Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Generated, not drawn — page 3

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.
A quasilattice down a 5-fold axis: 10-fold, from an axis of order 5. 153 points of a three-dimensional quasilattice, made by keeping the points of Z⁶ whose perpendicular image lies inside a window and projecting them into ordinary space, then viewed along one of its 5-fold axes. There is no lattice here and no unit cell, and the symmetry is nevertheless exact: all sixty rotations of the icosahedral group carry the set onto itself, on 153 points of its core, measured by applying them rather than assumed from the construction. Seen down this axis the set comes back to itself under a turn of a 10th and no finer turn, measured over every turn up to a twelfth — twice the order of the axis, and the factor of two is a centre of symmetry: this set equals its own negation, which is what a window centred on the origin produces, and that is checked here rather than assumed. A diffraction experiment would record the same 10 whatever: the measured intensity acquires a centre whether or not the structure has one, so the tenfold photograph of 1982 does not by itself distinguish a structure with a centre from one without. Order without repetition

Six integers, and the lattice that holds them

A fivefold rotation is not an integer matrix in three dimensions and is one in six. The icosahedral group permutes its own six fivefold axes, so in coordinates along those axes every one of its sixty rotations is a signed permutation — and Z⁶ is a lattice it maps onto itself.

 Every plane group from at most 4 operations. For each group, the fewest operations that generate the whole of it — the point operations and both lattice translations, since a group that does not reach its own translations is a different group. The floor is the abelianisation's number of invariant factors, which no group can beat, and the search is exhaustive over the operations within one cell of the origin. 14 of the seventeen meet their floor, which settles those exactly; the other 3 need more than the abelian argument can see, and p3m1 needs three where its abelianisation is cyclic. Operations

How few operations make a pattern

A plane group is infinite, and a handful of its operations is enough to rebuild all of it. How small a handful is a question with a floor from the abelianisation and a ceiling from an exhaustive search, and for fourteen of the seventeen the two numbers meet.

 Five shapes, and a lattice in space has no other. The five combinatorial types a Wigner–Seitz cell can have in three dimensions — cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, truncated octahedron — each drawn from a lattice that produces it. Fedorov proved in 1885 that there are no others, and that fourteen faces is the most any of them has, which is Minkowski's bound of 2(2ⁿ − 1) in three dimensions. Each solid here is cut out by the perpendicular bisectors of nearby lattice vectors and its volume checked against the primitive cell's, which is what catches a face that failed to appear. Lattices

Five parallelohedra, and no others

The cell that needs no basis and no convention has, in three dimensions, exactly five shapes. The fourteen Bravais lattices produce all five between them — and which one a lattice gives is not decided by which of the fourteen it is.

 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. The classification

Twenty-one vertices, eleven tilings

Regular polygons meeting at a point must fill exactly a turn, which is a Diophantine equation with seventeen answers and twenty-one cyclic arrangements. Ten of the twenty-one tile nothing at all — and the argument that kills them counts places round a polygon rather than measuring anything.

 glide: 3 mirrors. A glide of the plane, drawn together with the mirrors it is a product of. The first shape is the motif; the pale ones are what each mirror in turn produces; the last is the image the motion itself gives. There are 3 mirrors, which is the smallest number that can produce this kind of motion, and their product was formed and compared with the motion before the figure was drawn. Operations

Three reflections, and never four

Every motion of the plane is a product of mirrors, and the number needed is never more than three. That count is not a curiosity about mirrors — it is the classification of the four motions written as an integer, with the parity of the number deciding handedness and the geometry of the last two mirrors deciding everything else.

 3.4.6.4 and its dual. The tiling in pale outline with its dual drawn over it: one dual vertex at the centre of every tile, one dual edge across every shared edge, and one dual tile round every vertex. 3.4.6.4 has 3 kinds of tile and one kind of vertex; its dual has one kind of tile and 3 kinds of vertex, and the congruence of those tiles is checked rather than eyeballed — every dual face presents the same cyclic sequence of squared edge lengths, compared exactly. That swap is what the eleven duals are for: read one way the list classifies tilings with all vertices alike, read the other it classifies tilings with all tiles alike. The classification

Eleven duals, one tile each

Swap the vertices of a uniform tiling for its tiles and the eleven come back as eleven tilings by a single repeated shape. Three of those shapes are pentagons — which is worth pausing over on a site whose other essays prove that five-fold symmetry cannot exist.

 Which Schläfli symbols close. Every {p, q} with p polygons round each face and q faces round each vertex, from three to six of each. A solid exists only when 2p + 2q − pq is positive, which is the same statement as 1/p + 1/q > ½; the five that qualify carry their vertex, edge and face counts, and the three on the diagonal where the expression vanishes are the three regular tilings of the plane. Past them the expression is negative and the answer is the hyperbolic plane, where the list never ends. The five, the three and the infinity are one inequality read at its three signs. What a lattice forbids

Five solids from one inequality

Five families of rotation group in space, five regular solids, three regular tilings of the plane and an endless supply of hyperbolic ones — all of it is 1/p + 1/q compared with a half, read at its three signs.

 hexagonal: 0.5 and 0.577. The hexagonal lattice with both radii drawn together: the small circles are the largest that do not overlap and the large ones the smallest that leave no gap. The line runs from a lattice point to the deepest hole, which is a corner of the cell around it, and its length is the covering radius 0.5774 against a packing radius of 0.5. The deep hole was found by search on a grid of 24 and then refined, and checked afterwards against an independent grid. Lattices

Covering and packing want different lattices

A lattice has two natural radii — the largest spheres on its points that do not overlap, and the smallest that leave no gap — and both are radii of the same Voronoi cell. In the plane one lattice is best at both. In space the best packer and the best coverer are different lattices, and they are duals of one another.

