A Bravais lattice is a group of points that can fill a space with no gaps. Points can be centered in different ways, and each point is bounded by the same number of sides. Bravais lattices can be two or three-dimensional and have different shapes and symmetry rules. They are used in geometry and by researchers working with crystals.
The term lattice generally refers to a group of points, which can be part of a mathematical design or a physical crystal, for example. A Bravais lattice, in two or three dimensions, typically fills a space with no gaps, while the dots can be centered within the structure in four different ways. If the lattice points are placed only in the corners, this is called primitive centering. Body-centered points are in the middle of a lattice cell, while points can also be centered on the face or side of the cell; sometimes there are points in the center of all lattice faces.
Each point is normally bounded by the same number of sides as another in a lattice; the distance and direction from each other are also typically the same. The Bravais lattice, first studied by Auguste Bravais in the mid-1800s, can consist of an infinite number of dots, meaning there is no limit to the number of dots that can be included. It is often used in geometry and by researchers working with crystals, where each point typically represents an atom.
A two-dimensional Bravais lattice is usually square or rectangular in shape; the configuration is generally determined by the lengths of the lines. The lines are often at 90° angles to each other, but if they are at a 120° angle, a hexagonal lattice can form. If all sides are at right angles, lines can be drawn to show the symmetry of a shape formed by the Bravais lattice.
Shapes can have a dual axis of rotation if they include a symmetrical dividing line and are rotated 180°. Squares, for example, can be rotated 90° and folded, meaning they have a fourfold axis, while the hexagonal lattice, with threefold symmetry, can be rotated in 120° steps centered on each lattice point. A three-dimensional Bravais lattice generally has the same symmetry rules. Points can be attributed only to the corners, to the center of the cell, to the center of each face or to the center of the faces.
A cubic Bravais lattice is one of seven different shapes, which are typically defined by the presence of one or more alternating patterns of points. Shapes include the tetragonal lattice of Bravais as well as 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 naturally occurring substances.
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