Where does lattice collide?

In crystallography a lattice is the regular, endlessly repeating array of points that fixes where a crystal's repeating group of atoms is placed: every point has identical surroundings, and the whole pattern is built by stacking one unit cell along three directions. Bravais's 1850 Mémoire sur les systèmes formés par des points distribués régulièrement sorted these arrays into the fourteen space lattices now named after him. The referent is a geometric scaffold of points in physical space on which atoms sit, described by its symmetry and cell, and it is not an ordered set of elements ranked by a relation, nor an open framework of load-bearing struts.[1]

Unit

One repeating point pattern, set by a unit cell and three basis vectors

Recovery means

Not applicable

Measured by

Determined by X-ray or electron diffraction, which reveals the cell dimensions and the lattice symmetry

In order theory a lattice is a partially ordered set in which any two elements have both a least upper bound, called their join, and a greatest lower bound, called their meet: the structure is defined purely by the ordering relation, with no distances, coordinates, or points in space. Birkhoff's 1940 Lattice Theory gathered this algebra of order into a single discipline, covering distributive and Boolean lattices and their laws. The referent is an abstract algebraic object built from an ordering, such as the subsets of a set ordered by inclusion, and it has nothing in common with a crystal's array of atomic positions or a girder framework beyond the shared word.[2]

Unit

One ordered set in which every pair of elements has a meet and a join

Recovery means

Not applicable

Measured by

Verified by checking that each pair of elements has a least upper bound and a greatest lower bound

In the geometry of numbers a lattice is a discrete subgroup of a real vector space: the set of all whole-number combinations of a fixed collection of basis vectors, forming an evenly spaced grid of points that stretches out in every direction. Minkowski's 1896 Geometrie der Zahlen founded the study of these objects and used them to prove results about whole numbers, seeding today's lattice-based cryptography. The referent is an abstract arithmetic structure studied for its own counting and packing properties, and although a crystal's array is mathematically a close cousin, this meaning lives in pure number theory rather than in a physical solid, an ordering relation, or a load-bearing frame.[3]

Unit

One discrete grid of points, the whole-number span of a set of basis vectors

Recovery means

Not applicable

Measured by

Described by a basis matrix and quantities such as its determinant, shortest vector, and point density

In structural and materials engineering a lattice is an open framework of slender members, or struts, joined at nodes so that loads are carried mainly by stretching and compressing the members rather than bending them: the classic lattice truss and the modern architected lattice both trade solid material for a repeating skeleton of bars. Fleck, Deshpande, and Ashby's 2010 Micro-architectured materials: past, present and future reviewed how such lattices are designed to be stiff and strong while remaining extremely light. The referent is a physical, load-bearing structure of real struts and joints, sized by its stiffness and weight, not a set of atomic positions, an ordering of elements, or a subgroup of integer vectors.[4]

Unit

One repeating framework of struts and nodes, a chosen unit cell tiled through space

Recovery means

Not applicable

Measured by

Assessed by mechanical testing of stiffness, strength, and relative density against a solid of the same material