Crystallography & Symmetry
patterns derived from their groups
Depth

Series — page 2

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 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 Two half-turns make a translation. The half-turn about (0.25, 0.25) followed by the half-turn about (0.75, 0.5) is the translation by (1, 0.5) — twice the vector between the two centres, and not the vector itself. The open lens is a third centre, and it is not the midpoint of the two drawn: it is where the half-turn about the first lands when it is composed with one repeat vector of the lattice, which is half a repeat along. That is the step that puts two-fold centres on the half lattice and gives a p2 cell four inequivalent ones. Both the translation and the forced centre are computed from the operations and compared with the construction in exact rational arithmetic.

Composition

  1. 1 Where the product is
  2. 2 Three reflections, and never four
  3. 3 The four groups with a centre
  4. 4 Closing the plane from two centres
  5. 5 The axis a product lies on
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8 essays · operations p4: (1/2, 0) has a star of 2. The first Brillouin zone of the square lattice, with the reciprocal lattice points at its corners and centre, and the whole star of the wavevector (1/2, 0) under p4. The star has 2 members and the little group — the operations that leave the wavevector where it is, modulo the reciprocal lattice — has order 2. The two multiply to the order of the point group, which is the orbit–stabiliser theorem and is checked rather than displayed. Each member is drawn at whichever of its equivalent copies lies nearest the origin, because that is where a reader expects a wavevector to be.

K symmetry

  1. 1 The star of a wavevector
  2. 2 Where two levels must meet
  3. 3 A glide sticks two levels together
  4. 4 The crossing at the corner
  5. 5 Which levels join which, on the way out of a point
  6. +3 more
8 essays · space-groups The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry.

Lengths

  1. 1 How many vectors of each length
  2. 2 The lengths do not name the lattice
  3. 3 The sum that turns a lattice into its dual
  4. 4 The sum whose answer depends on the shape
  5. 5 No lattice in space wins at every width
  6. +3 more
8 essays · lattices 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 structure, and the vectors between its atoms. On the left, 4 atoms in a cell. On the right, every one of the 16 vectors between them, each drawn from a common origin: 13 distinct positions, with the 4-fold peak at the origin being each atom paired with itself. That right-hand picture is what a Patterson map shows, and it is the thing a diffraction experiment gives without phases. It has more peaks than the structure has atoms — n² against n — which is why interpreting one is hard, and why it is always symmetric about its centre.

The Patterson function

  1. 1 The map that needs no phases
  2. 2 Where symmetry stacks the vectors
  3. 3 Solving from the vector set
  4. 4 Seventeen groups, seven vector sets
  5. 5 A map of the atoms that break the law
  6. +3 more
8 essays · diffraction p4g in 4 letters and 8 relations. The presentation of p4g, derived from the group's own operations. The two translations commute; each conjugation relation is read off a column of a matrix; and the point group's relations are corrected by the translation they actually come back as, which is what makes this group an extension rather than a semidirect product. Every relator is evaluated where the group lives and must be the identity, and coset enumeration on the letters alone returns 8, which is the order of the point group.

Presentations

  1. 1 A group in four letters
  2. 2 What is left when the order is forgotten
  3. 3 How few operations make a pattern
  4. 4 Telling two words apart
  5. 5 How fast a group grows
  6. +3 more
8 essays · operations A rotation. The motif in the first colour, its images under a single rotation in the second, and the symmetry element marked where the operation itself says it lies.

What symmetry is

  1. 1 What a symmetry actually is
  2. 2 The four motions of the plane
  3. 3 Why it is a group and not a list
  4. 4 The orbit is the pattern
  5. 5 The same symmetry, somewhere else
  6. +3 more
8 essays · operations 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
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8 essays · lattices Subgroups of index 3, across the seventeen. Every subgroup of index 3 with cyclic quotient in each of the seventeen plane groups, sorted into the two kinds: 4 keep all the translations and lose operations, 22 keep all the operations and lose translations, and the total is 26. The split is decided by whether the homomorphism onto ℤ3 kills the two lattice translations, which is a property of the kernel and not a judgement. Every one of them is found by enumeration inside the finite quotient by 3Λ, and the count for the whole classification is a measurement.

