Crystallography & Symmetry
patterns derived from their groups
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Every pattern on this site is the orbit of a motif under a group, and the group is then rediscovered from the drawing. A picture that was drawn by hand can have symmetries nobody intended.
Nine frameworks, counted and then decided. Maxwell's count subtracts bars from twice the joints; the pebble game inserts the bars one at a time and discards any that cannot be paid for. The two agree on most of these frameworks and not on all, and where they differ the count is the one that is wrong — it assumes every bar is an independent constraint, and a bar added to a part that is already rigid is not. The redundant column is how many bars the game refused. Symmetry at work

A game that decides what counting only bounds

Maxwell's count subtracts bars from twice the joints and is a bound, not an answer, because it assumes every bar constrains something new. In the plane there is an exact repair: Laman's condition, run as a game in which each joint holds two pebbles and a bar is admitted only if four can be gathered at its ends. Two rigid bodies sharing a joint are what the count gets backwards.

 One hat patch laid out as hats, as equilateral tiles and as turtles. A patch of 36 hats found by exact cover on the kite grid, 4 of them reflected and drawn in the second colour, laid out three times. Every edge keeps its direction; short edges and long edges are given their own lengths. At short 1 and long √3 the tiles are hats, at equal lengths they are the equilateral member of the family, and at short √3 and long 1 they are turtles. In all three the same tiles touch the same neighbours along the same edges, and each layout was checked to be a tiling: 0 gaps and 0 overlaps, 0 gaps and 0 overlaps, 0 gaps and 0 overlaps among 1500 sample points, and every interior vertex surrounded by a full turn. Order without repetition

The hat and the turtle are one tiling

The hat has short sides and long sides; the turtle has the same turns with the two lengths exchanged, and looks nothing like it. Take a patch of hats, keep every edge pointing the way it points, stretch the short edges and shrink the long ones, and the patch becomes a patch of turtles — every tile touching the same neighbours along the same edges.

 Square ice scatters a pinch at the origin. The intensity scattered by the horizontal arrows of square ice, averaged over 100 configurations on a 32 by 32 torus, over the whole Brillouin zone with the origin at the centre; darker is more intense. Along the horizontal axis through the origin the intensity falls to zero — 2.9e-32 at the smallest wavevector — while along the vertical axis it stays near 1.75, so the two meet at the origin in a pinch. Order without repetition

The ice rule is a conservation law

Two arrows in and two out at every vertex is a statement that nothing flows in or out anywhere. That makes one half of the arrow field vanish identically, in every arrangement and not merely on average, and what is left scatters with a pinch at the origin: an intensity that approaches different values from different directions. Break the rule now and then and the pinch acquires a width Debye and Hückel predicted for a salt solution.

 A centre at every other ring, and never between. Two families of closed cage, each a tube of hexagons closed at both ends by a cap of six pentagons, taken from no rings of hexagons to 8. The top row has five faces to a ring and a pentagon at each pole; the bottom row has six and a hexagon. Each box holds the cage's number of atoms with its number of hexagons beneath, and a box is drawn solid with a dot under it when the cage has a symmetry that reverses orientation and fixes nothing — a centre, which is what lets the cage halve onto the projective plane. The five-family has one at even numbers of rings and the six-family at odd ones, so their hexagon counts are 0, 10, 20, 30 … and 8, 20, 32, 44 … — two arithmetic progressions rather than two rows. What a lattice forbids

A centre at every other ring

A census cannot settle an infinite row, and the construction proposed to settle it was a tube capped at both ends, lengthened a ring at a time. Carried out, it alternates: a centre appears at every other ring and never between, the two families it permits reach two arithmetic progressions rather than a row, and the first of them opens with exactly the cage the census found could not halve.

