Crystallography & Symmetry
patterns derived from their groups
Ladder

Lattice — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. hexagonalequal lengths, angle 120°the arrows are the basis; the shaded region is one unit cell

    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.

    rung 1 · lattices
  2. obliqueno constraint on lengths or anglerectangularangle 90°, lengths freecentred rectangularequal lengths, angle freesquareequal lengths, angle 90°hexagonalequal lengths, angle 120°the arrows are the basis; the shaded region is one unit cell

    Five lattices, and no others

    A repeating grid can be oblique, rectangular, centred, square or hexagonal. That is the complete list for the plane, and the argument that closes it is a page long.

    rung 2 · lattices
  3. centred rectangularequal lengths, angle freethe arrows are the basis; the shaded region is one unit cell

    The cell is a choice, the lattice is not

    Every lattice has infinitely many unit cells and infinitely many bases, and crystallography picks one by convention. Knowing which convention is in force is the difference between a symbol that means something and a symbol that means nothing.

    rung 3 · lattices
  4. as given — 3 reduction steps to go|a| = 5.831 |b| = 3.606reduced — shortest, then shortest independent|a| = 1.000 |b| = 1.000same lattice, same cell area, different descriptiondeterminant 1

    Reduction, and the shortest basis

    Every lattice has infinitely many bases and no arithmetic picks a preferred one — until a rule is imposed. Reduction is that rule, it terminates in a handful of steps, and it is what lets a database decide whether two reported crystals are the same crystal.

    rung 4 · lattices
  5. 5 families of rows1,00,11,11,-12,1one point per family, at 1 ÷ spacingthe geometric construction and the algebra agree exactly

    The dual lattice, as a construction

    The reciprocal lattice is usually introduced as a formula and then used as a fact. Building it instead — one point per family of lattice rows, at the inverse of the spacing — makes every property it has obvious rather than memorable.

    rung 5 · lattices

All ladders