 Every window returns within 3.0 n. For each window length, the largest distance between two consecutive occurrences of the same window, measured over 46,368 tiles. The gaps are Fibonacci numbers, and the ratio to the window length stays below 3.00 — the chain is linearly repetitive. That is a strong statement of uniformity: there is no stretch of the chain, however far out, in which a given patch fails to occur within a bounded multiple of its own size. Order without repetition

Every patch comes back

A chain that never repeats still repeats everything in it. Every block of tiles occurs again, and again, within a bounded multiple of its own length — and how large that multiple is turns out to be a fact about the continued fraction of a slope.

 a general quadrilateral tiles. A general quadrilateral — convex, with no equal sides and no parallel edges — with copies placed by half-turns about edge midpoints. The patch was checked by sampling 2000 points inside a disc: every one of them lies in exactly one tile, so there is no gap and no overlap anywhere in the region tested. The classification

Which shapes tile by themselves

Every triangle tiles the plane. So does every quadrilateral, convex or not. Six sides admits three families, seven sides admits nothing at all — and the five-sided case took a hundred years and finished with a computer search. The bound at seven needs no search: it is Euler's relation with the curvature set to zero.

 p4: a domain of 38 cells with 7 walls. The fundamental domain of p4 on a grid of 12ths, with the walls it shares with its neighbouring copies marked. Each wall names the element that carries this copy onto the copy across it, and there are 7 distinct such elements. Those elements generate the whole group — checked by closing them up and requiring every coset and the whole translation lattice to be reached, not assumed — which is Poincaré's theorem, and it means the generators of a wallpaper group can be read off a picture. The domain is pixelated rather than a polygon, so the wall count is a property of this domain and not of the group. Operations

Every wall names a generator

The copies of a fundamental domain tile the plane and stand in one-to-one correspondence with the elements of the group. So the elements that carry the home copy across a wall generate everything — and the generators of a wallpaper group can be read off a picture rather than looked up.

 5 units of 70.53°: 7.36° left. 5 tetrahedral units of face-centred cubic metal, each the mirror image of its neighbour in a {111} plane, arranged about a common ⟨110⟩ edge. The angle between two such planes is arccos(1/3) = 70.53°, computed from the plane normals rather than quoted, and 5 of them come to 352.64°. The shaded sector is what is left over: 7.36°, or 2.04 per cent of a full turn, which must be taken up by strain, by a gap, or by a defect along the axis. What a lattice forbids

Five copies, and the gap they leave

Gold, silver and silicon grow particles with a five-fold axis down the middle, out of a lattice that forbids one. Nothing is violated: five tetrahedral pieces of ordinary face-centred metal, each the mirror image of its neighbour, come to three hundred and fifty-two and a half degrees rather than three hundred and sixty — and the seven degrees left over have to go somewhere.

 P2₁/c from 27 marks. The marks of P2₁/c's plan, counted by kind, and what they rebuild to. Each mark is reduced to what a reader can see and handed to a closure with the matrices withheld: an axis gives its direction, its position and how far one turn advances along it; a plane gives its normal, its position and its slide. The lattice supplies the candidate matrices, the closure supplies the rest, and what comes back is the group — 4 operations against 4, with nothing missing and nothing extra. Into space

The plan contains the group

A space-group diagram has always been treated here as a picture of the group. It is more than that: hand back the marks alone — no matrices, no operations, not even the centring — and the group comes out exactly, forty-five times out of forty-five.

 Dropping one invariant of 3m makes two orbits agree. Every lattice point within four cells of the origin, coloured by the values a proper subset of 3m's invariants takes on it — the 2 generators with the first one removed, over a window of 4 cells. With the full set, the 25 orbits of the group take 25 distinct sets of values, one each, so the invariants are a complete set of coordinates on the quotient. With one removed, the two circled points — in different orbits, so no operation of the group carries one to the other — take the same values and become indistinguishable. That is the whole content of the statement that a complete set of invariants separates orbits: the completeness is what is doing the work. Operations

An orbit is what the invariants cannot tell apart

Two points of the plane lie in the same orbit of a group exactly when every invariant polynomial takes the same value on both. One direction of that is a definition; the other is a theorem, and it is checked here by comparing every pair of points in a window both ways.

 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. The classification

A tiling of the whole plane, decided on one tile's edge

Whether a shape tiles the plane is a question about an infinite object, and there is no procedure that answers it. There is a procedure that answers it *sometimes*, and it reads nothing but the shape's own boundary — a closed path of a few dozen steps, cut into six arcs. When the cut exists the tiling exists, and the cut names the group that makes it.

