Crystallography & Symmetry
patterns derived from their groups
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The theme: Order without repetition — page 2

Periodicity and order are not the same thing, and separating them is what quasicrystals forced. A pattern can be perfectly determined and never repeat.
heesch-two: surrounded 2 times. A shape that tiles nothing, with the rings of copies it does accept: the seed in the first colour and 2 coronas of 7 and 16 copies round it. The search that built this finished, so the shape's Heesch number inside this box is exactly 2, and it cost 3,097 placements. Every cell touching a tile of one ring, corners included, is covered by the next. The classification

Surrounded twice over, and covering nothing

A shape that tiles the plane can be surrounded by copies of itself for ever. A shape that tiles nothing cannot be surrounded for ever — but it can be surrounded once, and sometimes twice, and the number of times is a measurement of how much local success a global impossibility permits.

 two chains, periods 1 and 1.62. Two interpenetrating chains of atoms with periods 1 and 1.62, whose ratio is irrational, so no length is a whole number of both. Each chain is displaced from its own lattice by a wave with the other's period — the short ticks show each atom's displacement from where an unmodulated chain would put it — which is what makes this one crystal rather than two side by side. Nothing here is a unit cell: any length chosen contains a whole number of one chain's atoms and a fractional number of the other's. Order without repetition

Two lattices, one crystal, and no cell at all

A modulated crystal has a lattice and a wave running through it. A composite has neither host nor guest: two interpenetrating substructures with periods that share no common multiple, each modulating the other. Every reflection needs an index from both, and the unit cell a diffractometer reports belongs to whichever half scattered harder.

 9 approximants, period 2 to 89. The approximants of the Fibonacci chain: the n-th Fibonacci word taken as a unit cell and repeated. Each is a perfectly ordinary periodic crystal — it has a lattice, a cell and a space group — and each has the composition of the quasicrystal to the accuracy a ratio of Fibonacci numbers can manage, since its long and short tiles are consecutive Fibonacci numbers and their ratio is a convergent of the golden ratio. The last column is where the approximant stops agreeing with the infinite chain letter for letter: always past its own period, because the infinite word begins with every finite Fibonacci word, and never for ever. At order 9 the error in the composition is -3.87e-4, and it falls by a factor of τ² at every step up the sequence. Order without repetition

The crystal you get by rounding τ off

Everything aperiodic about a Fibonacci chain comes from one irrational number in the slope of a cut. Replace it by a fraction and the whole construction survives: the same lattice, the same strip, the same rule, and a chain that is periodic — agreeing with the quasicrystal for a length that grows with the denominator.

 The allowed energies of the Fibonacci chain, level by level. The set of energies at which a wave neither grows nor decays, for periodic approximants of the Fibonacci chain of 5, 8, 13, 21, 34 and 55 sites. Each row has exactly one band per site, and each band splits into smaller ones at the next level rather than growing. Nothing in the picture converges to an interval: the gaps opened at one level survive at every level after it, and the limit is a Cantor set — closed, containing no interval at all, and of measure zero, which is a theorem of Sütő's rather than something these six rows prove. Order without repetition

A spectrum that is a Cantor set

A wave in a periodic chain has bands with gaps between them. A wave in the Fibonacci chain has gaps inside the gaps, at every scale — and the traces that decide where they are obey a recursion with a quantity it cannot change.

 How fast a window's answer settles: 1/L on the chain, 1/√L on a shuffle. The largest error a window of each length makes about a block's frequency, over every position the window can take, on logarithmic axes. The upper line is a shuffle of the chain's own letters — same frequencies, no order — and its slope is close to −½, which is the random walk a sequence with no structure produces. The lower line is the Fibonacci chain itself and its slope is close to −1. The frequency of a block in the chain is therefore something a finite window measures rather than approaches: to know it to a part in a thousand needs a window of a thousand tiles, not a million. Order without repetition

The average is the same wherever it is taken

A measurement is made on a window somewhere, and the question is whether the answer belongs to the chain or to the window. For the Fibonacci chain the error falls as one over the window's length; for a shuffle of the same letters it falls as one over the square root, and the two exponents are fitted rather than asserted.

 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. Order without repetition

One tile, and no period

Every aperiodic pattern in this collection so far needs two shapes. A search over the eight-hundred-and-seventy-three ways of gluing eight kites together, filtered by nothing but whether a shape tiles and whether it repeats, returns exactly one — and it is the shape announced in 2023.

 Unreflected copies stop at 1 ring. Copies of the hat, all of the same handedness, covering a core of 1 ring of hexagons — 9 tiles, every cell covered once. At 2 rings the same search runs to exhaustion and returns nothing: there is no such covering, and the failure is a proof for that region rather than a search that gave up. The reflected copy is not a convenience of the drawing; the tiling cannot proceed without it. Order without repetition

The tile that needs no reflection

One shape tiles the plane and never repeats, and it does it with copies of both hands. Cut the tiles out of card and that is nothing; ask for it in a molecule, where handedness cannot be undone by turning something over, and it is the whole question.

