Enumeration — where it appears
Named by 57 essays across 9 fields — each of them below, with the objects they name alongside it.
The seventeen
Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon.
Thirty-two, and no others
There are exactly thirty-two ways a crystal can be symmetric about a point. Not thirty-two that anybody has catalogued — thirty-two that a finite search produces, from two starting groups, with every step of the reduction counted separately so that no two of them can quietly compensate.
A fingerprint that gave the right answer
The thirty-two classes were merged on a fingerprint — the census of operation types — and the fingerprint returned thirty-two, which is correct. Returning the correct answer is not the same as being entitled to it, and the difference took three wrong constructions to close.
Sixteen candidates, ten groups
One point group, one lattice, and every consistent way of attaching translations to it — enumerated in full. The count comes out at sixteen, and then at ten, and the step between the two numbers is a decision about what "the same group" means rather than an arithmetic fact.
Eleven ways to turn while climbing
Five rotation orders survive the crystallographic restriction, and each of them admits a screw for every whole number of cells its turns can amount to. That is a sum with four terms, and it is why there are eleven screw axes rather than some other number.
Reflect, then slide by half of something
A glide's slide must double to a lattice vector, which leaves three candidates in any plane and a fourth that exists only where centring has already made a half-diagonal into a lattice vector. Five letters, and the fifth is the one the enumeration explains.
Why sixteen become seven
Four extra operations give sixteen combinations and seven groups. The nine that vanish are not cases anybody forgot — each one comes back from the closure holding something it was never given, and one of them changes the lattice underneath it.
Why there is a list at all
Five lattices, seventeen groups, thirty-two classes, two hundred and thirty. Every one of those counts came out of a search that had to know when to stop, and the reason it could stop is a divisibility Minkowski proved in 1887.
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.
The lengths do not name the lattice
Seventeen hundred plane lattices, every one with a theta series shared with no other — the lengths determine the lattice, and an exhaustive search says so. In sixteen dimensions two different lattices have identical counts at every distance, and the example is sixty years old.
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.
Reduction modulo three
A finite group of integer matrices survives being reduced modulo three: no two of its operations collide. That single fact proves the classification finite without computing any bound — and modulo two it is false, refuted by the inversion centre.
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.
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.
Seventy-five ways to be a thread
Eighty layer groups and seventy-five rod groups are usually quoted, both as numbers from the literature. The second is derived here, class by class — and the total alone turned out to be no check at all, because two errors of six groups each give seventy-five as well.
Two structures, one Patterson
Eight arrangements of four atoms on a ring of eight sites, and only seven distinct sets of interatomic vectors between them. Two of the arrangements are genuinely different and no measurement of intensities can tell them apart — at any resolution, for ever.
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.
Where the pairs come from
A structure whose atoms are the sums of two smaller sets has a partner: reverse one factor and the interatomic vectors do not notice. The construction is Patterson's own, it explains why homometry exists, and the smallest structure it can build has nine atoms for a reason worth following.
The unknowns against the observations
A structure determination is a fit of some number of parameters to some number of measurements, and both counts can be worked out before any data exist. The ratio turns out not to depend on how large the crystal's cell is, or on how symmetric it is — only on the resolution, and on that as a cube.
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.
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.
The twelve belongs to the vertex
Twelve pentagons is read as a fact about closing a surface. It is not: it is a fact about three edges meeting at a point. Let four edges meet instead and the sphere charges eight triangles; let five meet and it charges twenty; let six meet and it cannot be paid at all.
The surfaces a count by genus skips
A count indexed by genus steps in twelves and lands only on even numbers. A closed surface can have any characteristic at or below two, and the odd ones belong to the surfaces that cannot be oriented — where the projective plane charges six pentagons, a bill no orientable surface ever presents.
Seven friezes round a cylinder
A point group with one principal axis belongs to one of seven infinite families, and there are seven frieze groups. They are the same seven. Draw a frieze on a strip, roll the strip into a cylinder, and every translation becomes a turn about the axis and every glide a rotoreflection.
A gap the sphere does not have
A net of pentagons and hexagons on the projective plane must have six pentagons, and the count permits any number of hexagons. Not every number happens. Lifting each net to the sphere turns the question into one about which cages have a centre — and the answer leaves two gaps where the sphere has one.
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 molecule size that hides a disorder
Two disorder models with the same occupancy leave averaged structures that differ only in a handful of partial atoms. The difference is 20% in structure factors for a ten-atom molecule and 3% for a sixty-atom one — so the data choose between the models for a small molecule and stop choosing for a large one, and seven pairs are identical at any size.
Going up costs the cell a parameter
The usual asymmetry — finitely many maximal subgroups below, infinitely many minimal supergroups above — is false in both halves for a plane group. Both directions are infinite and equinumerous index by index. The real asymmetry is that 17 of the 31 edges cost the lattice a parameter going up and nothing going down.
Symmetry does not rescue a Patterson
Every homometric pair found so far sits on a bare ring with no operations imposed, and a real crystal sits in a space group. Impose one and the ambiguity does not go away: 12 of the 13 groups searched still have pairs, and at six atoms the hexagonal groups are indistinguishable two to three times as often as the general position.
