Image Tile 3,6.svg thumb Triangular tiling . The vertices form a hexagonallattice with horizontal rows ... to point into two oblique directions. Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as hexagonallattice with horizontal rows like in the figure below , with triangles pointing up and down, and hexagonallattice with vertical rows , with triangles pointing left and right. They differ by an angle of 30 . The hexagonallattice with horizontal ... as high. Its symmetry category is wallpaper group p6m. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. Image Tile 6,3.svg thumb Hexagonal tiling , Honeycomb geometry honeycomb structure. The centers of the hexagons form a hexagonallattice with vertical rows for each color the centers of the hexagons of that color form a hexagonallattice with horizontal rows which is in linear scale 3 times as large. The vertices of all hexagons together with their centers form a hexagonallattice with horizontal rows which is in linear ... of 90 , or equivalently 30 . A honeycomb structure is in two ways related to a hexagonallattice the centers of the hexagons of a honeycomb form a hexagonallattice, with the rows oriented the same the vertices of a honeycomb, together with their centers, form a hexagonallattice, rotated by 30 or equivalently ... a corresponding hexagonallattice, or a structure which is not a lattice in the Group mathematics group ... bonds . With respect to a hexagonallattice we can distinguish two sets or three directions the directions ... lattice with horizontal rows one of the three directions is horizontal, and for a hexagonallattice ... 6 by 6 to form a hexagonal tiling . In each case the hexagons are oriented with two horizontal sides, and arranged in vertical rows, and in each case they have a dark triangle at the upper left. The hexagonallattice or equilateral triangular lattice is one of the five 2D Lattice group lattice types ... more details
wiktionary latticeLattice may refer to In art and design Latticework an ornamental criss crossed framework, an arrangement of crossing laths or other thin strips of material Lattice pastry In engineering A lattice shape truss structure In mathematics Lattice order , a partially ordered set with unique least upper bounds and greatest lower bounds Lattice group , a repeating arrangement of points Lattice discrete subgroup , a discrete subgroup of a topological group with finite covolume Lattice graph , a graph that can be drawn within a repeating arrangement of points Bethe lattice , a regular infinite tree structure Lattice multiplication , a multiplication algorithm suitable for hand calculation Lattice model finance , a method for evaluating stock options that divides time into discrete intervals In science A crystal structure fitting a lattice arrangement Lattice model physics , a model defined not on a continuum, but on a lattice Companies Lattice Semiconductor , a US based integrated circuit manufacturer Lattice, Incorporated, a software company and makers of Lattice C Lattice Group , a former British gas transmission business Other Lattice C , a compiler for the C programming language See also Grid disambiguation Mesh disambiguation Trellis disambiguation In title Lattice disambiguation ar da Gitter de Gitter Begriffskl rung es Ret culo fr Lattice it Reticolo he ka ja Lattice pl Krata pt Ret culo ru sk Mrie ka sv Gitter ... more details
Unreferenced date October 2007 Image HEXAGONALPHASE1.JPG thumb right Schematic of the molecular organisation in a type I hexagonal lyotropic liquid crystalline phase. A hexagonal phase of lyotropic liquid crystal is formed by some amphiphilic molecules when they are mixed with water or another polar solvent. In this phase, the amphiphile molecules are aggregated into cylindrical structures of indefinite length and these cylindrical aggregates are disposed on a hexagonal lattice, giving the phase long range orientational order. In normal topology hexagonal phases, which are formed by type I amphiphiles , the hydrocarbon chains are contained within the cylindrical aggregates such that the polar apolar interface has a positive mean curvature . Inverse topology hexagonal phases have water within the cylindrical aggregates and the hydrocarbon chains fill the voids between the hexagonally packed cylinders. Normal topology hexagonal phases are denoted by H sub I sub while inverse topology hexagonal phases are deonted by H sub II sub . When viewed under a polarising microscope , thin films of both normal and inverse topology hexagonal phases exhibit birefringence , giving rise to characteristic optical textures. Typically, these textures are smoke like, fan like or mosaic in appearance. The phases are highly viscous and small air bubbles trapped within the preparation have highly distorted shapes. gallery Image Hexagonal1a.jpg smokey optical texture of type I hexagonal phase Image Hexagonal2a.jpg mosaic optical texture of type I hexagonal phase Image Hexagonal3a.jpg mosaic focal conic optical texture of type I hexagonal phase gallery Category Liquid crystals ... more details
