Crystallography & Symmetry
patterns derived from their groups
Generator

One layer, and the two ways to sit on it

One layer, and the two ways to sit on it
One layer, and the two ways to sit on it. A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours.

A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours.

12 essays call close-packing. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

One layer, and the two ways to sit on it. A close-packed layer — large pale discs, each touching six others, which is as tight as one layer of equal spheres can be. Its hollows come in two sets, marked in the two smaller colours, and a second layer must take one set or the other; the two choices are mirror images and equally good. The third layer then faces the same choice again, and this time the two answers are genuinely different: over the first layer, or over the hollows the second did not use. That single decision, repeated, is the whole of close packing — and nothing in the geometry prefers either answer, because both give the same density and the same number of touching neighbours. Symmetry at work

Two stackings, one density

Stack spheres as tightly as they will go and the third layer has a free choice. Both answers fill exactly the same fraction of space and give every sphere the same twelve neighbours — and their space groups are Fm3̅m and P6₃/mmc, which is the only thing that tells them apart.

 The symmetry of a cut through Pnma. Every height in one cell, and the number of operations of Pnma that map the plane at that height to itself. The answer is 2 almost everywhere and rises to 4 at the special heights, where the plane group named above the spike is what a reader looking down at that surface would see. The rule is two conditions and no more: the operation must not tilt the plane, and the plane must come back to its own height — so a twofold axis lying in the plane survives at two heights per cell and a screw axis along the normal survives nowhere. The classification

What a cleave leaves

A surface is a crystal that has been cut, and the symmetry it presents is what the space group leaves of itself on that plane. Two conditions decide it — the plane must not tilt, and it must come back to its own height — and the answer changes with where the cut was made.

 How many close packings there are of each period. Every cyclic sequence over three letters with no two adjacent alike is a close packing, and two sequences describe the same structure when one becomes the other by rotating the cycle, reversing it, or relabelling the three positions. Counting the classes that remain gives 1 of period 2, 1 of period 3, 1 of period 4, 1 of period 5, and 38 altogether up to period 10. Period two is hexagonal close packing and period three is cubic; everything above them is a polytype, equally dense and equally close packed, and silicon carbide has been found in more than two hundred of them. Nothing in the geometry chooses. What chooses is an energy difference of a few thousandths of an electron volt per atom, and this site computes no energies. Symmetry at work

How many polytypes there are

One free choice per layer, repeated, gives a family of structures with the same composition, the same density and the same twelve neighbours — differing only in a sequence. Counting them up to rotation, reversal and relabelling turns "silicon carbide has hundreds of forms" into an enumeration.

 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.

 Arrangements per site, falling towards the exact value. The number of configurations of an L × L torus obeying the ice rule, taken to the power of one over the number of sites. The largest computed here is 4,484,823,396 configurations on a 7 × 7 torus. The values fall towards Lieb's exact 1.54 from above, and every one of them is above Pauling's estimate of 1.5 — which undercounts, because it treats the vertices as independent and they are not. Order without repetition

The arrangements a crystal keeps at absolute zero

Ice has a residual entropy, and the number a calorimeter measures is the logarithm of a count of arrangements. Pauling's one-line estimate of that count is out by two and a half per cent; the exact count in two dimensions is available, falls towards its limit from above, and the whole disorder is invisible to a diffraction experiment, which sees only the average.

 The whole space of plane lattices, and its corner. Every plane lattice appears exactly once in this picture. Scaling changes no density, so the leading coefficient is fixed at one; reduction then confines the other two to 0 ≤ b ≤ 1 ≤ c, and every lattice has exactly one reduced form. The curves are the levels of constant density, which are parabolas — a density d needs 4c − b² to equal (π/2d)². They crowd toward the corner b = c = 1, which is the hexagonal lattice at π/√12 ≈ 0.9069; the square lattice sits on the left edge at π/4 ≈ 0.7854. The picture is a search over a region rather than over a list, which is what makes the answer a decision: there is nowhere else for a lattice to be. Symmetry at work

The densest lattice in the plane

Which arrangement of equal discs covers the most floor is a question about infinitely many lattices, and reduction turns it into a question about a two-parameter region with a corner. The answer is at the corner, and the argument finishes.

 Four ways to lay a second row on the first. The same shape four times, with the upper row related to the lower one by a translation, a half turn, a glide and a mirror. In each case the upper row is pushed down until it touches, and the number is the density that results. The three that keep the shape the same way round come out within two per cent of one another; the mirror packs at 78 per cent of the best of them — 22 per cent less dense — because it presents a protrusion to a protrusion. Stated the other way round, the best of the four is 28 per cent denser than the mirror; the two percentages are the same measurement against two different bases, and neither is the other. Symmetry at work

The four plane groups a molecule packs in

A molecule is not a disc: it has bumps and hollows, and packing it tightly means getting one molecule's bump into another's hollow. A mirror puts a bump against a bump. Filter the seventeen by that one observation and four survive — and the space groups the structural literature is mostly made of are the three-dimensional version of the same four.

 square: 3 dislocations, 2 stable. The short lattice vectors of the square lattice, grouped into orbits under its own automorphism group of 8 operations. Two vectors of one orbit are the same defect seen from different directions, so the number of kinds of dislocation is the number of orbits — 3 out to 4 times the shortest squared length. Each orbit has its own colour. Solid arrows are stable: no pair of shorter lattice vectors adds to them with a smaller total of |b|². Dashed ones split, and 1 of the orbits do. Which splits is decided by comparing integers; that the energy goes as |b|² at all is Frank's rule and comes from elasticity, not from here. Symmetry at work

How many dislocations a lattice has

A circuit round a defect comes back to the wrong lattice point, and the amount by which it misses is a lattice vector. That much is quantised. The next question has a number for an answer: how many *different* dislocations are there? Two Burgers vectors related by an operation of the point group are one defect seen twice, so the answer is a count of orbits.

 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.

 How densely each shape packs, by translation alone. The densest lattice packing of each shape, as its area over the critical determinant of its difference body. The two that tile the plane by translation reach one and must, which is a check on the search rather than a result of it. The triangle reaches exactly two thirds because its difference body is a hexagon. The many-sided approximation to a circle reaches π/√12, which this collection computes a completely different way. And the pentagon is the worst of them, which is where the search is doing work nobody could do by inspection. Symmetry at work

The densest packing of a shape that is not a disc

Which lattice packs equal discs most densely has a proof that finishes. Replace the disc with a pentagon and the same question has no closed form, but it does have a reduction: translates overlap exactly when the difference of their positions lies inside the shape minus itself, so the question becomes the smallest determinant a lattice can have while avoiding one convex body — and that is a search with a resolution attached.

 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.

 How far apart points on a sphere can be kept. For each number of points, the largest smallest angle a search could find between any two of them. Two unit spheres touching a third do not overlap exactly when their contact points are 60° or more apart, so the largest count whose best arrangement still clears 60° is the kissing number. Twelve clears it with three degrees to spare and thirteen falls short by more than three. The circle column is the same problem in the plane, where the answer is exactly 360/n and needs no search at all. Symmetry at work

The room a thirteenth sphere would need

Twelve equal spheres touch one, and whether a thirteenth could was argued in 1694 and settled in 1953. The reason it took so long is measurable: the twelve leave three and a half degrees of slack, which is enough room to look promising and not enough to use — and in the plane, where the same question has no slack at all, nobody ever argued.

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