+1'' space with integral coordinates, permutations of the whole numbers (0,1,..,n). A5* lattice The A lattice (also called A) is the union of six A5 lattice, A5 lattices, and is the dual vertex arrangement to the ''omnitruncated 5-simplex honeycomb'', and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex. : ∪ ∪ ∪ ∪ ∪ = dual of Related polytopes and honeycombs Projection by folding The ''omnitruncated 5-simplex honeycomb'' can be projected into the 3-dimensional omnitruncated cubic honeycomb by a Coxeter–Dynkin diagram#Geometric folding, geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 5-simplex Honeycomb
In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1. Structure Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells. It can be constructed as six sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane. Related polytopes and honeycombs See also Regular and uniform honeycombs in 5-space: *5-cubic honeycomb *5-demicubic honeycomb * 5-simplex honeycomb In Five-dimensional space, five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-simplex Honeycomb
In Five-dimensional space, five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb (geometry), honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb. A5 lattice This vertex arrangement is called the A5 lattice, A5 lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the _5 Coxeter group. It is the 5-dimensional case of a simplectic honeycomb. The A lattice is the union of two A5 lattices: : ∪ The A is the union of three A5 lattices: : ∪ ∪ . The A lattice (also called A) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex. : ∪ ∪ ∪ ∪ ∪ = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-demicube Honeycomb
The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb. It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h and the alternated vertices create 5-orthoplex facets. D5 lattice The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex figure of the ''5-demicubic honeycomb'' reflect the kissing number 40 of this lattice. The D packing (also called D) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8). : ∠... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is the remainder of the Euclidean division of by , where is the dividend and is the divisor. For example, the expression "5 mod 2" would evaluate to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. Although typically performed with and both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of is 0 to inclusive ( mod 1 is always 0; is undefined, possibly resulting in a division by zero error in some programming languages). See Modular arithmetic for an older and related ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 5-demicube
In five-dimensional geometry, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes. Steric 5-cube Alternate names * Steric penteract, runcinated demipenteract * Small prismated hemipenteract (siphin) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of : (±1,±1,±1,±1,±3) with an odd number of plus signs. Images Related polytopes Stericantic 5-cube Alternate names * Prismatotruncated hemipenteract (pithin) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations: : (±1,±1,±3,±3,±5) with an odd number of plus signs. Images Steriruncic 5-cube Alternate names * Prismatorhombated hemipenteract (pirhin) (Jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-cube Honeycomb
In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an ''order-4 penteractic honeycomb''. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space. Constructions There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol . Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol 5. Related polytopes and honeycombs The ,33,4 , Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5-demicubic Honeycomb
The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb. It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of uniform facets. The 5-cubes become alternated into 5-demicubes h and the alternated vertices create 5-orthoplex facets. D5 lattice The vertex arrangement of the 5-demicubic honeycomb is the D5 lattice which is the densest known sphere packing in 5 dimensions. The 40 vertices of the rectified 5-orthoplex vertex figure of the ''5-demicubic honeycomb'' reflect the kissing number 40 of this lattice. The D packing (also called D) can be constructed by the union of two D5 lattices. The analogous packings form lattices only in even dimensions. The kissing number is 24=16 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