centered cubic lattice is optimal. However, the less regular Weaire Phelan structure is slightly better. Chicken wire consists of a hexagonallattice of wires. This structure exists naturally in the form ... tiling 3.3.3.3.6 See also HexagonallatticeHexagonal prismatic honeycomb Tilings of regular ...No footnotes date March 2011 Uniform tiles db Reg tiling stat table Uh In geometry , the hexagonal tiling ... of the plane . The other two are the triangular tiling and the square tiling . Applications The hexagonal ... states that the hexagonal tiling is the best way to divide a surface into regions of equal ... cylindrical surface gallery The hexagonal tiling appears in many crystals. In three dimensions, the face centered cubic and hexagonal close packing are common crystal structures. They are the densest ... parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way ... of the two. Pure copper , amongst other materials, forms a face centered cubic lattice. Uniform colorings There are 3 distinct uniform coloring s of a hexagonal tiling, all generated from reflective ... hexagonal distances as h first, and k second. class wikitable width 640 k uniform colspan 3 1 ... 100px File Hexagonal tiling 4 colors.png 100px File Hexagonal tiling 2 1.png 100px File Hexagonal ... Image Wallpaper group cmm 1.jpg thumb Bricks as nonregular hexagonal tiling The hexagonal tiling can ... pattern can be considered a nonregular hexagonal tiling. Each rectangular brick has vertices inserted ... symmetry groups Wallpaper groups 632 p6 symmetry . The herringbone pattern is also a distorted hexagonal tiling. class wikitable 4 color hexagonal tilings Regular hexagons Hexagonal weave p6m 632 p6 632 File Hexagonal tiling 4 colors.png 240px File Weaved hexagonal tiling.png 240px colspan 2 Brick pattern Herringbone colspan 2 p4g 4 2 File Bond brick hexagonal tiling.png 240px File Herringbone pattern as hexagonal tiling.png 240px Related polyhedra and tilings This tiling is topologically ... more details
A hexagonal number is a figurate number . The n th hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side. Image Hexagonal numbers.svg none The first four hexagonal numbers. The formula for the n th hexagonal number math h n 2n 2 n n 2n 1 2n times 2n 1 over 2 . , math The first few hexagonal numbers OEIS id A000384 are 1 number 1 , 6 number 6 , 15 number .... Every hexagonal number is a triangular number , but only every other triangular number the 1st, 3rd, 5th, 7th, etc. is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal ... 6 6 1 9 3 1 3 9 . Every even perfect number is hexagonal, given by the formula math M p 2 p 1 M p ..., hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2x3 6 the 4th ... as a sum of at most four hexagonal numbers is 130 number 130 . Adrien Marie Legendre proved in 1830 that any integer greater than 1791 can be expressed in this way. Hexagonal numbers can be rearranged into rectangular numbers of size n by 2 n &minus 1 . Hexagonal numbers should not be confused with centered hexagonal number s, which model the standard packaging of Wiener sausage s. To avoid ambiguity, hexagonal numbers are sometimes called cornered hexagonal numbers . Test for hexagonal numbers One can efficiently test whether a positive integer x is an hexagonal number by computing math n frac sqrt 8x 1 1 4 . math If n is an integer, then x is the n th hexagonal number. If n is not an integer, then x is not hexagonal. Other properties The n th number of the hexagonal sequence can ... to be 0. See also Centered hexagonal number External links Mathworld entry on http mathworld.wolfram.com HexagonalNumber.html Hexagonal Number Category Figurate numbers ar de Sechseckszahl es N mero hexagonal fr Nombre hexagonal it Numero esagonale nl Zeshoeksgetal ja pt N mero hexagonal ru sl estkotni ko tevilo fi Kuusikulmioluku sv Hexagonala tal ta ... more details
border 1 bgcolor ffffff cellpadding 5 align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 Hexagonal trapezohedron align center colspan 2 Image Hexagonal trapezohedron.png 180px Hexagonal trapezohedron br Click on picture for large version. bgcolor e7dcc3 Type Trapezohedron trapezohedra bgcolor e7dcc3 Faces 12 kite geometry kites bgcolor e7dcc3 Edges 24 bgcolor e7dcc3 Vertices 14 bgcolor e7dcc3 Face configuration 6,3,3,3 bgcolor e7dcc3 List of spherical symmetry groups Symmetry group D sub 6d sub , 2 ,12 , 2 6 bgcolor e7dcc3 Dual polyhedron hexagonal antiprism bgcolor e7dcc3 Properties convex, face transitive The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face uniform polyhedra which are dual polyhedron to the antiprism s. It has twelve faces which are congruence geometry congruent kite geometry kites . See also hexagonal bipyramid A similar 12 sided polyhedron with triangular faces. External links mathworld urlname Trapezohedron title Trapezohedron http www.georgehart.com virtual polyhedra vp.html Virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML model http www.georgehart.com virtual polyhedra vrml hexagonal trapezohedron.wrl 6 Polyhedron stub Polyhedron navigator Category Polyhedra eo Seslatera kajtopluredro ... more details
