Concept

Basis — where it appears

A pair or triple of vectors whose integer combinations are exactly the lattice. A lattice has infinitely many and no arithmetic picks a preferred one until a rule is imposed, which is what reduction supplies.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The hexagonal lattice. Every periodic pattern in the plane repeats on one of five lattices. The classification is by which point symmetries the lattice itself admits, and the five are exhaustive — a sixth would need a rotation order no lattice can carry.

The lattice underneath

Strip a pattern of everything but its repeats and a grid of points is left. That grid is not decoration — it is the object that decides which symmetries the pattern is permitted to have.

lattices · Lattice

Neumann's principle for elastic constants in 6/mmm. Each bar is one operation's contribution to the character of the representation the elastic constants live in — the stiffness of the crystal in every direction and shear. The identity contributes the unconstrained count of 21; every other operation of 6/mmm subtracts from it, and the average over all 24 is 5, which is how many independent components the property may have. The sum is exact in integers and is taken in the lattice basis, so no Cartesian frame is chosen anywhere in it.

A character does not know its basis

The number of independent elastic constants a hexagonal crystal has is computed here from integer matrices in a lattice basis, having never chosen a Cartesian frame. That looks wrong the first time: a physical tensor lives in an orthonormal frame and these matrices are not orthogonal.

point-groups · Neumann's principle

Named alongside it

The objects these essays reach for when they reach for this one.

Character Holohedry Invariant Lattice Neumann principle Representation Tensor Trace Translation group Unit cell

All concepts