 Equilibrium and growth are different shapes. Two predictions for the habit of the same cubic crystal, computed through the same intersection of half-spaces. The equilibrium shape puts each face at a distance proportional to its surface energy, which is Wulff's construction; the growth shape puts it at a distance proportional to its growth rate, taken here from this site's own spacing rule. They differ — the equilibrium shape carries {111}, {110}, {100} and the growth shape {100} — and the difference is between two rules rather than between two pieces of code. A crystal on a bench has grown; a crystal annealed long enough has relaxed; the two look different and neither picture is wrong. Symmetry at work

The fast faces are the ones that vanish

A crystal has two predicted shapes and they are not the same. One minimises surface energy and is what a crystal settles into; the other is what growth leaves behind, and in it a face that grows quickly grows itself out of existence.

 anisohedral: 2 orbits of congruent tiles. A tiling of the plane by 8 copies of one shape per cell of a lattice of index 64, drawn 1 cell across and 8 up, and coloured by which orbit of the tiling's own symmetry group each tile belongs to. The group has 4 operations per cell and 2 orbits: every tile is congruent to every other, and no motion of the whole pattern carries a tile of one colour to a tile of another. Congruence is a fact about the shapes; an orbit is a fact about the pattern, and they are different facts. The classification

One shape, two kinds of tile

A tiling by copies of a single shape looks as though it must be homogeneous — every tile is congruent to every other, so what could distinguish them? The symmetry group can. There are shapes that tile the plane and admit no tiling whose group carries any tile to any other, and the smallest of them has eight cells.

 A circuit that closes on the wrong point: (1, 0). A square lattice with one extra half-column, drawn as a graph: the rows above the core have one more site than the rows below, and the core is the site at the end of the extra column. The path is 4 steps east, 4 north, 4 west and 4 south — the same number out as back — and it ends one lattice vector from where it started. Every one of the 12 circuits in the survey that goes round the core fails by that vector, and all 10 that miss it close exactly. Symmetry at work

The circuit that does not close

A defect in a crystal is usually introduced as a picture — an extra half-row of atoms, a wedge taken out. What makes a defect a crystallographic object rather than a drawing is a closure failure: walk a closed circuit through the lattice and come back to the wrong point, by an amount the lattice itself decides.

 4mm: a degeneracy tuned into existence, and gone at the next weight. Four invariant operators on one orbit of 8 points under 4mm, differing only in the weight given to a single class of pairs. The first column is not a choice: it is the value that weight has to take for two levels of different symmetry to arrive at the same number, found by sweeping the weight and closing on the crossing, and the two levels there agree to 1.0e-9. The character table predicts levels of sizes 1, 1, 1, 1, 2, 2; the tuned column shows 1, 1, 1, 2, 3 and every other column shows the predicted pattern again. That is the whole of what an accidental degeneracy is — a property of one choice of weights, not of the group — and it is why the weights have to be moved before a degeneracy is called forced. A degeneracy the group requires would be in all four columns, because nothing respecting the symmetry can lift it. What symmetry decides

A coincidence the group did not ask for

Two levels sitting at the same value look identical whether symmetry required it or not. The difference is testable: move the numbers the symmetry does not decide and watch what survives, because a degeneracy the group forces cannot be shifted by anything the group leaves alone.

 p4m from 3 orbits of waves — detected p4m. A density built as a sum of 3 symmetry-adapted waves of p4m, each of them the average of a plane wave over the group, shaded from light to dark across one cell. The level set of this density — the darkest points of it — was handed to the same detector the pattern figures use, and it reports p4m, which is exactly the group the waves were built from. The waves are invariant by construction, so the density can never have less symmetry than the group; the interesting direction is the other one. The classification

How many waves a group permits

A pattern can be written as a sum of waves instead of as an orbit of a motif, and then the group ties the coefficients together and forbids some outright. Building a density from the permitted ones and handing it back to the detector closes the same loop through a different door — and at low resolution the density has symmetry the crystal has not.

 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.

 p6m: freezing Γ2 leaves p31m. The same crystal three times. On the left, a pattern with the full symmetry of p6m. In the middle, the displacement each atom takes under the order parameter — the arrows are the mode, drawn in the first colour for one set of atoms and the second for the other where there are two. On the right, the atoms moved by a small multiple of those displacements. The group of the right-hand pattern is p31m, of index 2 in the parent, and it was found by the detector from the point set alone. The prediction — which operations carry the displacement field to itself — is made separately, from the mode and not from the points, and the two lists of operations are identical. Into space

An order parameter is a representation

The quantity that measures a lost symmetry is not a number the physics chooses freely: the parent group mixes its components, so it carries a representation, and the symmetry that survives is what leaves its value alone. Every prediction here is checked by moving the atoms and asking the detector.

 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.

 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. Symmetry at work

A structure with the distances thrown away

Keep which atoms are joined and throw away where they are, and what is left is an infinite graph that can be written on a postcard: a few vertices, a few edges, and a pair of integers on each. Two things about that writing-down are free, and neither of them changes the net.

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