unreferenced date June 2011 border 1 bgcolor ffffff cellpadding 5 align right style margin left 10px width 250 bgcolor e7dcc3 colspan 2 Hexagonal Bifrustum align center colspan 2 Image Hexagonal bifrustum.png 180px Hexagonal bifrustum bgcolor e7dcc3 Type Bifrustum bgcolor e7dcc3 Faces 12 trapezoid s, 2 hexagon s bgcolor e7dcc3 Edges 24 bgcolor e7dcc3 Vertices 18 bgcolor e7dcc3 Symmetry group Symmetry group Three dimensions D sub 6h sub bgcolor e7dcc3 Dual polyhedron elongated hexagonal dipyramid bgcolor e7dcc3 Properties convex The hexagonal bifrustum or truncated bipyramid is the fourth in an infinite series of bifrustum polyhedra. It has 12 trapezoid and 2 hexagon al faces. This polyhedron can be constructed by taking a hexagonal dipyramid and truncating the polar axis vertices, making it into two end to end frustum s. A sapphire or ruby crystal is an example of a hexagonal bifrustum. It has been used for a decorative polyhedral monthly calendar , with one month on each trapezoid face. dodecahedron regular and rhombic dodecahedron rhombic dodecahedra have also been used for calendar months. Category Polyhedra Polyhedron stub eo Seslatera dutrunko ... more details
Prism polyhedra db Prism polyhedron stat table AP6 In geometry , the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even numbered sequence of triangle sides closed by two polygon caps. If faces are all regular, it is a semiregular polyhedron . See also antiprism Set of antiprisms Octahedron , a triangle capped antiprism Square antiprism Pentagonal antiprism Octagonal antiprism External links mathworld urlname Antiprism title Antiprism http polyhedra.org poly show 29 hexagonal antiprism Hexagonal Antiprism Interactive Polyhedron model http www.georgehart.com virtual polyhedra vp.html Virtual Reality Polyhedra www.georgehart.com The Encyclopedia of Polyhedra VRML http www.georgehart.com virtual polyhedra vrml hexagonal antiprism.wrl model http www.georgehart.com virtual polyhedra conway notation.html Conway Notation for Polyhedra Try A6 Category Prismatoid polyhedra Polyhedron navigator Polyhedron stub eo Seslatera kontra prismo es Antiprisma hexagonal fr Antiprisme hexagonal zh ... more details
directions they are linearly a factor 2 denser. See also square tiling hexagonallattice centered square number Gaussian integer DEFAULTSORT Square Lattice Category Euclidean geometry Category Lattice ...Unreferenced date December 2009 Image Square Lattice.svg thumb 300px An upright square lattice left and a diagonal square lattice right . Image Square Lattice Tiling.svg thumb Upright square tiling . The vertices of all squares together with their centers form an upright square lattice. For each color the centers of the squares of that color form a diagonal square lattice which is in linear scale &radic 2 times as large as the upright square lattice. In mathematics , the square lattice is a type of Lattice group lattice in a two dimensional Euclidean space . It is the two dimensional version of the integer lattice . It is one of the five types of two dimensional lattices as classified by their symmetry ... of an image of the lattice are by far the most common. They can conveniently be referred to as upright square lattice and diagonal square lattice . They differ by an angle of 45 . This is related to the fact that a square lattice can be partitioned into two square sub lattices, as is evident in the colouring of a checkerboard . Symmetry The square lattice s symmetry category is wallpaper group p4m. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be viewed as a diagonal square lattice ..., after adding the centers of the squares of an upright square lattice we have a diagonal square lattice with a mesh size that is 2 times as small as that of the original lattice. A pattern with 4 fold rotational symmetry has a square lattice of 4 fold rotocenters that is a factor 2 finer and diagonally oriented relative to the lattice of translational symmetry . With respect to reflection axes ... lattice of 4 fold rotocenters, with these rotocenters at the centers of the squares formed by the reflection ... more details
