Crystallography & Symmetry
patterns derived from their groups
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Series

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
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.

Lattice

  1. 1 The lattice underneath
  2. 2 Five lattices, and no others
  3. 3 The cell is a choice, the lattice is not
  4. 4 Reduction, and the shortest basis
  5. 5 The dual lattice, as a construction
  6. +6 more
11 essays · lattices m3̅m → 4mm: 6 domain states. The transition from class m3̅m to class 4mm loses 40 of the parent's 48 operations, so the child has index 6 and the crystal comes apart into 6 domain states. Each colour is one state — one coset of 4mm in m3̅m — and each holds the same 8 poles. The lost operations are what carries one state onto another, and they survive in the crystal as the relation between its domains rather than as symmetries of any part of it. The descent changes the crystal system, so the states differ in shape as well as in orientation and the transition is ferroelastic.

Domains

  1. 1 How many domains a transition makes is an index
  2. 2 The descent of symmetry is a lattice, not a tree
  3. 2 The domains a lost translation makes, which nothing optical can see
  4. 3 Two hundred and forty-seven descents, or two hundred and twelve
  5. 5 The strain that arrives with the transition
  6. +5 more
10 essays · applied The thirty-two crystal classes. Every crystallographic point group, as a stereogram. Each was found by enumerating the subgroups of m3̅m and of 6/mmm, and each diagram is the orbit of one general direction under the group, filled where the pole is in the upper hemisphere and open where it is in the lower — which is the only thing in the picture that tells a rotation from a rotoinversion.

Crystal classes

  1. 1 Thirty-two, and no others
  2. 2 A fingerprint that gave the right answer
  3. 2 What a trace decides
  4. 3 Reading a class off its own axes
  5. 3 The holohedry is the ceiling
  6. +4 more
9 essays · point-groups A lattice and its reciprocal. The reciprocal lattice is where a crystal scatters. A long axis in the crystal gives closely spaced spots and a short one gives widely spaced spots, so the diffraction pattern is the structure turned inside out.

Diffraction

  1. 1 The reciprocal lattice
  2. 2 Systematic absences
  3. 3 The phase problem
  4. 4 What a powder pattern loses
  5. 5 The reflections that are not there
  6. +4 more
9 essays · diffraction A fundamental domain for p4m. One representative from every orbit of p4m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

Fundamental domain

  1. 1 The fundamental domain
  2. 2 The points a group treats differently
  3. 3 Domains of a subgroup
  4. 4 One part in however many, and why it is never quite that
  5. 5 Every wall names a generator
  6. +4 more
9 essays · operations the honeycomb net, unfolded over 3×3 cells. The infinite graph the quotient graph names, drawn over 3 by 3 cells with the home cell outlined. Each edge of the quotient becomes one edge per cell, running to the cell its voltage names; the drawing adds coordinates the net does not have, and they are the placement in which every vertex sits at the average of its neighbours. two vertices, three edges, degree three — the graph of graphene and of every hexagonal mesh.

Nets

  1. 1 A structure with the distances thrown away
  2. 2 Counting outwards
  3. 3 The placement nobody chose
  4. 4 A net is a choice of what counts as a bond
  5. 5 The level that does not move
  6. +4 more
9 essays · applied The seventeen wallpaper groups. Every way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.

Seventeen

  1. 1 The seventeen
  2. 2 Reading Hermann–Mauguin
  3. 3 p3m1 and p31m
  4. 4 The classification proof, one branch at a time
  5. 5 Orbifold notation, the shorter language
  6. +4 more
9 essays · classification A Penrose tiling, 4 inflations. Two rhombs, subdivided into smaller copies of themselves over and over. The result covers the plane, has five-fold symmetry about its centre, and never repeats — there is no translation that maps it to itself.

Aperiodic

  1. 1 Penrose tilings
  2. 2 Inflation, and where the golden ratio comes from
  3. 3 Order is not periodicity
  4. 4 Matching rules, and what actually forces aperiodicity
  5. 5 Cut and project
  6. +3 more
8 essays · aperiodic 60 vertices, 12 pentagons. A closed net with three edges at every vertex: 60 vertices, 90 edges and 32 faces, of which 12 are pentagons and 20 are hexagons. The pentagons are picked out in the second colour. Their number is not a property of this cage — it is twelve for every closed trivalent net of pentagons and hexagons, at any size, and the hexagon count is free.

