What is "lattice"

Lattice \Lat"tice\, n. [OE. latis, F. lattis lathwork, fr. latte lath. See Latten, 1st Lath.]

1. Any work of wood, metal, plastic, or other solid material, made by crossing a series of parallel laths, or thin strips, with another series at a diagonal angle, and forming a network with openings between the strips; as, the lattice of a window; -- called also latticework.

   The mother of Sisera looked out at a window, and cried through the lattice.
   --Judg. v. 28.

2. (Her.) The representation of a piece of latticework used as a bearing, the bands being vertical and horizontal.

3. (Crystallography) The arrangement of atoms or molecules in a crystal, represented as a repeating arrangement of points in space, each point representing the location of an atom or molecule; called also crystal lattice and space lattice.

   Lattice bridge, a bridge supported by lattice girders, or latticework trusses.

   Lattice girder (Arch.), a girder of which the wed consists of diagonal pieces crossing each other in the manner of latticework.

   Lattice plant (Bot.), an aquatic plant of Madagascar ( Ouvirandra fenestralis), whose leaves have interstices between their ribs and cross veins, so as to resemble latticework. A second species is Ouvirandra Berneriana. The genus is merged in Aponogeton by recent authors.

Lattice \Lat"tice\, v. i. [imp. & p. p. Latticed; p. pr. & vb. n. Latticing.]

1. To make a lattice of; as, to lattice timbers.

2. To close, as an opening, with latticework; to furnish with a lattice; as, to lattice a window.

   To lattice up, to cover or inclose with a lattice.

   Therein it seemeth he [Alexander] hath latticed up C[ae]sar.
   --Sir T. North.

c.1300, from Old French latiz "lattice," from late "lath, board, plank, batten" (Modern French latte), from Frankish or some other Germanic source, such as Old High German latta "lath;" see lath).

n. 1 A flat panel constructed with widely-spaced crossed thin strips of wood or other material, commonly used as a garden trellis. 2 (context heraldry English) A bearing with vertical and horizontal bands. 3 (context crystallography English) a regular spacing or arrangement of geometric points, often decorated with a motif. 4 (context order theory English) A partially ordered set in which every pair of elements has a unique supremum and an infimum. 5 (context group theory English) A discrete subgroup of Rn which spans the real vector space Rn. vb. 1 To make a lattice of. 2 To close, as an opening, with latticework; to furnish with a lattice.

1. n. an arrangement of points or particles or objects in a regular periodic pattern in 2 or 3 dimensions

2. small opening (like a window in a door) through which business can be transacted [syn: wicket, grille]

3. framework consisting of an ornamental design made of strips of wood or metal [syn: latticework, fretwork]

Lattice may refer to:

Lattice pastry is a pastry used in a criss-crossing pattern of strips in the preparation of various foods. Latticed pastry is used as a type of lid on many various tarts and pies. The openings between the lattice allows fruit juices in pie fillings to evaporate during the cooking process, which can caramelize the filling.

In mathematics, in the field of ring theory, a lattice is a module over a ring which is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedded in a real vector space.

In mathematics, especially in geometry and group theory, a lattice in R is a subgroup of R which is isomorphic to Z, and which spans the real vector space R. In other words, for any basis of R, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell.

Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory and group theory. They also arise in applied mathematics in connection with coding theory, in cryptography because of conjectured computational hardness 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 structure, a 3-dimensional array of regularly spaced points coinciding in special cases with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics.

In mathematics, a lattice is one of the fundamental algebraic structures used in abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.

Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.

In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of R, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.

The theory is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields. In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of Grigori Margulis states that in most cases all lattices are obtained as arithmetic groups.

Lattices are also well-studied in some other classes of groups, in particular groups associated to Kac-Moody algebras and automorphisms groups of regular trees (the latter are known as tree lattices).