Infobox face uniform polyhedron Image File Hexagonale bipiramide.png Polyhedron Type bipyramid Face List 12 triangle s Edge Count 18 Vertex Count 8 Symmetry Group D sub 6h sub , 6,2 , 226 Face Type V4.4.6 Dual hexagonal prism Property List convex set convex , face transitive A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramid geometry pyramids joined at their bases. The resulting solid has 12 triangular face geometry faces , 8 vertex geometry vertices and 18 edges. The 12 faces are identical isosceles triangle s. It is one of an infinite set of bipyramid s. Having twelve faces, it is a type of dodecahedron , although that name is usually associated with the regular polyhedron regular polyhedral form with pentagonal faces. The term dodecadeltahedron is sometimes used to distinguish the bipyramid from the Platonic solid . The hexagonal bipyramid has a plane of symmetry which is Horizontal plane horizontal in the figure to the right where the bases of the two pyramids are joined. This plane is a regular hexagon . There are also six planes of symmetry crossing through the two apex geometry apices . These planes are rhombus rhombic and lie at 30 angle s to each other, perpendicular to the horizontal plane. See also hexagonal trapezohedron A similar 12 sided polyhedron with a twist and Kite geometry kite faces. Snub disphenoid Another 12 sided polyhedron with 2 fold symmetry and only triangular faces. External links mathworld urlname Dipyramid title Dipyramid GlossaryForHyperspace anchor Bipyramid title Bipyramid http www.georgehart.com virtual polyhedra vp.html Virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML model http www.georgehart.com virtual polyhedra vrml hexagonal dipyramid.wrl hexagonal dipyramid Polyhedron stub Category Polyhedra Category Pyramids and bipyramids eo Seslatera dupiramido nl Hexagonale bipiramide ... more details
Image Glinski Chess Setup.png right thumb 300px W adys aw Gli ski Gli ski s hexagonal chess , starting position The term hexagonal chess designates a group of chess variant s played on hexagonal boards. The most popular one is Gli ski s hexagonal chess which was invented in 1936 by W adys aw Gli ski of Poland . Gli ski s hexagonal chess Gli ski s hexagonal chess is the most popular of many hexagonal ... is played on a hexagonal board with hexagonal cells having three colors light, dark, and mid tone ... of Hexagonal Chess isbn 0 904195 00 7 ref The usual set of chess pieces is increased by one Bishop .... The Queen chess queen moves as Rook chess rook plus bishop. There is no castling in Gli ski s hexagonal ... notation , except there is no castling. Dateline 1976 br June First Hexagonal Chess Congress at Bloomsbury Centre Hotel, London, which included the inauguration of the British Hexagonal Chess Federation and the first British Hexagonal Chess Championship. David Springgay took the title. December First issue of Hex Press Hexagonal Chess News published. 1977 br Hexagonal Chess was topic for many ... visited Poland. Successful Hexagonal Chess event staged. Wide publicity in Eastern Europe. Wspolna Sprawa ... which includes a regular column on Hexagonal Chess . Whilst in Poland, Simon Triggs played the first mixed 6 square and 6 hexagonal boards simultaneous display. Hexagonal Chess clubs formed in Poland ..., which included Inauguration of Internatienal Hexagonal Chess Federation First European Championship ... Hungarian Hexagonal Chess Championship in Szekszard. Title taken by Laszlo Sziraki. Fourth British ... production of a new complete Hexagonal Chess boxed game with roll up double sided board hexagonal square and wooden pieces to be distributed in the West European market. July An open Hexagonal Chess ... was organized by the President of the Hungarian Hexagonal Chess Association, Mihaly Gelencser, and sponsored ... newspapers took photographs. LBC Radio and BBC Radio London broadcast interviews. McCooey s hexagonal ... more details
Hexagonal water is a term used in a fraud marketing scam ref name wired cite web publisher Wired Science last Rowe first Aaron title Video Hexagonal Water is an Appalling Scam date 2008 03 17 accessdate 2011 10 18 url http www.wired.com wiredscience 2008 03 chem lab hexago ref ref name alabama cite web title Drinking Water and Water Treatment Scams publisher Alabama Cooperative Extension System date 2003 10 22 accessdate 2011 10 18 url http docs.google.com viewer?a v&q cache QAUI82HS Q4J www.aces.edu waterquality articles new WATER SCAMS3 HD.pdf hexagonal water scam&hl en&gl us&pid bl&srcid ADGEESjQa9CicZi5TfufUfNj0XOu2L1VnS9zhGpMJxHkzwnMDW3NUv3KFVQ9PgAqGO9eJq wr3VyFTayWhqYLW xnU91J75ZbJ7tDLcSLK8kyOI2cYFsZg8bLuRKbPDOuOXfo7F9NRDL ... cite web publisher Aqua Technology title Understanding Hexagonal Water accessdate 2011 10 18 url http www.aquatechnology.net hexagonalwater.html ref The term hexagonal water refers to a water cluster cluster of water forming a hexagonal shape that supposedly enhances nutrient absorption, removes metabolic ... cite web publisher Frequency Rising title Hexagonal Water accessdate 2011 10 18 url http www.frequencyrising.com ... claims central to hexagonal water include the following Hexagonal Water is composed of six individual ... organisms prefer the six sided hexagonal ring structure, found naturally in snow water ref name aquatechnology The percentage of hexagonal units appears to depend on a number of factors, including ... to create an oxygen rich, alkaline, energized and uniquely structured Hexagonal Water. ref name aquatechnology Hexagonal water is associated with aging. ref name aquatechnology ref name frequency this kind ... with science The concept of hexagonal water clashes with several established scientific ... the hexagonal water model s claim that the particular structure of water consumed is the same structure used by the body. Similarly, the hexagonal water model claims that this particular structure ... be created through bombardment of energy of these frequencies. ref name alabama In addition, the hexagonal ... more details