 656 sets, every one decided. Every set of one, two, three and four tiles over two colours — sixteen tiles exist in all, so these are complete lists rather than samples — reduced by relabelling the two colour alphabets, and each set decided by the two half-searches. The last column is the one that matters: it is empty. At these sizes there is no room for a set that tiles the plane and admits no periodic tiling, which is the residue undecidability lives in. The smallest aperiodic set is known to have eleven tiles and four colours. The classification

How much room a hard question needs

No algorithm decides whether a set of tiles covers the plane. Every set of four or fewer tiles over two colours is nevertheless decided here, exhaustively, in under a second — because the sets that defeat the two half-searches have nowhere small to live.

 One crystal, two rows: one streaks and one does not. Two rows of reflections from the same faulted crystal, at a fault rate of 0.05, each drawn against the same row from a perfect one. The upper row has h − k not divisible by three, so each layer contributes a different cube root of unity and the sequence of layers enters the sum: the sharp peaks collapse into a streak. The lower row has h − k divisible by three, the phase factor is one, every layer scatters in step, and the peaks are exactly as sharp as in the perfect crystal. The sorting is an integer condition — a reflection either can see the stacking or cannot, decided by h − k modulo three — which is the most direct evidence there is that the disorder is in the stacking and not in the layers. The profiles are averaged over 16 independently faulted crystals, because one crystal gives speckle rather than a diffuse profile. How it is known

The streaks a faulted stack makes

Close packing settles two directions and leaves the third to chance. A crystal that chooses wrongly now and then has a lattice in the plane of its layers and none across them — and its diffraction pattern says so, with some rows of spots as sharp as ever and others smeared into streaks, sorted by an integer condition.

 A peak that grows, and not fast enough. The strongest peak of three chains, divided by the square of the number of letters, as each chain is lengthened. A Bragg reflection is a sum of terms in phase, so its intensity grows as the square of the count and this number settles: the Fibonacci chain and the period-doubling chain both do, at exponents of about two. The Thue–Morse chain does neither — its strongest peak grows, so it is not diffuse scattering, and it grows more slowly than the square, so it is not a Bragg peak. The fitted exponents are printed beside each curve and no threshold enters the comparison. Order without repetition

Neither a peak nor a bump

A chain whose strongest reflection grows as the length to the power one and a half. A Bragg peak grows as the square and a diffuse bump grows as the length itself, so this is neither — and the essay that ruled out the first possibility could only say so by quoting a theorem.

 How often each block of 5 occurs. Every block of length 5 in the fibonacci chain, with its frequency from the Perron eigenvector of the block substitution and again from a count over a chain of 46368 letters. The two share nothing: one is a linear algebra problem over a matrix of integers, the other a loop over a string. The eigenvalue of the block matrix is the inflation factor of the letter matrix, which is a second check and a stronger one — a chain inflates at one rate whatever length of window is being counted. Order without repetition

How often each patch occurs

That a patch has a frequency at all is the ergodic theorem. What the frequency is turns out to be an eigenvector: the substitution acts on blocks as well as on letters, the block matrix has a Perron vector, and its entries are the frequencies exactly. For the Fibonacci chain those entries take three values at every length, and the three values are the three gaps of a rotation.

 One local configuration, and a number of structures that doubles. How many kinds of adjacent pair a close-packed stack has, how many kinds of triple, and how many stackings of each period there are. The first column never moves: every pair of layers is congruent to every other, at every period, which is what an order-disorder family is. The last two agree with 2ⁿ + 2(−1)ⁿ, which is the chromatic polynomial of a ring of n layers at three colours, because a stacking of period n is exactly a proper three-colouring of that ring. The gap between the first column and the last is the whole subject. Symmetry at work

A stack with no space group

Every pair of layers in a close-packed stack is congruent to every other pair, and the number of stacks doubles with every layer added. A family whose local configuration is completely determined and whose global structure is not determined at all has no single symmetry group — what it has is a set of operations that compose only when their ends match, which is a groupoid.

 21 superspace groups in (2+1) dimensions, from 31 names. The whole count, in the order it is built. Thirteen arithmetic classes of the plane; six of them admit an incommensurate wavevector; those six give ten sign assignments; each assignment contributes the plane cohomology times the internal cohomology, which is thirty-one names; and the names are merged by the changes of basis that are relabellings — a change of the plane basis, which moves the sign assignment and the internal cocycle with it, and the choice of q against −q. The last row is what the count would be if the two factors were quotiented separately, which over-counts because the merge is not independent of the internal part. Order without repetition

Superspace groups in the plane

A modulated crystal has no space group, and in a space of one more dimension it has one. Counting them in the plane is the seventeen's own extension arithmetic with a third coordinate on which the point group acts by a sign — and the sign has to be plus or minus exactly, which kills the three-fold, four-fold and six-fold classes before a single extension is counted.