A thread's hand is not a choice
A sheet's handedness in space depends on a sign that the plane pattern does not fix, so one plane group carries several sheets and exactly one of them is chiral. A thread has no such freedom: 32 of the 75 rod groups are chiral, they sit over 9 of the 27 axial classes, and which they are is settled before any structure is drawn. Only its direction depends on how the class lies along it.
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.
A line carries one screw
A plane can hold two glide operations at once, and in 1992 the International Tables invented a letter for the case where neither has a claim. The same question put to an axis has the opposite answer: across 3,388 axes, not one line carries two — and the reason is that a slide has two dimensions to be ambiguous in and an intrinsic translation has one.
A rolled sheet is never one of a pair
Rolled up along every lattice vector that gives a crystallographic tube, the seventeen plane groups reach fifty-four of the seventy-five rod groups: every achiral one and eleven of the chiral. Not one of the sixteen screws that come in left- and right-handed pairs is among them, and the reason is a single fact about how far a rolled lattice can climb.
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.
Everything except the hexagons
Three counts of what a closed net must carry end on the same admission: an arithmetic saying what a net must charge does not say that a net exists. Eberhard's theorem says how close the charge comes to being enough, and the answer has a shape nobody would guess — it fixes every face count except the hexagons, and the hexagons are exactly the entry it cannot see.
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.
Finitely many is not few
Bieberbach's third theorem says each dimension holds finitely many crystallographic groups and gives no idea how many. The counts are 2, 17, 230, 4783, 222018 and 28927922, and dividing them by the number of arithmetic classes says which of the classification's three steps supplies the explosion — the step that attaches translations, not the one that finds the matrix groups.
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.
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.
Two hundred and nineteen over seventy-three
The space groups average three to an arithmetic class, and an average of three could be a few classes carrying dozens or every class carrying a handful. Derived class by class from the cohomology and its normaliser, the answer is the second: twelve classes carry one group, the median carries two, and the largest carries sixteen. Sixty-one of the seventy-three admit a non-symmorphic group, and the step that merges relabellings, which the plane barely uses, removes more than a quarter of the list.
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.
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.
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.
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.
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.
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.
An experiment sees a Farey sequence
A devil's staircase has a step at every fraction, and no experiment sees them all. Which ones it sees is not a matter of luck: every step with the same denominator has the same width, so a finite resolution shows every fraction up to one denominator and nothing beyond it — a Farey sequence, whose length is a sum of Euler's totient. How that denominator grows with the resolution is the one thing the interaction decides, and what is left unseen is steps, never an incommensurate phase.
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 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.
Eighty is seventeen, seventeen and forty-six
The eighty layer groups are usually quoted, and an earlier count here reached sixty-three and blamed the translations. The eighty follow in three lines from things already counted: a layer group is a plane group with a sign on each operation, and the signs are either all plus, all doubled by a mirror in the layer, or a two-colouring of the plane group — seventeen, seventeen and forty-six. The sixty-three had missed the twenty colourings that turn a layer over on a translation, which are glide planes lying in the layer itself.
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.
The third survives a rectangle
On every even square torus, exactly a third of the square-ice arrangements have both windings divisible by three, and the explanation offered was a theta function that is its own dual. A test was proposed: on a torus twice as long as it is wide the two windings have different stiffnesses, the explanation gives a different number, and the exact third should fail. It does not fail on any of eight rectangles. But the explanation, worked through properly for a rectangle, never predicted that it would. The two directions of a rectangle are each other's duals, and their factors cancel to a third at every shape.
Ninety-six four-colour designs
The seventeen plane groups have 281 four-colourings, counted as subgroups of index four, and many of them are the same design described from a different origin, on different axes or from a neighbouring cell. Identified under every affine map that carries the pattern's group to itself, they make ninety-six designs. The same computation run at two and three colours gives forty-six and twenty-three, and all three numbers match the classical counts of coloured plane groups.
Every band group is a ribbon
A ribbon cut from a sheet has a symmetry group with translations along one direction only, one of 31 band groups. The 31 can be counted from three signs: whether an operation reverses the ribbon's direction, its width, or the sheet's faces, and whether it slides half a period. Cutting the eighty layers every way reaches all 31, seven of them as single edges and twenty-four only as ribbons cut symmetrically about a special line. Five of those twenty-four have no improper operation at all.
Five colours, seven colours, and the primes that split
Four colours gave the seventeen plane groups 281 colourings, five different groups permuting the colours and ninety-six designs. Five colours give 104 colourings, three groups and fourteen designs, and from five upwards every prime behaves the same way. A prime number of colours that the point group cannot divide forces every colouring onto a sublattice the rotations keep, and whether the square and hexagonal patterns have one is decided by whether the prime splits in the Gaussian or the Eisenstein integers.
Twenty-five colours, and the mirrors come back
With a prime number of colours from five up, five of the seventeen plane groups can never be coloured at all: the ones with a mirror across a rotation. With the square of a prime they all can. A colouring with twenty-five colours may meet the lattice in the whole plane modulo five, which every group keeps, and the colours then form a plane the point group acts on by its matrices. The designs come out as the prime's count again, with one more for each of the seventeen groups.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
CountingCensusCombinatorial curvatureCrystallographic restrictionOrbitPoint groupEntropyThe Euler characteristicInversion centreLocal rulesScrew axisStructure factor