Infobox Polyhedron Image File Hexagonal pyramid.png Polyhedron Type Pyramid geometry Pyramid Face List 6 triangle s br 1 hexagon Edge Count 12 Vertex Count 7 Symmetry Group C sub 6v sub , 6 , 66 Vertex List 6 3 sup 2 sup .6 br 3 sup 6 sup Dual Self dual polyhedron self dual Property List convex set convex In geometry , a hexagonal pyramid is a Pyramid geometry pyramid with a hexagon al base upon which are erected six triangle triangular faces that meet at a point the apex . Like any pyramid , it is self dual polyhedron dual . A right hexagonal pyramid with a regular hexagon base has List of spherical symmetry groups Other C sub 6v sub symmetry . See also Hexagonal bipyramid A right regular pyramid is the one which has a regular hexagon at the base, it is called as a right regular pyramid because the line joining the centre of the base to the apex of the pyramid, the line joining the centre of the hexagon base and any of the corners of the hexagon and the line joining that corner of the hexagon and the apex form a right angled triangle. External links Mathworld urlname Hexagonal Pyramid title Hexagonal pyramid http www.georgehart.com virtual polyhedra vp.html Virtual Reality Polyhedra www.georgehart.com The Encyclopedia of Polyhedra http www.georgehart.com virtual polyhedra conway notation.html Conway Notation for Polyhedra Try Y6 Category Pyramids and bipyramids Category Self dual polyhedra Category Prismatoid polyhedra Polyhedron stub ... more details
, c and angles between the sides given by , , ref The lattice constant or lattice parameter refers to the constant distance between unit cell s in a crystal lattice . Lattices in three dimensions generally have three lattice constants, referred to as a , b , and c . However, in the special case of cubic crystal structures, all of the constants are equal and we only refer to a . Similarly, in hexagonal .... A group of lattice constants could be referred to as lattice parameters . However, the full set of lattice parameters consist of the three lattice constants and the three angles between them. For example the lattice constant for a common carbon diamond is a 3.57 at 300 K . The structure is equilateral although its actual shape can not be determined from only the lattice constant. Furthermore, in real applications, typically the average lattice constant is given. As lattice constants have the dimension of length, their SI unit is the meter . Lattice constants are typically on the order of several angstrom s i.e. tenths of a nanometre . Lattice constants can be determined using techniques such as X ray diffraction or with an atomic force microscope . In epitaxy epitaxial growth , the lattice constant is a measure of the structural compatibility between different materials. Lattice ... epitaxy epitaxial growth of thicker layers without defects. Lattice matching Matching of lattice structures ... , and aluminium arsenide have almost equal lattice constants, making it possible to grow almost arbitrarily thick layers of one on the other one. Lattice grading Typically, films of different materials grown on the previous film or substrate are chosen to match the lattice constant of the prior layer to minimize film stress. An alternative method is to grade the lattice constant from one value ... layer will have a ratio to match the underlying lattice and the alloy at the end of the layer growth will match the desired final lattice for the following layer to be deposited. The rate of change ... more details
Unreferenced date February 2011 Prism polyhedra db Prism even polyhedron stat table P6 In geometry , the hexagonal prism is a Prism geometry prism with hexagon al base. This polyhedron has 8 faces, 18 edges, and 12 vertices. Since it has eight Face geometry faces , it is an Octahedron Other octahedra octahedron . However, the term octahedron is primarily used to refer to the regular octahedron , which has eight triangular faces. Because of the ambiguity of the term octahedron and the dissimilarity of the various eight sided figures, the term is rarely used without clarification. As a semiregular or uniform polyhedron If faces are all regular, the hexagonal prism is a semiregular polyhedron , more ... sides and two regular polygon caps. It can be seen as a truncation geometry truncated hosohedron hexagonal ... dual of a hexagonal prism is a hexagonal bipyramid . The symmetry group of a right pentagonal ... h math . math V frac 3 sqrt 3 2 a 2 