 Four of the 980 piles in a three-cube box. A stack of unit cubes in the corner of a box, seen down the body diagonal. Every visible face is one of three rhombi and the picture is a tiling of one fixed hexagon — the same hexagon for every pile, because a pile in an a×b×c box always shows ab+bc+ca faces however it is stacked. The four here are taken at even intervals through the enumeration, from the empty box to the full one. Order without repetition

A facet with no energy in it

Stack cubes into the corner of a box and look down the body diagonal: the pile is a tiling of a hexagon by three rhombi, and the number of piles is a product MacMahon wrote down in 1916. Because the count is exact, so is the average pile — and the average has a flat corner meeting a rounded middle, which is the shape of an equilibrium crystal, arrived at by counting with no surface energy anywhere in the argument.

 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.

 A patch of hats scatters a pattern that repeats. The diffracted intensity of the 1217 points of a patch of 183 tiles laid out as the hat, over 2 by 2 cells of the kite grid's reciprocal lattice, whose edges are the faint lines. Every local maximum above a hundredth of the central peak is a disc with area proportional to its intensity; 12 reach the central peak's full height. 72 maxima are drawn. Adding a reciprocal lattice vector to the scattering vector changes the intensity by at most 1.1e-15 of the central peak, so each cell holds the same pattern. Order without repetition

How much of the hat is a crystal

Put a scatterer on every corner of a patch of hats and the diffraction pattern repeats exactly, because every corner sits on a lattice. Inside each repeat the strongest reflections are those of an ordinary crystal with partly filled sites, and by Parseval's identity they carry sixty-three per cent of what the pattern holds. The aperiodicity the hat is famous for lives in the remaining third, in reflections a hundred times weaker.

 Whether a rolled sheet ever comes back round. Three plane lattices, each with the same rolling vector C = 3a₁ + a₂ drawn from the origin and the line through the origin perpendicular to it. A translation of the rolled pattern straight up the tube, with no turn, is a lattice vector on that line. The square lattice has one, marked T, and the tube repeats every 10 turns. The general rectangular lattice has none in this direction — only along its cell edges — and the general oblique lattice has none in any direction at all, so its rolled pattern climbs forever without returning to the same angle. What a lattice forbids

Most sheets roll into a tube that never repeats

Rolling the honeycomb along a lattice vector always gives a tube with a repeat, and that is a property of the honeycomb rather than of rolling. Over the seventeen plane groups, 567 of 1,008 rolling directions give a tube with no translation along its axis at all — and every direction of an oblique pattern is one of them.

 The window of a three-letter chain. The cut-and-project window of the tribonacci chain: every prefix of the chain, projected onto the plane spanned by the two complex roots of x³ = x² + x + 1, and coloured by the letter that follows it. 223,317 points. It is a bounded region in three pieces whose areas are 0.542, 0.296, 0.162 of the whole, which are the frequencies of the three letters; it fills 71.5 per cent of its bounding box, and a straight cut across it meets up to 6 separate pieces. A window for a two-letter chain is an interval. Order without repetition

A window that is not an interval

The usual cut-and-project construction takes a strip through a lattice and keeps the points falling within an interval. Add a third letter and the window stops being an interval: the tribonacci chain's window is a fractal in three pieces, and a straight cut across it meets up to six.

 Every rational holds a window, and there is nothing in between. The ground state density of a chain of particles with a convex repulsion, against the chemical potential that sets how many of them there are. Every density with denominator up to 24 is a flat step of positive width — 177 of them — and the steps with the simplest fractions are the widest: a half takes 19 per cent of the whole range on its own. The risers between them are not smooth stretches; they are where the densities with larger denominators sit, and a finer computation fills them with more steps. What is left after every rational has taken its window is the irrational densities, which are the genuinely incommensurate ground states and have no width at all. Order without repetition

Every fraction holds a window

Three essays here name the devil's staircase and none computes one. A chain of particles with any convex repulsion has a ground state at every rational density holding an interval of chemical potential to itself — 709 of them computed, the widest taking 19% of the axis and the narrowest two parts in a million million — and the incommensurate densities are what is left over.

 Identical layers, each turned by an angle no number of turns undoes. Plan views of 4 layers of a stack. Each layer is the same square lattice with one cell shaded and one direction drawn, and each is turned from the one below through the angle whose cosine is three fifths, about 53.13 degrees. That angle is not a rational part of a full turn, so no number of layers brings the drawn direction back to where it started. A tiling of space with this structure has a symmetry that turns one layer onto the next and climbs one layer, and it has no translation. Order without repetition

Aperiodic is two words in space

A tile is aperiodic when none of its tilings is periodic, and periodic has been read two ways: a tiling with a translation, or a tiling with infinitely many symmetries. In the plane those are one condition, provably. In space they come apart, and a prism found in 1988 sits exactly in the gap.

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