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Topic: Square lattice

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	Square (geometry) - Wikipedia, the free encyclopedia
		In plane geometry, a square is a polygon with four equal sides and equal angles.
		Squares are regular quadrilaterals, rectangles, rhombi, kites, parallelograms, and isosceles trapezoids/isosceles trapezia.
		The diagonals of a square are equal and conversely, if the diagonals of a rhombus are proven to be equal, then that rhombus must be a square.
	en.wikipedia.org /wiki/Square_(geometry) (129 words)

	Centered square number - Wikipedia, the free encyclopedia
		A centered square number is a centered figurate number that represents a square with a dot in the center and all other dots surrounding the center up to a certain city block distance, i.e.
		a diagonal square region of a square lattice, or an upright square region of a "centered square lattice".
		A centered square prime is a centered square number that is prime.
	en.wikipedia.org /wiki/Centered_square_number (277 words)

	The Geometry Junkyard: Geometry of Numbers
		More formally, a lattice can be defined as a discrete subgroup of a finite-dimensional vector space (the subgroup is often required not to lie within any subspace of the vector space, which can be expressed formally by saying that the quotient of the space by the lattice is compact).
		Other types of lattices arise in crystallography and in sphere packing, where they are used to describe the locations of atoms or spheres.
		Polyominoes, figures formed from subsets of the square lattice tiling of the plane.
	www.ics.uci.edu /~eppstein/junkyard/lattice.html (669 words)

	Wallpaper Groups: lattices
		That's because in that case, the points in the lattice nearest any one point in the lattice are the vertices of a regular hexagon.
		Lattices are special kinds of patterns, and their symmetry groups of five of these 17 groups.
		Since a square is simultaneously a rectangle and a rhombus, square lattices have all the symmetries of both rectangular lattices and rhombic lattices.
	www.clarku.edu /~djoyce/wallpaper/lattices.html (864 words)

	Flexibility of a lattice (Site not responding. Last check: 2007-10-09)
		A stiffened square of a lattice is said to be redundant if the removal of the stiffness of this square leads to an equivalent lattice.
		The number of squares that may be removed from the lattice is actually less than the number of redundant squares, but equal to the number of loops of the graph.
		The set E × E can be represented by a square lattice of suitable size, where each row and column represent an element among E, in such a way that the row and column of same rank represent the same object.
	www.eleves.ens.fr /home/ollivier/treillis/texte/treillis.html.en (2246 words)

	Bravais Lattices (Site not responding. Last check: 2007-10-09)
		For example, the hexagonal lattice is an example of a 2-dimensional Bravais lattice, but the honeycomb lattice is not since neighboring points are not equivalent (it is an example of a compound structure, which we will discuss next).
		For the bcc lattice, a point is added to the center of each cube; alternatively, this lattice can be viewed as two sets of sc sublattices, A and B, that inter-penetrate (however, note that there is no way to distinguish these two sets).
		This is completely obvious in 2-D, where the W-S cell is a square for a square lattice, a hexagon for a hexagonal lattice, and a rectangle for a rectangular lattice.
	carini.physics.indiana.edu /p615/lattices.html (773 words)

	Appendix 6: Site and Bond Percolation Thresholds (Site not responding. Last check: 2007-10-09)
		Consider a finite square lattice in which some, but not necessarily all, sites are occupied (by spins), and in which any two spins occupying nearest neighbour sites are connected by an open bond.
		Consider a finite square lattice in which all sites are occupied (by spins), but in which any two spins occupying nearest neighbour sites may or may not be connected by an open bond.
		The concept of a "cluster of spins" arose in the consideration of a pure lattice, in which all lattice sites are occupied by spins (of various spin values) and all spins are connected by energetic bonds.
	hermetic.nofadz.com /compsci/thesis/app6.htm (1042 words)

	TOP Trap Experiments
		The subsequent evolution of this pseudo-spin-1/2 superfluid towards a state of offset square lattices involves an intriguing interplay of phase-separation and -mixing dynamics, both macroscopically and on the length scale of the vortex cores, and a stage of vortex turbulence.
		Stability of the square lattice structure is confirmed via the application of shear perturbations, after which the structure relaxes back to the square configuration.
		While undisturbed lattices form a regular hexagonal lattice structure, large-amplitude quadrupolar shape modes are shown to induce a wealth of nonequilibrium lattice dynamics.
	jilawww.colorado.edu /bec/top_trap_macromedia (1196 words)

	Lattice Geometries
		The method of representation of lattice geometries described in this article was used in the software developed by the author to simulate the behavior of magnetic material by means of Ising and Potts spin models, as described in detail in his M.Phil.
		The dimensionality of a lattice is the dimensionality of the Cartesian space used to define that lattice.
		The "diamond" lattice, of dimensionality 3, is the geometry formed by carbon atoms in a diamond crystal.
	www.hermetic.ch /compsci/lattgeom.htm (1805 words)