times h math Other images Image Hexagonal Prism.svg 240px BR A translucent ... wikitable width 400 align center Hexagonal prismatic honeycomb BR CDD node 1 6 node 3 node 2 node 1 infin node Triangular hexagonal prismatic honeycomb BR CDD node 6 node 1 3 node 2 node 1 infin node Snub triangular hexagonal prismatic honeycomb BR CDD node h 6 node h 3 node h 2 node 1 infin node Rhombitriangular hexagonal prismatic honeycomb BR CDD node 1 6 node 3 node 1 2 node 1 infin node align center File Hexagonal prismatic honeycomb.png 100px File Triangular hexagonal prismatic honeycomb.png 100px File Snub triangular hexagonal prismatic honeycomb.png 100px File Rhombitriangular hexagonal ... Spherical truncated trigonal prism.png 100px BR Hexagonal prism 4.6.4 BR CDD node 1 2 node 1 3 node ... prism External links mathworld urlname HexagonalPrism title Hexagonal prism http polyhedra.org poly show 24 hexagonal prism Hexagonal Prism Interactive Model works in your web browser Category Prismatoid ... fr Prisme hexagonal nl Hexagonaal prisma ... more details
space vectors. Simple hexagonallattice The reciprocal to a simple hexagonal Bravais lattice with lattice constants c and a is another simple hexagonallattice with lattice constants math begin matrix ...In physics , the Multiplicative inverse reciprocal lattice of a lattice usually a Bravais lattice is the lattice in which the Fourier transform of the spatial wavefunction of the original lattice or direct lattice is represented. This space is also known as momentum space or less commonly k space , due ... lattice of a reciprocal lattice is the original or direct lattice . Mathematical description Consider a set of points R constituting a Bravais lattice, and a plane wave defined by math e i mathbf ... has the same periodic function periodicity as the Bravais lattice, then it satisfies the equation math ... mathbf R 1 math Mathematically, we can describe the reciprocal lattice as the set of all vector geometric vector s K that satisfy the above identity for all lattice point position vectors R . This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces. For an infinite three dimensional lattice, defined by its primitive cell primitive vector s math mathbf a 1 , mathbf a 2 , mathbf a 3 math , its reciprocal lattice can be determined by generating its three reciprocal ... s definition, comes from defining the reciprocal lattice to be math e 2 pi i mathbf K cdot mathbf R 1 math which changes the definitions of the reciprocal lattice vectors to be math mathbf b 1 frac ... manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency . It is a matter of taste which definition of the lattice is used, as long as the two are not mixed. Each point hkl in the reciprocal lattice corresponds to a set of lattice planes hkl in the real space lattice. The direction of the reciprocal lattice vector corresponds to the normal mathematics normal to the real ... more details
symmetrically staggered rows File Rhombic Lattice.svg 200px a hexagonallattice or equilateral triangular ... boundary, with the hexagonallattice as vertex, and i for the square lattice. The rectangular lattices ...File Equilateral Triangle Lattice.svg thumb right 250px A lattice in the Euclidean plane . In mathematics , especially in geometry and group theory , a lattice in R sup n sup is a discrete subgroup of R sup n sup which linear span spans the real number real vector space R sup n sup . Every lattice in R ... linear combination s with integer coefficients. A lattice may be viewed as a regular tiling of a space ... of several lattice problems , and are used in various ways in the physical sciences. For instance, in materials science and solid state physics , a lattice is a synonym for the frame work of a crystalline ... positions in a crystal . More generally, lattice model physics lattice models are studied in physics , often by the techniques of computational physics . Symmetry considerations and examples A lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. A lattice in the sense of a 3 dimension al array of regularly spaced points coinciding with e.g. ... translational symmetry, is a translate of the translation lattice a coset , which need not contain the origin, and therefore need not be a lattice in the previous sense. A simple example of a lattice in R sup n sup is the subgroup Z sup n sup . A more complicated example is the Leech lattice , which is a lattice in R sup 24 sup . The period lattice in R sup 2 sup is central to the study of elliptic ... of abelian function s. Dividing space according to a lattice A typical lattice in R sup n ... , ..., v sub n sub is a basis for R sup n sup . Different bases can generate the same lattice, but the absolute ... by d . If one thinks of a lattice as dividing the whole of R sup n sup into equal polyhedron ... more details