 The fewest contacts twelve pentagons can have, by size. For every cage of pentagons and hexagons up to forty-four atoms, the number of pairs of pentagons sharing a bond. The lower line is the fewest any cage of that size achieves — 30, 24, 21, 18, 17, 15, 14, 12, 11, 10, 9, 8 — the upper line the most, and the dashed line the bound that counting edges gives: the twelve pentagons carry sixty edges between them, a contact uses two and an edge to a hexagon uses one, so the contacts cannot fall below 30 − 3h with h hexagons. The bound is attained while the hexagons are few and goes loose at five, after which each extra hexagon removes about one contact rather than three. The number of cages at each size is printed beneath, and it is the least rather than the average that the bound is about. What a lattice forbids

How close the twelve must be

The charge fixes twelve pentagons and says nothing about where they go, because it is a sum over faces and cannot see which face touches which. What it cannot see is a graph on twelve points, and the fewest edges that graph can have falls from thirty to eight over the cages a census reaches — then keeps falling at a rate that puts its first zero exactly where the truncated icosahedron is.

 An orbit on a parabola, discrete and cocompact. The images of the origin under the group generated by two commuting affine maps of the plane: A slides one step along x and lifts y by the x it started at plus a half, and B is the translation by one in y. The images are the points with whole-number first coordinate and second coordinate a whole number above half the square of it, so the large dots lie on the dashed parabola and the small ones are the rest of the orbit. No two distinct images come closer than 1.000, and no point of the square between the axes lies farther than 0.610 from one — so the action is discrete and its quotient is compact, which is exactly what Bieberbach's first theorem asks for. What a lattice forbids

Straight lines, and no distances

Every finiteness met so far rests on the motions preserving a metric, because the trick that produces one is an average and an average needs something to average over. Keep the straight lines and drop the distances, and Bieberbach's first theorem is false in the plane — by an example two lines long, whose group is the plane's own translations and whose translations have rank one.

 Every arrangement on a torus 4 across, sorted by defects. The transfer matrix that counts ice arrangements chooses, at each vertex, the one horizontal arrow the rule permits. Enumerating both choices instead and carrying a polynomial that records how many vertices end up with three arrows in or three out gives the number of arrangements at every defect count at once. The first column, drawn solid, is the ice count — 2970 arrangements with no defect at all, which is the number the earlier transfer matrix gives and is checked against it. The second column is empty: no arrangement has exactly one defective vertex, because a defect carries a charge and the charges on a closed surface must cancel. The columns together add to two raised to the number of edges, which is every assignment of arrows whatever. Order without repetition

What a defect costs the count

Each broken vertex relaxes the rule and so adds arrangements — the question left standing was whether each adds a fixed amount or the cloud around it costs some back. The exact count at every defect number at once answers the first half outright: almost all of the rise is the freedom to choose which vertices break, and what is left once that is taken out exactly is small, uneven, and lowest at the first count that needs a four.

 One rule, one lattice, two entropies. The number of arrangements per vertex for square ice, counted two ways on the same lattice with the same rule. On a torus the count falls towards Lieb's exact value of 1.5396 from above. Inside a domain wall — every arrow on the top and bottom edges pointing in, every arrow on the left and right pointing out — the count rises towards 3√3/4, which is 1.2990, from below. A residual entropy is supposed to be a bulk quantity that forgets the boundary; these two differ by sixteen per cent and the only difference between them is the boundary. Order without repetition

The count that depends on the edge

A residual entropy is supposed to be a bulk number: so much per vertex, whatever surrounds the lattice. Square ice has two of them. On a torus the count per vertex heads for 1.5396 and inside a domain wall it heads for 1.2990, with the same rule on the same lattice — and the sixteen per cent between them is sitting in the corners.

 A colouring, and the arrows it writes. A proper three-colouring of the cells of a four-by-four torus — no two cells sharing an edge carry the same colour — with an arrow drawn on each shared edge by the difference of the two colours it separates. The difference is one or two modulo three, never nought, so every edge gets a direction. At each corner four cells meet and their four differences go round a cycle and add to nothing modulo three, which forces two of the arrows in and two out. That is the ice rule, arrived at from a colouring with no arrows in its statement. Order without repetition

Three colours on a chessboard

Colour the cells of a board in three colours so that no two sharing an edge agree. The number of ways is the number of ice arrangements on the same board — the same integer, to the last digit, at every even size — so a residual entropy a calorimeter reads is also the answer to a colouring problem with no physics in it at all. At odd sizes the two counts part company, and why they do is a condition on going round.