Curvature

  1. 1 Twelve pentagons, and no way round them
  2. 2 As many heptagons as pentagons
  3. 3 The twelve belongs to the vertex
  4. 4 The surfaces a count by genus skips
  5. 5 A gap the sphere does not have
  6. +3 more
8 essays · restriction Why the seventeen is a number at all. The classification is finite because three counts in a row are finite, and the first two are where the work is. Finitely many lattice types, because a lattice's symmetry group is a finite group of integer matrices; finitely many such groups, by Minkowski's lemma and his bound; and finitely many ways to attach translations to each, which is the extension problem. Every step is a count this site makes elsewhere — five, thirteen, seventeen — and this is the reason each of those searches was allowed to stop.

Finiteness

  1. 1 Why there is a list at all
  2. 2 Reduction modulo three
  3. 3 The average that makes it finite
  4. 4 The same group means the same pattern
  5. 5 What forces a lattice
  6. +3 more
8 essays · restriction At which indices a group contains a copy of itself. A filled circle where the group has a subgroup of that index which is the same plane group again. The groups with no rotation past a half-turn take every index — the lattice can be stretched along one direction by any factor. The four-fold groups take the sums of two squares and the three- and six-fold groups take the Loeschian numbers, because a sublattice invariant under a quarter or a third of a turn is an ideal in the Gaussian or Eisenstein integers and its index is a norm. The groups with mirrors take fewer still, and p4g takes only the squares.

Isomorphic subgroups

  1. 1 The same group in a bigger cell
  2. 2 A bigger cell, and sometimes the mirror
  3. 3 A screw that contains its own mirror image
  4. 4 The primes a cell can grow by
  5. 5 An ideal across and a prime along
  6. +3 more
8 essays · space-groups A 5-fold cluster in a crystal that has no 5-fold axis. A cluster of 10 points with an exact 5-fold axis at the centre of each cell, repeated by the lattice. Two measurements, on the same points. The cluster is carried onto itself by a turn of 72° to within 2e-16 of a cell — exact, as far as the arithmetic goes. The pattern is not: applying the same turn about a lattice point sends some atoms 1.19 of a cell from the nearest atom, which is most of the way across it. Both are true at once. The axis is a symmetry of the contents of one cell and not of the crystal, which is what non-crystallographic symmetry means and why a virus with a sixty-fold capsid can crystallise in an ordinary space group.

Local symmetry

  1. 1 A fivefold axis in an ordinary crystal
  2. 2 The symmetry of an average
  3. 3 The most of an icosahedron a crystal can keep
  4. 4 Five copies, and the gap they leave
  5. 5 What a molecule gives up to sit in a crystal
  6. +3 more
8 essays · restriction One layer, and the two ways to sit on it. A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours.

Packing

  1. 1 Two stackings, one density
  2. 2 How many polytypes there are
  3. 3 The densest lattice in the plane
  4. 4 The four plane groups a molecule packs in
  5. 5 The densest packing of a shape that is not a disc
  6. +3 more
8 essays · applied A diffraction pattern with tenfold symmetry. Sharp spots, arranged with a symmetry that no periodic crystal can have. When the ten-fold case was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered. The star of reciprocal vectors is a parameter here, so the eight- and twelve-fold patterns that were found afterwards come out of the same call.

Quasicrystals

  1. 1 What Shechtman measured
  2. 2 The smallest quasicrystal
  3. 3 Icosahedral symmetry
  4. 4 The freedom a crystal has not
  5. 5 Six integers, and the lattice that holds them
  6. +3 more
8 essays · aperiodic The five rotations a lattice will carry. One motif and every rotation a plane lattice permits: orders 1, 2, 3, 4, 6, and nothing else up to 12. Each panel turns the motif by its own operation as many times as the order allows, on the lattice that operation requires — oblique for the identity and the half turn, hexagonal for the third and the sixth of a turn, square for the quarter. The trace printed under each is the sum of the diagonal of the operation's matrix written in the lattice's own basis, and it is a whole number in every panel, which is the entire content of the crystallographic restriction. The list of orders is produced twice, once from that trace condition and once from the degree of a cyclotomic polynomial, and the figure refuses to draw if the two disagree.

