Depth

Series — page 4

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.

P2₁/c under every cell choice. One group, described in each of the 6 bases that keep its cell the shape its system requires, with the symbol derived from the operations each time. The distinct symbols are P2₁/c, P2₁/a, P2₁/n — 3 names for one group. Nothing about the crystal has changed: the basis changes all have determinant ±1, so the lattice is untouched, and the census of rotations, screws, mirrors and glides is identical in every row, since conjugation cannot turn one kind into another. What changes is which lattice vector a glide's translation is half of, and the glide letter names exactly that.

Settings

1 One group, three symbols
2 Two origins for one group
3 Six ways to name one group
4 One matrix, four rules
5 Five hundred and thirty rows, four hundred and ninety-nine descriptions

5 essays · space-groups

p3, twinned. p3 twinned by a rotation. To the left of the composition line the motif sits where p3 puts it; to the right every copy has been carried over by the twin law, which is one of the 3 operations the hexagonal lattice has and p3 does not. 24 images on the left, 24 on the right, and the lattice runs through the line unbroken — which is exactly why a twinned crystal looks like a single one.

Twinning

1 A twin is a symmetry the lattice has and the crystal does not
2 Twenty-five of the thirty-two can twin, and seven cannot
3 Quartz has exactly three twin laws, and its lattice is why
3 A merohedral twin moves no spot at all
5 The index and the angle a twin misses by

5 essays · applied

8 tiles over 5 colours. Wang tiles: unit squares with a colour on each edge, which may be laid side by side only where the touching edges agree, and which may never be turned or reflected. That last restriction is what makes them a computational object rather than a jigsaw — an edge colour is a symbol passed from one tile to its neighbour, and turning a tile would let a symbol change direction. The set here was generated from a stated seed.

Decidability

1 Nothing decides whether a set of tiles tiles the plane
2 Surrounded twice over, and covering nothing
3 How much room a hard question needs
4 The argument that closes eleven

4 essays · classification

Defects

1 The circuit that does not close
2 How many dislocations a lattice has
3 The defect that needs two laps
4 The point defect whose charge has no sign

4 essays · applied

Three lattices at 2 forms each: 6, 14, 12 faces. The shape each cubic lattice predicts, built as the solid bounded by its top 2 forms, with each face's distance from the centre inversely proportional to its interplanar spacing. The three lattices have the same metric and the same list of indices; every difference between these solids comes from which reflections are systematically absent. Pm-3m leads on {100} and comes out with 6 faces; Fm-3m leads on {111} and comes out with 14 faces; Im-3m leads on {110} and comes out with 12 faces. Taking more than the leading form matters only where the extinction correction has moved something: in a cubic metric a form's planes are placed at a distance proportional to the root of the sum of the squares of its indices, which is exactly where the corresponding corner of the cube already is, so an uncorrected second form arrives tangent and cuts nothing off.

Growth

1 Which faces a crystal shows
2 The fast faces are the ones that vanish
3 The step that never runs out
4 Which faces are flat

4 essays · applied

7 cells explain the lines; one of them is right. A line list from a face-centred cubic cell of 5.64 Å, with a realistic error added, handed to a sweep over every cubic cell between 2 and 12 Å in all three centrings. 7 distinct cells explain every line within the tolerance, and each is a genuine solution rather than a numerical accident. The true cell comes top by de Wolff's figure of merit — the last Q over twice the mean discrepancy times the number of lines the candidate says should have been visible — which punishes a candidate for predicting lines nobody saw. That is the whole of what makes indexing decidable in practice: not the arithmetic, which has many answers, but a criterion for preferring one.

Indexing

1 Indexing a powder pattern
2 A cell from a bag of spots
3 The figure of merit a supercell always beats
4 Every alias is a supercell

4 essays · applied

The seven frieze groups. Every way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.

Friezes

1 Seven friezes
2 Why sixteen become seven
3 The friezes inside the seventeen

3 essays · classification

Y-pentomino: A B C D E F, with 6 arcs. The boundary of the Y-pentomino cut into six arcs. A runs from one corner to another and D is the same arc traversed backwards, so D is a translate of A and the translation is (3, 1) cells. Each of B, C, E and F is carried onto itself by the half turn about its own midpoint, and those midpoints are the four marked dots — That is Conway's criterion, and a shape meeting it tiles the plane by translations and half turns.

Isohedral

1 A tiling of the whole plane, decided on one tile's edge
2 One shape, two kinds of tile
3 One tile, two lattices, and a row that slides

3 essays · classification

How often a dot gives the wrong group. Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it.

Ornament

1 The groups ornament actually uses
2 Two colours, and a symmetry that swaps them
3 The Alhambra question

3 essays · classification