 The closure is a lattice exactly when the orders allow one. Twelve pairs of rotation orders, with a centre of each order placed one unit apart and the group they generate closed out to words of length 6. The linear parts reached are exactly the least common multiple of the two orders, every time — two rotations generate rotations, and the angles they generate are the multiples of the smaller of two fractions of a turn. A lattice admits rotations of order one, two, three, four and six and no others, so the closure can be a plane group exactly when that multiple is one of those five. The pairs where it is not are the pairs where the translations keep getting shorter. Operations

Closing the plane from two centres

Put two rotation centres down and close under composition: the result is a plane group or is not discrete, and nothing in between. What decides it is the least common multiple of the two orders, because two rotations generate rotations and the angles add — so the crystallographic restriction arrives as a condition on a closure rather than as one on a lattice.

 A screw out of two rotations that have none. Four pairs of located rotations of space, composed, with the result read back as a screw: its angle, and its pitch, which is the part of its translation lying along its own axis. Axes that meet give a rotation and no translation at all, because the point where they meet is fixed by both. Parallel half-turns give a translation. Skew axes give a screw — a motion with a translation in it, out of two motions with none — and the translation is twice the distance between the two axes. Nothing in either factor moves anything along the product's axis, and the product does. Operations

The axis a product lies on

Two rotations of space about axes that do not meet compose to a screw — a motion with a translation in it, out of two that have none. The translation is twice the distance between the axes and the angle twice the angle between them, and the screw's own axis is not somewhere arbitrary: it lies on the two axes' common perpendicular, at a place the arithmetic gives.

 The turns that keep the join discrete. Two copies of p4 on one square lattice, one turned against the other, with the shortest translation their union generates. At a turn whose cosine and sine are both rational the translations are a lattice, and its shortest vector is one over the square root of Σ — where Σ is the odd part of p² + q² for the rational point (p, q) — which the measurement reproduces to six places at every one tried. At a whole number of degrees other than a multiple of ninety there is no such point, and the search finds shorter translations the further it runs. Operations

Two patterns laid over one another

Compose two plane groups rather than two operations. The group generated by both is one of the seventeen or is not discrete at all, with nothing between — and it takes two conditions, one on the rotation orders and one on the turn between the lattices. The turns that work have rational cosines, which by a theorem of Niven's means none of them is a whole number of degrees.

 The fewest pentagon contacts, from twenty atoms to sixty. The least number of pairs of pentagons sharing a bond, over every cage of pentagons and hexagons at each size from 20 to 60 atoms, with two lower bounds: 30 − 3h from counting edges, and 20 − h from counting the pentagons round each hexagon, which is also the prediction of one fewer contact for each hexagon added. The measured values past 44 atoms are 8, 8, 7, 5, 5, 4, 4, 3, 0; the bound is met at 20, 24, 26, 28, 36, 38, 40, 42, 44, 50, 60 atoms and nowhere else. What a lattice forbids

The last three contacts go at once

The fewest pentagon contacts a cage can have fell by one for every hexagon added over eight sizes up to forty-four atoms, which pointed straight at the truncated icosahedron at sixty. Carried to sixty by a faster search, the fall does not continue evenly. It pauses, jumps, pauses again, and loses its last three contacts in a single step. The steady fall was not a trend. It was a lower bound, twenty minus the number of hexagons, which a short count round each hexagon proves, and the census now shows where cages meet it and where they cannot.

 How often a broken vertex carries four. Of all the broken vertices in all the arrangements with a given number of them, the fraction whose four arrows all point the same way — on tori four, five and six vertices across. If the vertices were independent that fraction would be one in five at every count. With only two defects it is far lower, 0.085, 0.075 and 0.066 on the three tori, and falling with the size; as the count rises it climbs, oscillating, to one in five and stays there. The odd counts sit above the even ones because an odd number of defects cannot be made of twos alone — three defects must hold exactly one four, which is a share of one third. Order without repetition

A four is not two twos

A vertex of square ice can break the rule with three arrows one way or with all four, and a census that counts the two apart finds them behaving as charges of two and four should: the fours are rare while defects are few, a pair of fours is squeezed about four times as hard as a pair of twos, and the dip an earlier census reported turns out to be nothing but the first count that cannot be made without a four.

