One can distinguish positional order ( whether molecules are arranged in any sort of ordered lattice ) and orientational order ( whether molecules are mostly pointing in the same direction ), and moreover order can be either short-range ( only between molecules close to each other ) or long-range ( extending to larger, sometimes macroscopic, dimensions ).

Icosahedral quasicrystals in three dimensions were projected from a six-dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984 .< ref > The tiling is formed by two tiles with rhombohedral shape.

Two stimuli that will change the lattice dimensions of a material are force and temperature.

Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions.

Detail of a bond percolation on the square lattice in two dimensions with percolation probability p = 0. 51 In some cases p < sub > c </ sub > may be calculated explicitly.

For example, for the square lattice Z < sup > 2 </ sup > in two dimensions, p < sub > c </ sub > = 1 / 2, a fact which was an open question for more than 20 years and was finally resolved by Harry Kesten in the early 1980s, see.

A limit case for lattices in many dimensions is given by the Bethe lattice, whose threshold is at p < sub > c </ sub > = 1 /( z − 1 ) for a coordination number z.

For example, p < sub > c </ sub > is not known for bond-percolation in hypercubic lattice in two dimensions.

In three dimensions, the face-centered cubic lattice offers the best lattice packing of spheres, and is believed to be the optimal of all packings.

In this way, the reciprocal lattice of the crystal can be mapped in three dimensions.

The second generation are all subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:

Many of the cross-sections of the Leech lattice, including the Coxeter – Todd lattice and Barnes – Wall lattice, in 12 and 16 dimensions, were found much earlier than the Leech lattice.

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.

This property is also true in 1, 2 and 8 dimensions, with 2, 6 and 240 unit balls, respectively, based on the integer lattice, hexagonal tiling and E8 lattice, respectively.

showed that the Leech lattice is the unique laminated lattice in 24 dimensions.

One of these is the exceptional Leech lattice which has many surprising properties ; due to its existence, the answers to many problems such as the kissing number problem and densest lattice sphere-packing problem are known in 24 dimensions but not in many lower dimensions.

The internal energy of the cubic polytype is slightly lower than that of the hexagonal form and growth rates from molten material in both natural and bulk synthetic diamond production methods are slow enough that the lattice structure has time to grow in the lowest energy ( cubic ) form that is possible for sp < sup > 3 </ sup > bonding of carbon atoms.

Although the neutral ground-state chemical compounds of argon are presently limited to HArF, argon can form clathrates with water when atoms of it are trapped in a lattice of the water molecules.

The crystal lattice of solid americium and its compounds contains intrinsic defects, which are induced by self-irradiation with alpha particles and accumulate with time ; this results in a drift of some material properties.

For example, the allotropes of carbon include diamond ( where the carbon atoms are bonded together in a tetrahedral lattice arrangement ), graphite ( where the carbon atoms are bonded together in sheets of a hexagonal lattice ), graphene ( single sheets of graphite ), and fullerenes ( where the carbon atoms are bonded together in spherical, tubular, or ellipsoidal formations ).

Half or more of the trench is filled with " green " ( unfired ) bricks which are stacked in an open lattice pattern to allow airflow.

Sheet metal or boards are used to route the airflow through the brick lattice so that fresh air flows first through the recently burned bricks, heating the air, then through the active burning zone.

For example, in crystalline solids, these correspond to phonons, which are quantized versions of lattice vibrations.

These electrons are bound to the metal lattice but no longer to an individual atom.

Metals are particularly conductive because there are a large number of these free electrons, typically one per atom in the lattice.

Point defects are defects that occur only at or around a single lattice point.

* Vacancy defects are lattice sites which would be occupied in a perfect crystal, but are vacant.

* Dislocations are linear defects around which some of the atoms of the crystal lattice are misaligned.

In mineralogy, diamond ( from the ancient Greek αδάμας – adámas " unbreakable ") is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice.

Its distinguishing feature is that the schema, viewed as a graph in which object types are nodes and relationship types are arcs, is not restricted to being a hierarchy or lattice.

They are effectively solidified as an ultrahigh density deuterium lump ( Pycnodeuterium ) inside each octahedral space within the unit cell of the palladium host lattice.

The arguments of the lattice theory operations meet and join are elements of some universe A.

NpNi < sub > 2 </ sub > undergoes a similar lattice distortion below T < sub > C </ sub > =

In this model, we have spins arranged on a-dimensional lattice with only nearest neighbor interactions similar to the Ising model.

Something similar can be done for periodic systems, such as electrons moving in the crystal lattice of metals and semiconductors, using the so called quasi-momentum or crystal momentum ( Bloch wave ).

More generally, if L is any positive definite unimodular lattice of dimension 25 with at least 4 vectors of norm 1, then the Weyl vector of its norm 2 roots has integral length, and there is a similar construction of the Leech lattice using L and this Weyl vector.

The Ewald's sphere analysis is similar to that for bulk crystals, however the reciprocal lattice for the sample differs from that for a 3D material due to the surface sensitivity of the RHEED process.

Crystal systems, crystal families, and lattice systems are similar but slightly different, and there is widespread confusion between them: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term " crystal system " is sometimes used to mean " lattice system " or " crystal family ".

In similar fashion, we can define a modulatory space for seven-limit just intonation, by representing 3 < sup > a </ sup > 5 < sup > b </ sup > 7 < sup > c </ sup > in terms of a corresponding cubic lattice.

Therefore, the design of a lattice model more fully reflects the substantive characteristics of a particular employee share option or similar instrument.

The Jordan River Bridge, finished in 1861, employed the patented Remington bridge lattice similar to other bridges he helped construct in Pennsylvania.

Design of aluminum lattice towers is similar to that for steel, but must take into account aluminum's lower Young's modulus.

Two types of strainers are used: lattice towers similar to the suspension towers, but equipped with strainer insulators and of stronger design and much more often pole-type towers with a central pole supported by two diagonal bars.

The aerodynamics are novel, combining vestigial cruciform wings with tail control surfaces of a lattice configuration ( similar devices are used on the R-400 Oka ).

Furthermore, stage ( b ) corresponds to Chris Alexander's " tree " city, while ( c ) is similar to his " lattice " system ( following his dictum " the city is not a tree ").

The largest towers of the Zhoushan Island Overhead Powerline Tie are of a very similar design, but are tubular lattice steel towers.

In contrast, network with a single well-defined scale are somewhat similar to a lattice in that every node has ( roughly ) the same degree.

Its basic format is similar to that of Pac-Man: the player moves around a fixed rectilinear lattice, attempting to visit each location on the board while avoiding the enemies.