Crystallography & Symmetry
patterns derived from their groups
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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.
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. +2 more
7 essays · space-groups 11 frameworks, 3 where the count is wrong. Every net in this collection read as a framework of rigid bars and free joints, with the cell free to change shape. Maxwell's count and the number of mechanisms agree on most of them and not on all: a framework with a state of self-stress has a bar the count treats as removing a freedom that the others had already removed, and it has a mechanism the count cannot see. Here that is fes, snb, ring5, where the count says 2, -1, -4 and the rank says 3, 0, 9. The identity Maxwell is always right about — count equals mechanisms minus self-stresses — holds on every row.

Rigidity

  1. 1 The count that promises a mechanism
  2. 2 A fold that keeps its symmetry
  3. 3 A mechanism that is a wave
  4. 4 The mechanisms a count cannot see
  5. 5 The polyhedra that can flex
  6. +2 more
7 essays · applied P2₁/c, in the two diagrams the Tables print. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

Space groups

  1. 1 The step a flat surface has no room for
  2. 2 Forgetting a group in three dimensions
  3. 3 The half of a translation that is not a choice
  4. 4 Sixteen candidates, ten groups
  5. 5 Eleven groups that are their own reflection's rival
  6. +2 more
7 essays · space-groups Sublattices of index n in the plane. For each index up to 12: the number of sublattices found by building every Hermite normal form of that determinant, and the number the Dirichlet series ζ(s)ζ(s−1) predicts — the sum of the divisors in the plane, and a longer sum in space. The two columns are computed by routines that share no code, and the figure does not appear at all if any row disagrees.

Sublattices

  1. 1 How many ways there are to thin a lattice
  2. 2 The sublattices that stay square
  3. 3 The reflections a superlattice adds
  4. 4 Every way down, and no way round
  5. 5 The sublattices that are the same shape
  6. +2 more
7 essays · lattices 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. +2 more
7 essays · operations 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
  6. +1 more
6 essays · operations The angles between the faces of {102̅}. The form {102̅} of class 3̅m in section, with each face labelled by its indices. Its 6 faces make 15 pairs and only 3 distinct angles, the smallest being 76.43°. Every value is computed from the cell's metric — the one calculation in this family that is not integer arithmetic, because an angle is a real number and a lattice does not constrain it.

Forms

  1. 1 Why a crystal face carries small whole numbers
  2. 2 The angles belong to the substance, the shape to the specimen
  3. 2 A form is an orbit, and whether it closes is an integer question
  4. 3 Five classes grow the same cube
  5. 5 A zone is a vanishing dot product
  6. +1 more
6 essays · applied 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. +1 more
6 essays · lattices 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. +1 more
6 essays · operations 6mm: 6 irreducible characters on 6 classes. The character table of the plane point group 6mm, constructed rather than quoted. The columns are its 6 conjugacy classes, with the number of operations in each; the rows are its 6 irreducible representations, of dimensions 1, 1, 1, 1, 2, 2. The dimensions satisfy 1² + 1² + 1² + 1² + 2² + 2² = 12, which is the order of the group, and that identity together with the count of classes is what proves the list complete. Every entry is an exact element of the twelfth cyclotomic ring; the decimals shown are the numerical value of an integer vector, not a computed approximation.

Representations

  1. 1 What a group does to a function
  2. 2 How large a degeneracy may be
  3. 3 A coincidence the group did not ask for
  4. 4 The shell that splits into kinds
  5. 5 What a group forbids to happen
  6. +1 more
6 essays · point-groups 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. +1 more
6 essays · space-groups N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does.

Statistics

  1. 1 Whether there is a centre is a statistic
  2. 2 The zones that behave as if there were a centre
  3. 3 The average that knows the atoms and not where they are
  4. 4 A twin hides in the statistics
  5. 5 A translation that is nearly there
  6. +1 more
6 essays · diffraction Every way regular polygons can fill a turn. The seventeen multisets of regular polygons whose interior angles add to exactly 360°, listed with the sum that qualifies each of them. They are found by a search over sizes from three upward: the largest polygon that can appear is the forty-two-gon, which needs a triangle and a heptagon beside it, and the search stops there because the smallest interior angle is a third of a turn so at most six polygons can meet. Nothing here is a table looked up — the list is the output of the search, and every count on the page downstream of it is counted from this one.

Tilings

  1. 1 Twenty-one vertices, eleven tilings
  2. 2 Eleven tilings, five groups
  3. 3 Eleven duals, one tile each
  4. 4 Which shapes tile by themselves
  5. 5 Three answers in whole numbers
  6. +1 more
6 essays · classification The fourteen Bravais lattices. All fourteen lattices: triclinic P, with 2 symmetries; monoclinic P, with 4 symmetries; monoclinic C, with 4 symmetries; orthorhombic P, with 8 symmetries; orthorhombic C, with 8 symmetries; orthorhombic I, with 8 symmetries; orthorhombic F, with 8 symmetries; tetragonal P, with 16 symmetries; tetragonal I, with 16 symmetries; rhombohedral P, with 12 symmetries; hexagonal P, with 24 symmetries; cubic P, with 48 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

The fourteen Bravais lattices

  1. 1 Twenty-five cells, and fourteen lattices
  2. 2 Forty-eight becomes sixteen
  3. 3 A lattice described on somebody else's axes
  4. 4 Thirty-two from fourteen matrices
  5. 5 Seventy-three, without a search
5 essays · lattices 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
5 essays · space-groups ⟨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
5 essays · diffraction Two structures on 8 sites with the same vectors. Two arrangements of 4 atoms on a ring of 8 positions. They are not the same arrangement — no rotation of the ring and no reflection carries one onto the other — and every interatomic vector occurs the same number of times in both. The bars below are the shared vector counts, which is the Patterson function of each: the tall one at the origin is the atom count and carries no information, and everything else is what a diffraction experiment measures. Their diffraction patterns are identical in every intensity, so no measurement of intensities, at any resolution, distinguishes them.

Homometry

  1. 1 Two structures, one Patterson
  2. 2 Where the pairs come from
  3. 3 Two structures on a torus, and one Patterson
  4. 4 When the atoms are not all the same
  5. 5 Symmetry does not rescue a Patterson
5 essays · diffraction 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
5 essays · point-groups 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
5 essays · aperiodic The hat: eight kites, thirteen sides. The shape a search over the eight-kite polykites returns, drawn on the kite grid it lives in — the Laves tiling [3.4.6.4], in which every hexagon is cut into six kites. The eight kites of the shape are tinted and its outline is drawn heavy. Thirteen sides result, of two lengths only: a half and root three over two, in units of the hexagon's circumradius, with one side of twice the shorter length where two kite edges lie in a line. Its interior angles are 90, 120, 240 and 270 degrees. Nothing about the shape was chosen: it is the one octakite that clears every filter in the search.

Monotile

  1. 1 One tile, and no period
  2. 2 The tile that needs no reflection
  3. 3 The hat and the turtle are one tiling
  4. 4 How much of the hat is a crystal
  5. 5 Aperiodic is two words in space
5 essays · aperiodic

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