 Pentagon contacts as pieces of the icosahedron. The icosahedron drawn with one vertex at the centre, two rings of five, and the twelfth vertex as the outer circle; each vertex is a pentagon and each edge a possible contact. Left: the dodecahedron, whose twelve pentagons all touch five others, uses every edge. Centre: the commonest contact graph among the 5,770 cages to sixty atoms, 6 contacts in clusters of 3 + 3 + 2 + 2 + 1 + 1, carried by 179 cages, placed on the icosahedron with every contact on an edge. Right: the one cage with no contacts at all. All but two of the 340 contact graphs can be placed this way. What a lattice forbids

Three hundred and forty ways for twelve pentagons to touch

Every cage of pentagons and hexagons has twelve pentagons, and which of them share a bond is a graph on twelve points. Over all 5,770 cages from twenty atoms to sixty there are only 340 such graphs, and the graph almost never tells one cage from another: the commonest is shared by 179 cages and one recurs at thirteen different sizes. And every one of them except two, both from small cages, fits inside a single graph, the icosahedron's, which is what the dodecahedron's twelve pentagons make when every one touches five others.

 Every arrangement on two tori, by how it winds. The ice arrangements on tori eight and nine vertices across, each placed at the pair of numbers it winds by — how much the height on the faces gains going once round the torus across and once round it down — with a disc whose area grows with the number of arrangements there. On the even torus every winding is even and the arrangements crowd the centre; on the odd torus every winding is odd and there is no centre to crowd, so they sit at plus and minus one. Solid discs are the sectors whose windings are both multiples of three, which are exactly the arrangements that are also three-colourings. On the even torus that is exactly a third of everything; on the odd torus it is the four sectors at plus and minus three, and they hold about one arrangement in eighty. Order without repetition

A winding does not dilute

Three-colourings of an odd torus are a small and shrinking fraction of the ice arrangements on it, and the natural guess is that the fraction goes to nothing. It does not. It settles near one in twenty-eight, because the windings that decide it are a property of the whole surface rather than of its edge — and on even tori the same windings put exactly a third of the arrangements in one class, which in the limit is Jacobi's theta identity.

 The h = 1 anomaly moves with the length of the chain. For a size-coupled chain with ε = 0.03, the ratio of the second-order pair's tilt, divided by k², to the first-order tilt at h = 1, from plain sums over the sites and from sums with a smooth window, for 22 chain lengths from 10,000 to 480,000 sites. The plain ratio swings between -0.08 and 5.60; at 30,000 and 120,000 sites, the two lengths first tried, it sits near 0.31 both times. The windowed ratio is one half at every length. Order without repetition

The satellite beside the zero-order reflection

An earlier essay found that the second-order satellites of a modulated chain tilt half as far as the first-order ones at every reflection but the first, where they tilted a third as far, the same on chains of 30,000 and 120,000 sites. It left the departure unexplained and proposed a higher order of the expansion. The expansion was never wrong. The weak satellite at h = 1 sits a quarter of a reciprocal unit from the strongest reflection a chain has, and a plain sum over a finite chain carries that reflection's tail, which does not shrink steadily as the chain grows. It swings, and the two lengths tried happened to swing to the same place.

 The second-order satellites alone cannot measure the curvature of a size effect. The Cramér–Rao error in the quadratic size coefficient ε₂, as a fraction of its true value, from six sets of satellites of a modulated chain of 30,000 sites with 2% intensity error, and the correlation between the errors in ε₂ and the linear coefficient ε₁. From second-order satellites alone the error is thousands of times ε₂ at one reflection and still a hundred times it at three, with the two coefficients correlated at 1.000. With the first-order satellites added, one reflection gives ε₂ to about a third of itself and three to about a tenth. Order without repetition

The second order cannot see a curved size effect alone

When the spacing between atoms depends on composition along a curve rather than a line, the displacement of a modulated chain gains a second harmonic, and the second-order satellites should measure it. They cannot do it on their own. At every reflection a second-order satellite sees the straight part and the curved part of the size effect in almost the same proportion, so adding reflections adds copies of one equation. The first-order satellites, which see only the straight part, are what separates them.