	[No title]
		Thus, we have identified 5 types of 2-D lattices: the general oblique lattice that does NOT have reflection or anything beyond 360o and 180o rotational symmetries; the square lattice, the hexagonal lattice, the rectangular lattice, and the centered rectangular lattice.
		For a hexagonal lattice, (a1(= (a2(, and that the angle, (, between a1 and a2, is 120o (or we could choose 60o) For a rectangular lattice, (a1(((a2(, and that the angle, (, between a1 and a2, is 90o.
		In this lattice, in addition to having lattice points at all the corners of a cube, there is a lattice point right in the middle of the volume of the cube.
	www.cbu.edu /~jholmes/P353/N100CrysStr.doc (2065 words)

	Four Pegs That Form a Square
		Four pegs on a square grid form a square, i.e., the pegs are located at the corners of a grid square.
		In part it is. We may think of the original square as being the unit - the smallest - square of the grid.
		Now since the minimal lattice of a square is the standard square lattice with that square as one of the cells, it is a trivial consequence of the above lemma that it is impossible to obtain a square of a different size or orientation from the original by peg-jumping moves.
	www.cut-the-knot.com /proofs/PegSquare.shtml (628 words)

	3.3- Local Topologies
		In contrast to the rapid, exponential growth of populations in homogeneous systems, the population growth on the square lattice is much slower, and quadratic in form.
		The boundary of the expanding infected region is roughly circular (despite the fact that the infection neighborhood is square) and fairly sharp.
		Suppose that a single node i in the square lattice is infected with A.
	www.research.ibm.com /antivirus/SciPapers/Kephart/ALIFE3/alife3-node6.html (1104 words)

	Chapter 7: Summary of Results and Comparison with Published Results
		The diamond lattice has the same coordination number as the square lattice (4), but the critical temperatures are different (2.269 and 2.704).
		Sections 4.3 and 4.4 present details of the dependence of the critical temperature of a dilute square lattice on the degree of dilution (for both site-diluted and bond-diluted lattices).
		For the Ising model the slope of the log-log plot of the magnetization against time was measured using a square lattice of size 128 and initial magnetization of 0.02 at the critical temperature, and the slopes of the log-log plots of the autocorrelation and the 2
	www.hermetic.ch /compsci/thesis/chap7.htm (951 words)

	Prof. Buks van Rensburg
		A maze is a special kind of random tree: in particular, it is a spanning tree of a square in the square lattice (such a maze would have only one way out from the center).
		The interior of a closed loop in the square lattice (or a polygon) is a Disk.
		In this example, a square lattice polygon was randomized by subjecting it to a Metropolis Monte Carlo algorithm.
	www.math.yorku.ca /Who/Faculty/Rensburg/menu.html (355 words)

	I
		Beginning with a single lattice point, all of the other lattice points in the array can be generated by shifting the lattice point by the translation vector.
		Note that a lattice point is not a physical object, it is simply a point in space upon which a real object may be placed.
		An alternative description would be to draw the unit cell as a rectangle, which contains a lattice point at each corner of the unit cell and one in the center of the unit cell (for a total of two lattice points per unit cell and an area twice that of the rhombus unit cell).
	www.chemistry.ohio-state.edu /~woodward/ch754/sym_2d.htm (1524 words)

	Lattice Square (Site not responding. Last check: 2007-10-09)
		This pattern was traditionally done with each quarter having four trapezoids and an inset square.
		Cut apart on the square lines and on the marked diagonals, between the rows of stitching.
		Piece four quarter blocks, using squares and half squares, then sew two rows of two quarter blocks each, alternating seam pressing directions by row.
	www.quilterscache.com /J_L/LatticeSquareBlock.html (187 words)

	[No title]
		Using an infinite number of copies of given prototiles, one can tile the plane completely (centers of tiles form the square lattice) and without overlaps (except boundaries of tiles) such that all matching rules are satisfied.
		If a smaller square is a 1-square (with $n/2$ in (19)), then we call it a {\bf bad} square.
		Let $k$ be the smallest number such that all squares of size $n/2^{k}$ which are not final ones, are connected to three bad squares and one good square.
	www.ma.utexas.edu /mp_arc/papers/97-381 (5218 words)

	: Package edu.clarku.tools (Site not responding. Last check: 2007-10-09)
		A two-dimensional square lattice with a fixed size.
		Thrown to indicate the bounds specified for the lattice are invalid: for example when the minimum value is greater than the maximum value.
		Thrown to indicate that an error that occured relating to the fact that the lattice size is fixed.
	webphysics.davidson.edu /WebTalks/clark/docs/edu/clarku/tools/package-summary.html (179 words)

	Redwood Gazebos by Sylvan Woods
		A lattice skylight insert is an option, along with free-standing bar and two standard, 24" high bar stools.
		Constructed with 8' long lattice privacy panels on two sides, three 4x4 corner posts, and a flat, 1x4 slat roof, the cabana goes with any size spa.
		The roof is 10' square and the walls are 98" between corner posts.
	www.sylvanwoods.com /sylvanwoods/gazebos.html (545 words)

