Crystallography & Symmetry
patterns derived from their groups
Concept

Higher-dimensional lattice — where it appears

A periodic lattice in more than three dimensions, used to describe an aperiodic structure as a slice of a periodic one. The extra dimensions are bookkeeping rather than a claim about where the material is, and they are what make the arithmetic decidable.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

An integer matrix of order 5. The companion matrix of the 5th cyclotomic polynomial has whole-number entries and order exactly 5, so it is a genuine 5-fold symmetry of a 4-dimensional lattice. The plane it rotates sits at an irrational angle to that lattice, and the lattice's shadow on it is dense — which is why a projection needs a window before it becomes a pattern.

Where five-fold becomes legal

A five-fold rotation with whole-number entries exists — in four dimensions, as a four-by-four matrix that can be written down. The plane forbids it because the plane is too small, and knowing which dimension is large enough changes what a quasicrystal is.

restriction · Restriction The icosahedral group, counted. The sixty rotations of an icosahedron, found by trying every map that sends one adjacent pair of vertices to another and keeping those that carry the whole vertex set onto itself. They fall into 6 axes of order 5, 10 axes of order 3, 15 axes of order 2 — and the fivefold axes are the reason this group cannot be the point group of any crystal, since no three-dimensional lattice admits a rotation of order five. Quasicrystals have it anyway, which is what made 1982 an argument rather than a measurement.

Icosahedral symmetry

Sixty rotations, six fivefold axes, and no lattice in three dimensions that can hold any of them. It is the point group a crystal is forbidden, and the one the first quasicrystal turned out to have.

aperiodic · Quasicrystals 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.

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.

aperiodic · Quasicrystals In the plane, the lengths do name the lattice. Every reduced binary form with coefficients up to 20 — 1750 lattices — with its theta series computed to 120 terms. No two of them agree. That is Schiemann's theorem for binary forms, which says the theta series determines the lattice in two dimensions and in three, confirmed here as far as the search reaches rather than proved. The closest pair is worth the space: two lattices whose shortest vectors both have squared length twenty agree for 38 terms — because neither has any vector before then — and part at the next one.

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.

lattices · Lengths 66 squares and 106 rhombs. The Ammann–Beenker tiling, built by keeping the points of a four-dimensional lattice whose companion image falls inside an octagon and projecting them into the plane. Every tile has the same edge length; the squares and the forty-five degree rhombs are told apart by their diagonals. Nothing was placed — the faces were found among the projected points.

Eight-fold, with the golden ratio taken out

Every quasicrystal on this site has been built on five: Penrose's rhombs, the Fibonacci chain, the ten-fold pattern Shechtman measured. A method that works only on the golden ratio is a method tuned to its answer — so here is the same construction run on eight, where the irrational is √2 and nothing else changes.

aperiodic · Quasicrystals Two, seventeen, two hundred and thirty, and then. The number of arithmetic crystal classes and the number of crystallographic groups in each of the first six dimensions, with the second divided by the first. The classes multiply by between five and fourteen a dimension; the groups multiply by much more, and the quotient — how many groups an average class carries — goes 1.00, 1.31, 3.15, 6.74, 36.5 and 339. The last column says what is derived on this page and what is quoted: the plane in full, six of the seventy-three classes in space, and nothing at all above three dimensions, where the counts come from machine enumerations of the 1970s onwards.

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.

restriction · Finiteness One group without an axis, and as many as the order with one. For a rotation of each order that an integer matrix can have in a small dimension, the number of space groups its arithmetic class admits — computed from the cohomology rather than enumerated. A rotation acting on the smallest lattice that will hold it fixes no direction and admits exactly one group: the symmorphic one, with no screw. Add a direction it leaves alone and the count becomes the order of the rotation, and the extra groups are its screws. The four-fold with an axis gives four, which are P4, P4₁, P4₂ and P4₃; the five-fold with an axis gives five, in five dimensions, where no published table exists to check it against.

The screw a dimension does not have

The extension count is a machine that runs in any dimension, and the seventeen were the case where every step could be checked against a list arrived at four other ways. Run on a cyclic point group it has a closed form two lines long — and it says a five-fold screw axis does not exist in four dimensions, which is a prediction rather than a check.

classification · Cohomology

Named alongside it

The objects these essays reach for when they reach for this one.

QuasicrystalArithmetic crystal classClassificationCrystallographic restrictionCut-and-projectCyclotomic polynomialsEnumerationGroup extensionIndexingPerpendicular spacePoint groupAmmann beenker

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