Colour

  1. 1 Three colours, and why most patterns cannot have them
  2. 2 Seventy-four colourings, forty-six groups
  3. 3 What a half-turn does to three colours
  4. 4 Four colours, five ways to permute them
  5. 5 Ninety-six four-colour designs
  6. +2 more
7 essays · classification p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

Flat space

  1. 1 The two that fold into a surface
  2. 2 Ten ways for space to be flat
  3. 3 Every net folds onto a torus
  4. 4 Crystallography in a box
  5. 5 A whole star needs a cubic box
  6. +2 more
7 essays · classification 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
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7 essays · applied 4mm: how many independent invariants there are at each degree. The number of independent polynomial invariants of the plane point group 4mm, one bar per degree from 0 to 8. The heights are the coefficients of the Molien series, computed from an integer recursion on the trace and determinant of each operation. A dot above a bar marks a degree where the same number has been computed a second way, by averaging every monomial of that degree over the group and taking the rank of the result — a computation sharing no code with the first. The two agree at every degree checked. Degrees where the bar is absent hold no invariant at all: for 4mm that is every odd degree below the first invariant, and it is a statement about which functions the group refuses to leave alone.

Invariants

  1. 1 How many invariants of each degree
  2. 2 The groups whose invariants are free
  3. 3 Three invariants and one relation
  4. 4 The cubic term that forbids a continuous change
  5. 5 Twelve of the thirty-two are free
  6. +2 more
7 essays · point-groups One hundred and twenty-two magnetic point groups. The three kinds, counted. Thirty-two ordinary groups, which contain no primed operation; thirty-two grey groups, which contain time reversal on its own and are the symmetry of anything magnetically disordered; and fifty-eight black-and-white groups, one for each way of splitting a class into a subgroup of index two and its complement. The last number is the one that has to be computed: the index-two subgroups are found by closure inside each class, reduced up to conjugacy, and reduced once more by an equivalence that needs a rotation no lattice may have. 32 + 32 + 58 = 122, and every term is a measurement.

Magnetic

  1. 1 The operation that reverses time
  2. 2 Which magnetism a class permits
  3. 3 The halving a lattice will not permit
  4. 4 Magnetic reflections land where nuclear ones cannot
  5. 5 Where a powder goes blind
  6. +2 more
7 essays · point-groups The region every plane lattice lands in. The shape of a plane lattice is one complex number, τ, and every lattice can be brought by a change of basis into the region shaded here: the strip between 0 and a half, outside the unit circle. Its interior is the oblique lattices. Its left edge is the rectangular ones, its arc and its right edge the centred rectangular ones, and its two corners are the square lattice at i and the hexagonal lattice at ρ. Five kinds, and they are a region, three arcs and two points rather than five things of one sort. The region is unbounded upwards, where the cell gets longer and thinner without limit.

Moduli

  1. 1 The space every lattice lives in
  2. 2 Two moves reach every basis
  3. 3 How far one lattice is from another
  4. 4 How many lattices share a determinant
  5. 5 The lattice that minimises a sum
  6. +2 more
7 essays · lattices Neumann's principle for elastic constants in 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 mmm subtracts from it, and the average over all 8 is 9, 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.

Neumann's principle

  1. 1 Neumann's principle, as one sum
  2. 2 A character does not know its basis
  3. 3 Twenty-one, thirteen, nine, three
  4. 4 Permitted is not present
  5. 5 The parts a property splits into
  6. +2 more
7 essays · point-groups The origins of p2 that change nothing. One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique.

Normalisers

  1. 1 The same pattern, described twice
  2. 2 One crystal, and sixteen coordinate lists
  3. 3 The normaliser is not a function of the group
  4. 4 The same site under two names
  5. 5 Three of them, and they are equivalent
  6. +2 more
7 essays · operations A spontaneous vector in 222. Averaging each of the three axes over the 4 operations of 222 leaves 0 independent components. No direction survives, so the class permits no spontaneous polarisation at all — which is a statement about what is forbidden, not about any measurement.

Properties

  1. 1 The ten with a direction of their own
  2. 2 Twenty of the twenty-one
  3. 3 Each permits what the other forbids
  4. 4 A filter of great precision and no predictive power
  5. 5 The strain the atoms do not follow
  6. +2 more
7 essays · point-groups

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