In mathematics , the term lattice group is used for two distinct notions a lattice group lattice , a discrete subgroup of R sup n sup and its generalizations. a lattice ordered group , a group that with a partial ordering that is a lattice order lattice order. mathdab ... more details
math a 3 sqrt 1 3 cos 2 alpha 2 cos 3 alpha math Hexagonallattice system Hexagonal math frac ...In geometry and crystallography , a Bravais lattice , studied by harvs txt first Auguste last Bravais ... the lattice. This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R , the lattice looks exactly the same. A crystal is made up of a periodic arrangement of one or more atoms the basis repeated at each lattice point. Consequently, the crystal looks the same when viewed from any of the lattice points. Two Bravais lattices are often considered ... space there is just one type of Bravais lattice. In two dimensions, there are five Bravais ... 1 oblique, 2 rectangular, 3 centered rectangular rhombic , 4 hexagonal, and 5 square Bravais lattices ... lattice system s or axial systems with one of the lattice centerings. Each Bravais lattice refers to a distinct lattice type. The lattice centerings are Primitive centering P lattice points on the cell corners only. Body centered I one additional lattice point at the center of the cell. Face centered F one additional lattice point at center of each of the faces of the cell. Base centered A, B or C one additional lattice point at the center of each of one pair of the cell faces. Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total ... equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A or B centered lattices can be described ... Bravais lattices, shown in the table below. align left border 1 style margin left 1em The 7 lattice ... 80px Tetragonal, body centered rowspan 2 align center rhombohedral lattice system rhombohedral br align center P Image Rhombohedral.svg 80px Rhombohedral rowspan 2 align center Hexagonallattice system hexagonal align center P Image Hexagonal lattice.svg 80px Hexagonal rowspan 2 align center Cubic ... more details
on the integer lattice , hexagonal tiling and E8 lattice , respectively. It has no root system and in fact is the first unimodular lattice with no roots vectors of norm less than 4 , and therefore ... , sub 2 sub is to the hexagonallattice , and sub 3 sub is the face centered cubic packing. harvtxt ...In mathematics , the Leech lattice is an even unimodular lattice sub 24 sub in 24 dimensional Euclidean ... year 1967 . History Many of the cross sections of the Leech lattice, including the Coxeter&ndash Todd lattice and Barnes&ndash Wall lattice , in 12 and 16 dimensions, were found much earlier than the Leech lattice. harvtxt O Connor Pall 1944 discovered a related odd unimodular lattice in 24 dimensions, now called the odd Leech lattice, whose even sublattice has index 2 in the Leech lattice. The Leech lattice was discovered in 1965 by harvs txt authorlink John Leech mathematician first John last ... group of the Leech lattice, and discovered three new sporadic group s as a by product the Conway ... he found in 1940 was the Leech lattice. See his collected works harv Witt 1998 loc p. 328 329 for more comments and for some notes Witt wrote about this in 1972. Characterization The Leech lattice sub 24 sub is the unique lattice in E sup 24 sup 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 ... lattice is isometric to the set of simple roots or the Dynkin diagram of the reflection group of the 26 dimensional even Lorentzian unimodular lattice II sub 25,1 sub . By comparison, the Dynkin diagrams of II sub 9,1 sub and II sub 17,1 sub are finite. Constructions The Leech lattice can be constructed ... , a 24× 24 matrix with determinant 1. Using the binary Golay code The Leech lattice can be explicitly ... in a construction for the 196560 minimal vectors in the Leech lattice. Using the Lorentzian lattice II sub 25,1 sub The Leech lattice can also be constructed as math w perp w math where w is the Weyl ... more details
In mineralogy , atomic lattice refers to the arrangement of atoms into a crystal structure . In order theory , a lattice order lattice is called an atomic lattice if the underlying partial order is atomic order theory atomic . disambig ... more details
class wikitable align right colspan 2 align center Image Berillo.jpg 240px BR An example of the hexagonal crystals, beryl Image Hanksite.JPG 180px BR Hexagonal Hanksite crystal image Hexagonal lattice.svg 120px BR Hexagonallattice cell In crystallography , the hexagonal crystal system is one of the 7 crystal system s, the hexagonallattice system is one of the 7 lattice systems, and the hexagonal ... other, but they are not the same. The hexagonallattice system consists of just one Bravais lattice type the hexagonal one. The hexagonal crystal system consists of the 7 point groups such that all their space groups have the hexagonallattice as underlying lattice. The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonallattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal ... is an example of