 What each kind of satellite of a two-wave chain says about the curvature of the size effect. The Cramér–Rao error in the quadratic size coefficient ε₂, as a fraction of its true value, from seven sets of satellites of a chain modulated by two incommensurate waves, with 2% intensity error: with the two modulation amplitudes refined as well (upper bar of each pair) and with them known (lower bar). The sum or the difference satellites alone say almost nothing either way. Together, with the amplitudes known, they fix the curvature to under one per cent from a single reflection; with the amplitudes free they do not, and adding the first-order satellites, which fix the amplitudes, restores it. Order without repetition

Two waves measure the curvature one wave hides

A chain modulated by one wave cannot tell a curved size effect from a straight one at second order: every second-order satellite sees the two in the same proportion. Modulate it by two incommensurate waves and the curvature puts the product of the waves into the displacement, at their sum and at their difference. The satellites there see the curvature with opposite signs. Two equations of opposite sign separate two unknowns, and once the two waves' amplitudes are known, the mixed satellites of a single reflection fix the curvature to under one per cent.

 At the sum of all three wavevectors the cubic and the quadratic paths nearly cancel. The intensity of the satellite at h + q₁ + q₂ + q₃ for the first four reflections, computed with only one size coefficient switched on at a time and with all three. The cubic coefficient alone gives about 3 × 10⁻⁷ at h = 1, the quadratic alone about 2 × 10⁻⁷, the linear alone less than 10⁻¹⁰. With all three the satellite is ten times weaker than with the cubic alone, because the cubic term's displacement and the quadratic term's pair harmonic times a first-order wave arrive with opposite signs. Order without repetition

Three waves measure the cube through the slope

A size effect that bends twice puts the product of three modulation waves into the displacement, at the sums and differences of all three wavevectors, and nowhere else does a cubic term reach so directly. The satellites there were expected to measure it. They see it, but the strongest of them is ten times weaker than the cube alone would make it, because the quadratic term arrives at the same place with the opposite sign, and against any realistic background they say nothing. The cube is measured instead by the strongest satellites of all, the first-order ones, because it makes each wave's slope depend on how strong the other two waves are.

 Where the two-fold axis sits decides what the coloured optic axes keep. Interference-figure positions of the two optic axes from 400 to 700 nm for four model crystals seen down the acute bisectrix. Orthorhombic: both mirrors survive and only the angle between the axes changes. Two-fold axis along the optic normal: the axes stay on one line and slide unequally (inclined dispersion), one mirror along that line. Along the obtuse bisectrix: the axial plane stays parallel and shifts (horizontal), one mirror across it. Along the acute bisectrix: the axial plane turns about the centre (crossed), a half-turn and no mirror. What symmetry decides

Three places for the two-fold axis

A monoclinic crystal fixes one direction of its indicatrix, the two-fold axis, and an ellipsoid has three principal directions for it to be. Which one it is decides what an interference figure keeps in white light: a single mirror along the plane of the optic axes, a single mirror across it, or a half-turn and no mirror at all. Mineralogy calls the three inclined, horizontal and crossed dispersion. They are one subgroup computed three times, and in light of a single colour all three disappear.

 Down one optic axis, at three colours, for two crystals. Interference figures of two model monoclinic crystals, inclined above and crossed below, each a plate cut across the optic axis at the sodium line, at 450, 589 and 650 nm. In the inclined crystal the melatope slides along the horizontal mirror and the figure stays symmetric across it. In the crossed crystal the melatope leaves the line away from sodium, and the pattern loses the mirror. What symmetry decides

The mirror a plate keeps in one colour

Cut a plate of a crystal across one optic axis, and in the light the plate was cut for its interference figure has a perfect mirror, whatever the crystal's class. Change the colour and the mirror survives only if the crystal really has one. Computed ring by ring, five model crystals give the prediction exactly, including the part the eye would miss: the mirror does not come back when the line is moved to follow the melatope.

 The error in the count, against tiles chosen at random. For clusters from about twenty-five hats to nearly two trillion, the distance of the reflected count from the exact share of all hats. For the hat it stays under one half. For independent random choice at the same share the expected error grows as the square root of the size, past 400,000 at the largest cluster. Order without repetition

The count of reflected hats is never one out

In a hat tiling one tile in about eight is reflected, and that ratio was only ever quoted here. Four clusters and a table of small integers give it exactly: seven plus three root five over two, the fourth power of the golden ratio. The same table says something the ratio does not. Grow a cluster to two trillion hats and its reflected count is still within half a tile of the exact share, where the same share chosen at random would be out by nearly half a million.

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