|
Kissing Number Problem
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a Lattice (group), lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another. Other names for kissing number that have been used are Newton number (after the originator of the problem), and contact number. In general, the kissing number problem seeks the maximum possible kissing number for n-sphere, ''n''-dimensional spheres in (''n'' + 1)-dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space. Finding the kissing number when centers of spheres a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
N-sphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cubic Close-packed
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is :\frac \approx 0.74048. The same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by Thomas Hales. The highest density is so far known only for 1, 2, 3, 8, and 24 dimensions. Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
E6 Lattice
In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol . It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex. Its vertex arrangement is the '' E6 lattice'', and the root system of the E6 Lie group so it can also be called the E6 honeycomb. Construction It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 6-dimensional space. The facet information can be extracted from its Coxeter–Dynkin diagram, . Removing a node on the end of one of the 2-node branches leaves the 221, its only facet type, The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 122, . The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, ''t''2, . The face figure is the vertex figure of the edge figure, here being a triangular duoprism, &tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
D5 Lattice
D5 or D05 may refer to: Arts and fiction * ''Dedication 5'', a 2013 mixtape by American rapper Lil Wayne * D5 class Klingon starships, in the ''Star Trek'' science fiction series * Dimension Delete and Deadly Destruction Day, featured in '' Digimon Xros Wars'' Aviation, transport, vehicles, communications * Albatros D.V, a German World War I fighter plane * Caterpillar D5 * Citroën D5 * Dauair (IATA code D5) * Dublin 5, an Irish postal district * Identification symbol for ''Nachtjagdgeschwader'' 3, a Luftwaffe night fighter-wing of World War II * Land Rover D5 - Land Rover Discovery 5 * Trident D5 ballistic missile * Volvo D5 engine, an engine family made by Volvo Cars Roads * D5 motorway (Czech Republic) * D5 road (Croatia), a state road * Detour for the M5 South Western Motorway, Sydney, Australia Substances * Vitamin D5 * D5, also known as DRD5, a subtype of dopamine receptors * A 5% dextrose solution, a type of intravenous fluid * Decamethylcyclopenta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
D4 Lattice
In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and constructed by a 4-dimensional packing of 16-cell facets, three around every face. Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B4, D4, or F4 lattice. Alternate names * Hexadecachoric tetracomb/honeycomb * Demitesseractic tetracomb/honeycomb Coordinates Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even. D4 lattice The vertex arrangement of the 16-cell honeycomb is called the D4 lattice or F4 lattice. The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003. The related D lattice (also called D) can be constructed by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
A3 Lattice
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedron, octahedra and tetrahedron, tetrahedra in a ratio of 1:2. Other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation. John Horton Conway calls this honeycomb a tetroctahedrille, and its dual a Rhombic dodecahedral honeycomb, dodecahedrille. R. Buckminster Fuller combines the two words octahedron and tetrahedron into octet truss, a rhombohedron consisting of one octahedron (or two square pyramids) and two opposite tetrahedra. It is vertex-transitive with 8 tetrahedra and 6 octahedra around each Vertex (geometry), vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge. It is part of an infinite family of uniform honeycombs called alternated hypercubic honeycombs, formed as an Alternati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
A2 Lattice
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of English mathematician John Conway called it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Kissing Growth
A kiss is the touching or pressing of one's lips against another person, animal or object. Cultural connotations of kissing vary widely; depending on the culture and context, a kiss can express sentiments of love, passion, romance, sexual attraction, sexual activity, sexual intercourse, sexual arousal, affection, respect, greeting, peace, or good luck, among many others. In some situations, a kiss is a ritual, formal or symbolic gesture indicating devotion, respect, or a sacramental. The word comes from Old English ('to kiss'), in turn from ' ('a kiss'). History Anthropologists disagree on whether kissing is an instinctual or learned behaviour. Those who believe kissing to be an instinctual behaviour cite similar behaviours in other animals such as bonobos, which are known to kiss after fighting - possibly to restore peace. Others believe that it is a learned behaviour, having evolved from activities such as suckling or premastication in early human cultures passed on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. In 2020, most of the editorial board of ''JCTA'' resigned to form a new, |
Doklady Akademii Nauk SSSR
The ''Proceedings of the USSR Academy of Sciences'' (, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), ) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. The journal is indexed in Russian Science Citation Index. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dating from 1726. ''Doklady Akademii Nauk SSSR-Comptes Rendus de l'Académie des Sciences de l'URSS, Seriya ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Leech Lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Ernst Witt in 1940. Characterization The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties: *It is unimodular lattice, unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix (mathematics), matrix with determinant 1. *It is even; i.e., the square of the length of each vector in Λ24 is an even integer. *The length of every non-zero vector in Λ24 is at least 2. The last condition is equivalent to the condition that unit balls centered at the points of Λ24 do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can kissing number, simultaneously t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
