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.
A chain modulated at q = 0.211. The lower row is the lattice: 34 sites, evenly spaced. The upper row is the structure: the same sites displaced by a wave of amplitude 0.12 of a spacing and wavevector 0.211, drawn through them. Because 0.211 is not a ratio of small whole numbers, no cell of any size holds the structure — the displacement pattern never repeats — and yet the atoms are nowhere near random: each one is exactly where a single sine wave says it should be. That is what an incommensurately modulated crystal is, and its diffraction pattern is sharp.

Modulation

  1. 1 The satellites that need a second integer
  2. 2 The extra dimension that makes it periodic
  3. 3 Two lattices, one crystal, and no cell at all
  4. 4 Superspace groups in the plane
  5. 5 Every fraction holds a window
  6. +8 more
13 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. +8 more
13 essays · restriction 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. +7 more
12 essays · restriction One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does.

Centring

  1. 1 Centring, and why cm is not pm
  2. 2 Centring, counted as a sublattice
  3. 3 Why the bigger cell wins
  4. 4 Only two and three can centre a lattice
  5. 5 The body-centred hypercube is the checkerboard lattice
  6. +6 more
11 essays · lattices 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. +6 more
11 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. +6 more
11 essays · restriction 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. +6 more
11 essays · operations 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. +6 more
11 essays · space-groups 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 The 4₁ screw axis. 1 of the eleven screw axes a lattice permits, each drawn as the helix it is: 4₁. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

Screws and glides

  1. 1 Turning and climbing at once
  2. 2 Eleven ways to turn while climbing
  3. 3 Reflect, then slide by half of something
  4. 4 The operations nobody put in
  5. 5 The plane that carries two glides
  6. +6 more
11 essays · space-groups 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. +5 more
10 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. +5 more
10 essays · restriction ⟨cos Φ⟩ against κ, 379 triplets. The mean cosine of the triplet, binned by the concentration κ = 2|E₁E₂E₃|/√N, for the 379 triplets of a structure of 24 atoms whose reflections all exceed |E| = 1.2. The curve is Cochran's I₁(κ)/I₀(κ), computed from the distribution and not fitted to anything; the points are measured, with the number of triplets in each bin printed above. They agree to 0.1 root-mean-square. The measured points sit slightly above the curve throughout, which is the finite structure showing: Cochran's derivation assumes atoms placed at random and there are only 24 of them.

Direct methods

  1. 1 Three phases that do not move when the origin does
  2. 2 The formula that has the answer already
  3. 3 The solver that knows no symmetry
  4. 4 The relation that can say no
  5. 5 The relation that is an equality
  6. +5 more
10 essays · diffraction 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 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. +5 more
10 essays · aperiodic 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. +5 more
10 essays · aperiodic p4m: 4 and 4. The standard motif — three points in no particular arrangement — repeated by p4m, with each copy coloured by the sign of the area of the triangle it makes. 4 copies have one sign and 4 the other, because the group contains an operation that reverses orientation. A structure built from one enantiomer cannot sit here: the group would put its mirror image in the same crystal. The colours were computed from the coordinates rather than assigned.

Chirality

  1. 1 The groups a single hand may sit in
  2. 2 How chiral, as a number
  3. 3 A hand made of pieces that have none
  4. 4 Chiral in the plane is not chiral in the room
  5. 5 A thread's hand is not a choice
  6. +4 more
9 essays · space-groups What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Counting

  1. 1 Counting what a group cannot tell apart
  2. 2 Every colour count at once
  3. 3 An orbit is what the invariants cannot tell apart
  4. 4 The table that decides every action
  5. 5 Swapping the species does not halve the count
  6. +4 more
9 essays · operations 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

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