a crystal that crystallizes in the hexagonal crystal system. Hexagonallattice system The hexagonallattice system is one of the seven lattice system s, consisting of the hexagonal Bravais lattice. It is associated with 45 space group s whose underlying lattice has point group of order 24. It is often confused with the smaller hexagonal crystal system, which consists of the 27 space groups such that all space groups with the same point group are in the hexagonallattice system, or with the larger hexagonal crystal family , consisting of the 52 space groups in either the hexagonal or rhombohedral lattice system s. Hexagonal crystal system The crystallographic point group ... of their space groups has the hexagonallattice as underlying lattice, and is the union of the hexagonal ... are exactly those whose Bravais lattice is either hexagonal or rhombohedral. See also Crystal structure ...., pp. 78 89, ISBN 0 471 80580 7 ref In some cases, it is useful or instructive to redraw a hexagonal ... Schoenflies notation Schoenflies Orbifold notation Orbifold Coxeter notation Coxeter 168 173 Hexagonal ... more details
In crystallography , a lattice plane of a given Bravais lattice is a plane or family of parallel planes whose intersections with the lattice or any crystalline structure of that lattice are periodic functions periodic i.e. are described by 2d Bravais lattices and intersect the Bravais lattice equivalently, a lattice plane is any plane containing at least three noncollinear Bravais lattice points. ref name Ash76 Neil W. Ashcroft and N. David Mermin, Solid State Physics Harcourt New York, 1976 . ref All lattice planes can be described by a set of integer Miller indices , and vice versa all integer Miller indices define lattice planes . ref name Ash76 Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystal s this is known as a cut and project construction of a quasicrystal and is typically also generalized to higher dimensions . ref J. B. Suck, M. Schreiber, and P. H ussler, eds., Quasicrystals An Introduction to Structure, Physical Properties, and Applications Springer Berlin, 2004 . ref References references Category Crystallography Category Geometry geometry stub de Gitterebene it Piani reticolari ... more details
Unreferenced date December 2009 In mathematics , the n dimensional integer lattice or cubic lattice , denoted Z sup n sup , is the lattice group lattice in the Euclidean space R sup n sup whose lattice points are n tuples n tuples of integer s. The two dimensional integer lattice is also called the square lattice , or grid lattice. Z sup n sup is the simplest example of a root lattice . The integer lattice is an odd unimodular lattice . Automorphism group The automorphism group or group of congruence relation congruence s of the integer lattice consists of all permutation s and sign changes of the coordinates, and is of order 2 sup n sup n nowiki nowiki . As a matrix group it is given by the set of all n × n signed permutation matrices . This group is isomorphic to the semidirect product math mathbb Z 2 n rtimes S n math where the symmetric group S sub n sub acts on Z sub 2 sub sup n sup by permutation this is a classic example of a wreath product . For the square lattice, this is the group of the square, or the dihedral group of order 8 for the three dimensional cubic lattice, we get the group of the cube, or octahedral group , of order 48. Diophantine geometry In the study of Diophantine geometry , the square lattice of points with integer coordinates is often referred to as the Diophantine plane . In mathematical terms, the Diophantine plane is the Cartesian product math scriptstyle mathbb Z times mathbb Z math of the ring of all integers math scriptstyle mathbb Z math . The study of Erd s Diophantine graph Diophantine figures focuses on the selection of nodes in the Diophantine plane such that all pairwise distances are integer. Coarse geometry In coarse structure coarse geometry , the integer lattice is coarsely equivalent to Euclidean space . See also Regular grid DEFAULTSORT Integer Lattice Category Euclidean geometry Category Lattice points Category Diophantine geometry ... more details
notability Products date April 2012 primary sources date April 2012 The Lattice Project combines computing resource s, Grid middleware, specialized scientific application software and web services into a comprehensive Grid computing system for scientific analysis. A major aspect of the project makes use of the Berkeley Open Infrastructure for Network Computing BOINC platform. The Lattice Project maintains a separate http boinc.umiacs.umd.edu BOINC web site . External links http lattice.umiacs.umd.edu The Lattice Project web site Category Berkeley Open Infrastructure for Network Computing projects Lattice Project, The BOINC topics DEFAULTSORT Lattice Project science software stub pt The Lattice Project es The Lattice Project ... more details
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