	2 - Lecture notes for GEOL3010
		In the 2-D example of a square or isoclinic plane lattice, we use the rule or geometric operation of a translation to move the snail an equal distance in an XY coordinate system.
		The area (or volume in the case of 3-D) that comprises the unit length's distances between lattice points is known as the unit cell.
		A primitive lattice may also be described as a rectangular lattice with a centered (c) unit cell.
	www.gly.uga.edu /schroeder/geol3010/3010lecture02.html (632 words)

	Hexagonal to square lattice conversion in bilayer systems (Site not responding. Last check: 2007-10-09)
		The results clearly show that the character of the reconstructive hexagonal to square lattice conversion in bilayer colloid systems is potential dependent.
		In this case the hexagonal to square lattice transition is first order.
		from the square lattice by strong first order transitions, and from the hexagonal lattice by either weak first or second order transitions.
	mrsec.uchicago.edu /Publications/Rice/rice039.html (249 words)

	George Bell's Triangular Peg Solitaire Page
		It is common to number the holes consecutively, however we use a notation that is derived from that used for square lattice boards.
		The board Rhombus(6) is in fact the familiar 6x6 square board with the addition of moves along one diagonal.
		For it has been shown in the case of a square lattice, the general problem of determining if an arbitrary pattern of pegs can be reduced to one peg is NP complete [4].
	www.geocities.com /gibell.geo/pegsolitaire/tindex.html (7118 words)

	node4.html (Site not responding. Last check: 2007-10-09)
		Fig 4 portrays a portion of a square lattice upon which a site-percolation process has been superimposed.
		The three frames represent the same region of the lattice at three different concentrations: The fraction of filled sites (p) grows from 0.25 to 0.50 to 0.75 in going from Fig 4(a) to (b) to (c).
		The growth in volume of the infinite cluster, for bond percolation on the square lattice(Fig 1), is shown as the heavy curve in Fig 5.
	www.people.fas.harvard.edu /~wu2/paper1/node4.html (908 words)

	Square Lattice 3-Part Specifications
		Roof deck shall be roll formed 3004H36 Aluminum tube 1-1/2” square, spaced 4.5” on centers with vinyl end caps,
		Lattice shall be attached with mechanical fasteners field applied to provide structural integrity for the completed assembly.
		Confirm that surrounding area is ready for the shade structure installation.
	www.americana.com /commercial/sqlatspecs.php (340 words)

	Polymorf - Knowhere - Square spaceframe lattices (Site not responding. Last check: 2007-10-09)
		Square spaceframe lattices are comprised of a pair of two-way square grid layers that are
		embedded in the lattice in half thereby exposing their square shaped cross-sections.
		The Square-2 lattice design is the most common spaceframe structure in use today.
	www.polymorf.net /engineer66.htm (124 words)

	Wallpaper classification
		In the glide reflection column, an asterisk * means that there is a glide mirror which is not a reflection mirror of the pattern.
		The lattice type of a pattern denotes the most general translation lattice which can occur for the pattern.
		A pattern with square lattice type can occur only with a square translation lattice.
	www.math.uga.edu /~clint/2005/5210/wallpaper.htm (210 words)

	[No title]
		Equivalently, this is asking for the densest packing of the plane by disjoint equilateral triangles translated to the vertices of the integer square lattice.
		I think this is the maximum possible size for a square contained in a triangle that avoids the integer points, but have no proof.
		Since it doesn't contain the unit square, it in particular doesn't contain a unit square aligned with the coordinate axes in this placement, and therefore there is enough room for its top corner to poke between the integer points on the line y=1.
	www.ics.uci.edu /~eppstein/junkyard/tripack.html (1124 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		A square lattice is a simple two-dimensional grid in which each point is surrounded by four other evenly spaced points.
		The original hypothesis was that the average distance grew proportionately to N3/4 (N3/2 for distance squared, due to the fact that na nb is equal to na + b), where N is the number of steps.
		The long-distance behavior of the mean square distance is consistent with the simple power law described in Equation (1) where 2g is approximately equal to 1.141.
	home.comcast.net /~mikey-soft/introduction.doc (2674 words)

	Citebase - An integrable generalization of the Toda law to the square lattice
		An integrable generalization of the Toda law to the square lattice
		The exponential solutions describe the evolution of one and two smooth two-dimensional shock waves on the square lattice.
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:nlin/0409050 (502 words)

	Crystals
		In describing this structure we must distinguish between the pattern of repetition (the lattice type) and what is repeated (the unit cell).
		For a square lattice as shown on the left, the coordination number is 4 (the number of circles touching any individual).
		If we stack the cells into a lattice we notice that the atoms form diagonal layers - the reason for the colors is to make these stand out.
	www.chemistry.nmsu.edu /studntres/chem116/notes/crystals.html (1888 words)

	Dayna's Crochet (Site not responding. Last check: 2007-10-09)
		12 inch square (with 7" and 8" options), easy and light.
		Round 12: Sl st in corner loop, ch 2, 4 hdc in corner, [2 hdc in each loop across, 5 hdc in corner] 3 times and * to * once more, join to 2nd ch of beg ch 2.
		Round 12 can be done with sc or dc depending on the size of your square.
	www.texasdg.net /daynascrochet/pattern.asp?id=58 (380 words)

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