Restriction

  1. 1 The crystallographic restriction
  2. 2 Why five-fold is impossible
  3. 3 The restriction in three dimensions
  4. 4 Where five-fold becomes legal
  5. 5 Thirteen ways to hold a lattice
  6. +3 more
8 essays · restriction The Wigner–Seitz cell of the hexagonal lattice. Every point closer to the central lattice point than to any other. The faint lines run to the 6 neighbours whose perpendicular bisectors bound the region; every other lattice point is cut off by one of them. The cell has exactly the area of a unit cell — asserted while the figure is drawn, against √det G computed from the metric — and it carries all 12 of the lattice's symmetries, which a conventional cell need not. Nothing was chosen to build it: no basis, no axes, no convention. Two people who agree about the lattice cannot disagree about this cell.

Wigner–Seitz cells

  1. 1 The cell nobody chose
  2. 2 The zones above the first
  3. 3 Five parallelohedra, and no others
  4. 4 Covering and packing want different lattices
  5. 5 A bigger cell, a smaller zone
  6. +3 more
8 essays · lattices The Fibonacci chain: p(n) = n + 1. The number of distinct windows of each length in the Fibonacci chain, measured by sliding a window along 46,368 tiles. Every count is checked against the same count on half the chain, and only lengths where the two agree are drawn — a factor count on a finite word is otherwise a lower bound wearing the clothes of an answer.

Complexity

  1. 1 n plus one, and no fewer
  2. 2 Every patch comes back
  3. 3 How many patches of each size
  4. 4 Three gaps, and never four
  5. 5 The average is the same wherever it is taken
  6. +2 more
7 essays · aperiodic Order 5: 32,768 arrangements. An Aztec diamond of order 5, with every possible dimer drawn at an opacity equal to the fraction of arrangements it appears in — a probability computed exactly, by counting the arrangements of the region with that dimer's two sites removed, rather than sampled. The four corners come out nearly certain and the middle nearly even, with a circle between them. The most certain dimer here occurs in 0.97 of the arrangements, which is 1 − 2⁻5 exactly, so nothing is frozen at any finite size.

Entropy

  1. 1 How many arrangements one rule allows
  2. 2 The arrangements a crystal keeps at absolute zero
  3. 3 A facet with no energy in it
  4. 4 The ice rule is a conservation law
  5. 5 What a defect costs the count
  6. +2 more
7 essays · aperiodic Every solution of the axis equation. The integer solutions of 2 − 2/N = Σ(1 − 1/nᵢ), which is what counting the pairs (rotation, fixed pole) two ways gives. Two classes of axis force n₁ = n₂ = N and give the cyclic groups; three classes give the dihedral family and exactly three sporadic answers — (2, 3, 3), (2, 3, 4) and (2, 3, 5), of orders 12, 24 and 60, which are the rotation groups of the tetrahedron, the octahedron and the icosahedron. Four classes are impossible, because four terms of at least a half already exceed the left-hand side. Nothing about crystals has been used.

Finite groups

  1. 1 Before the lattice has a say
  2. 2 Five solids from one inequality
  3. 3 Eleven, eleven and ten
  4. 4 Seven friezes round a cylinder
  5. 5 The tube has a screw no lattice allows
  6. +2 more
7 essays · restriction Σ5: two square lattices at 36.87°. Two square lattices, one turned through 36.87° about a shared point. At this angle one point in 5 lands exactly on a point of the other lattice — 29 of the 149 drawn — and those shared points are themselves a lattice, the coincidence site lattice, of index 5. The angle comes from tan(θ/2) = 1/3, and Σ is the odd part of 3² + 1² = 10. Nothing here is measured: whether a point is shared is decided by an integer congruence.

Interfaces

  1. 1 Turn a lattice against itself and almost nothing lines up
  2. 2 Every coincidence index is odd, and in the plane most of them do not exist
  3. 3 Two different lattices never coincide, and the question becomes how nearly
  4. 4 A small angle is a row of dislocations
  5. 5 The dislocations a boundary allows
  6. +2 more
7 essays · applied

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