Lattices are of interest in many areas of mathematics: geometric group theory (as particularly nice examples of discrete groups), in differential geometry (through the construction of locally homogeneous manifolds), in number theory (through arithmetic groups), in ergodic theory (through the study of homogeneous flows on the quotient spaces) and in combinatorics (through the construction of expanding Cayley graphs and other combinatorial objects).

In musical tuning, a lattice "is a way of modeling the tuning relationships of a just intonation system. It is an array of points in a periodic multidimensional pattern. Each point on the lattice corresponds to a ratio (i.e., a pitch, or an interval with respect to some other point on the lattice). The lattice can be two-, three-, or n-dimensional, with each dimension corresponding to a different prime-number partial" or chroma. (Note that "partial" in the above quote is a misnomer, because partials are sinusoidal components of complex tones, as defined by Helmholtz.)

The points in a lattice represent pitch classes (or pitches if octaves are represented), and the connectors in a lattice represent the intervals between them. The connecting lines in a lattice display intervals as vectors, so that a line of the same length and angle always has the same intervalic relationship between the points it connects, no matter where it occurs in the lattice. Repeatedly adding the same vector (repeatedly stacking the same interval) moves you further in the same direction. Lattices in just intonation (limited to intervals comprising primes, their powers, and their products) are theoretically infinite (because no power of any prime equals any power of another prime). However, lattices are sometimes also used to notate limited subsets that are particularly interesting (such as an Eikosany illustrated further below or the various ways to extract particular scale shapes from a larger lattice).

Examples of musical lattices include the Tonnetz of Euler (1739) and Hugo Riemann and the tuning systems of Ben Johnston. Musical intervals in just intonation are related to those in equal tuning by Adriaan Fokker's Fokker periodicity blocks. Many multi-dimensional higher-limit tunings have been mapped by Erv Wilson. The limit is the highest prime number used in the ratios that define the intervals used by a tuning.

Thus Pythagorean tuning, which uses only the perfect fifth (3/2) and octave (2/1) and their multiples ( powers of 2 and 3), is represented through a two-dimensional lattice, while standard (5-limit) just intonation, which adds the use of the just major third (5/4), may be represented through a three-dimensional lattice though "a twelve-note 'chromatic' scale may be represented as a two-dimensional (3,5) projection plane within the three-dimensional (2,3,5) space needed to map the scale. (Octave equivalents would appear on an axis at right angles to the other two, but this arrangement is not really necessary graphically.)". In other words, the circle of fifths on one dimension and a series of major thirds on those fifths in the second (horizontal and vertical), with the option of imagining depth to model octaves:

A----E----B----F#+
| | | |
F----C----G----D
| | | |
Db---Ab---Eb---Bb

equals the pitch classes (ignoring octaves and notating as ratios relative to an origin of 1/1):

5/3-- 5/4- 15/8-- 45/32
| | | |
4/3-- 1/1-- 3/2--- 9/8
| | | |
16/15- 8/5- 6/5--- 9/5

Erv Wilson has made significant headway with developing lattices than can represent higher limit harmonics, meaning more than 2 dimensions, while displaying them in 2 dimensions. Here is a template he used to generate what he called a “Euler“ lattice after where he drew his inspiration. Each prime harmonic (each vector representing a ratio of 1/n or n/1 where n is a prime) has a unique spacing, avoiding clashes even when generating lattices of multidimensional, harmonically based structure.

Another feature of the template is worth pointing out. Harmonically generated intervals (those with primes in the numerator) will always appear above the fundamental (1/1) and the subharmonics (those with primes in the denominator) below. With a the direction will be both above and to the right, leaving the other quadrant for more complex ratios with the opposite with the subharmonic. This makes it quite easy to understand what is being represented.

Erv Wilson would commonly use 10-squares-to-the-inch graph paper. That way, he had room to notate both ratios and often the scale degree, which explains why he didn't use a template where all the numbers where divided by 2. The scale degree always followed a period or dot to separate it from the ratios. Numerous examples appear throughout the Wilson Archives