Generator

What each group leaves distinct

The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

8 essays call orbit-count. 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 522 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.

Operations

Counting what a group cannot tell apart

Sixty-five thousand ways of putting two species on sixteen sites; eight hundred and five structures. The difference between those numbers is not a division, because the symmetric arrangements have short orbits — and the count that gets it right is an average of fixed points.

Operations

Every colour count at once

Eight hundred and five structures is the answer for two species on sixteen sites. For three species it is a different sum, and for four another. Averaging cycle counts instead of fixed-point counts turns the answer into a polynomial — and refining the same average says how many structures there are at each composition, which is the number anybody actually needs.

$Order 5: 32,768 arrangements. An Aztec diamond of order 5, with every possible dimer drawn at an opacity equal to the fraction of arrangements it appears in — a probability computed exactly, by counting the arrangements of the region with that dimer's two sites removed, rather than sampled. The four corners come out nearly certain and the middle nearly even, with a circle between them. The most certain dimer here occurs in 0.97 of the arrangements, which is 1 − 2⁻5 exactly, so nothing is frozen at any finite size.$ Order without repetition

How many arrangements one rule allows

Every count in this collection so far has been a count of symmetries, or of orbits under one. Here is a different count: the arrangements a purely local rule permits on a fixed lattice, with no symmetry quotient anywhere in it. The answers are enormous, they are exact, and the useful quantity is not the number but its growth per site.

A structure that is its own negative, and one that is not. Two arrangements of two species on a four-by-four block with p4m symmetry, each with eight sites of each species, beside the arrangement that exchanging the species produces. On the left the exchanged arrangement is the original one moved by an operation of the group, so the two are one structure: it is self-complementary, and its balance of species is forced by a symmetry. On the right the exchanged arrangement is a genuinely different structure, and the balance is a coincidence of count. Counting structures with the species interchangeable merges the pair on the right into one and leaves the one on the left as it was, which is why the count does not halve.

Operations

Swapping the species does not halve the count

Let the two species on a lattice be interchangeable and every structure pairs with its negative, so the count should halve. It does not, because some structures are their own negative: an operation of the group carries them onto the arrangement with the species exchanged. Those are counted by the operations whose every cycle is even, they all sit at exactly half composition, and read as spins they are the arrangements whose zero magnetisation a symmetry enforces.

Every square-lattice superstructure, by the size of its cell. The number of distinct arrangements of two species on the square lattice whose primitive cell holds exactly n sites, for n from one to twelve, logarithmic, counted over every cell shape at once: every sublattice of index n up to the lattice's own symmetry, every colouring of it up to translations and the sublattice's stabiliser, and only those whose true period is that sublattice. Odd prime sizes are low, because only a few cell shapes exist; sizes with many divisors are high.

Operations

Every superstructure counted once

The 805 arrangements of two species on a four-by-four cell were never the structures with sixteen atoms: forty-nine of them repeat in a smaller cell, and every sixteen-atom structure whose cell is not a four-by-four square is missing. Counting structures rather than arrangements means counting over every shape of cell at once, and discarding every arrangement that repeats sooner. On the square lattice that gives 2, 2, 4, 11, 16, 40, 48, 148 structures for cells of one to eight atoms, and 22,413 with sixteen.

$How many superstructures a diffraction pattern tells apart. For two-species superstructures of the square lattice with one to twelve sites in the primitive cell: the number of structures, and the number of classes that the positions of their superlattice spots separate them into, the positions together with the composition, and every intensity. At twelve sites 2,012 structures give 167 sets of spot positions, 504 with the composition and 1,815 sets of intensities. At five, seven and eleven sites the positions distinguish only the cells; the intensities distinguish everything. The bar shows the share resolved by positions, dark, and by intensities, light.$ Operations

The spots say the cell and little else

Of the 2,012 two-species superstructures of the square lattice with twelve sites in the cell, more than half light every superlattice spot their cell allows, so the positions of the spots say which cell and nothing more. Positions separate the 2,012 into 167 classes, the composition takes that to 504, and every intensity to 1,815. At a prime number of sites no spot is ever missing, because a sum of distinct p-th roots of unity cannot vanish.

When a three-species superstructure's spots can go dark. Every structure of three species on the square lattice with three to eight sites in its primitive cell, using all three species, and every superlattice spot of each, sorted by whether the spot is dark for every choice of scattering contrasts, dark at exactly one ratio of the two contrasts, or never dark; with the number of distinct ratios and their degree over the rationals. At the primes 3, 5 and 7 no spot is dark for all contrasts and no dark ratio is rational: none at three sites, quadratic ones at five and cubic at seven. At four and six sites every dark ratio is rational, at eight some are and some are not, and at six and eight some spots are dark for every contrast.

Operations

Three species hide a spot at the golden ratio

With two species and a prime number of sites in the cell, no superlattice spot can ever be missing, because a sum of distinct roots of unity of prime order vanishes only when it is all of them. With three species the structure factor weighs two sums against each other with two scattering contrasts, and the argument changes shape without breaking. At a prime size no spot is dark for every contrast and none at a rational ratio of them, but a spot can be dark at one irrational ratio: at five sites, a power of the golden ratio. It happens only in structures with a centre of symmetry.

Three-species arrangements whose sums are parallel at every spot, with and without a centre. Every arrangement of three species, all present, on a cyclic cell of four to nineteen sites. The third column counts those whose two phase sums are parallel at every superlattice spot, the fourth those among them with a centre of inversion, the fifth those without one, and the last how many kinds the exceptions make up to translation, inversion, relabelling the spots and renaming the species. Primes are shaded. At 5, 7 and 11 sites every parallel arrangement has a centre; at 13 sites 936 do not, and they are all one arrangement; at 17 there are 17 kinds and at 19 there are 48.

Operations

Thirteen sites obey Friedel's law without a centre

At five and seven sites the earlier census found that a three-species structure has a contrast that darkens a spot exactly when it has a centre, and asked whether that is a theorem. It is not. The two phase sums are parallel exactly when the A–B pairs are as common at every separation forwards as backwards, and exactly when no anomalous scatterer can produce a Bijvoet difference anywhere in the pattern. At 5, 7 and 11 sites only a centre achieves that. At 13 sites one arrangement does it without one.

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