The term lattice generally refers to a cluster of points, which can be part of a mathematical drawing or a physical crystal, for example. A Bravais lattice, whether it is in two or three dimensions, typically fills a space without any gaps, while the points can be centered within the structure in four different ways. If the lattice points are placed only in the corners, it is called primitive centering. Body centered points are located in the middle of a lattice cell, while points can also be centered on the cell face, or side; sometimes there are points in the center of all faces of the lattice.
Each point is normally bordered by the same number of sides as another in a lattice; the distance and direction of each relative to one another is typically the same as well. The Bravais lattice, first studied by Auguste Bravais in the mid-1800s, can consist of an infinite number of points, which means there is no limit to how many can be included. It is often used in geometry as well as by researchers working with crystals, in which each point typically represents an atom.
A two-dimensional Bravais lattice is usually either square or rectangular in shape; the configuration is generally determined by the lengths of the lines. The lines are often at 90° angles to one another, but if they are at a 120° angle, a hexagonal lattice can be formed. If all of the sides are at right angles, then lines can be drawn to show the symmetry of a shape formed by the Bravais lattice.
Shapes can have a two-fold rotation axis if they include a symmetrical dividing line and are turned 180°. Squares, for example, can be turned 90° and folded, which means they have a four-fold axis, while the hexagonal lattice, with a three-fold symmetry, can be rotated in 120° steps centered on each lattice point. A three-dimensional Bravais lattice generally features the same rules concerning symmetry. Points can be attributed to the corners only, the cell center, the middle of each face, or the center of the faces.
A cubic Bravais lattice is one of seven different forms, which are typically defined by the presence of one or several alternate patterns of points. The forms include the tetragonal Bravais lattice as well as the orthorhombic, hexagonal, trigonal, monoclinic, or triclinic types. In addition to their graphical and mathematical representations, each of these is often attributed to the crystalline structure of specific substances found in nature.
By Andrew Kirmayer