Crystallography & Symmetry
patterns derived from their groups
Concept

Invariant — where it appears

A quantity computed from an object that no permitted change alters, and therefore something two objects must share to be the same object. A trace, an abelianisation and a reduced form are the three this collection uses most.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

The merge, and what witnesses it. The two holohedries are enumerated separately and their class lists merged where the element signatures agree. That returns the right total, which is not the same as being right: 33 merges are made and every one of them is checked by constructing an explicit change of basis carrying one group onto the other.

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.

point-groups · Crystal classes The crystal classes 4, 4̅, 3̅, 6̅. 4, 4̅, 3̅, 6̅: the orbit of a general direction under each group, giving 4, 4, 6, 6 poles, with general positions and symmetry elements. Filled marks are poles above the plane of the page and open ones below it.

What a trace decides

Ten kinds of operation, and two integers tell them apart. The determinant and the trace name a symmetry operation completely — which is why the classification can be run on integer matrices in a lattice basis and never once ask what angle anything turns through.

point-groups · Crystal classes Neumann's principle for elastic constants in mmm. Each bar is one operation's contribution to the character of the representation the elastic constants live in — the stiffness of the crystal in every direction and shear. The identity contributes the unconstrained count of 21; every other operation of mmm subtracts from it, and the average over all 8 is 9, which is how many independent components the property may have. The sum is exact in integers and is taken in the lattice basis, so no Cartesian frame is chosen anywhere in it.

Neumann's principle, as one sum

A physical property of a crystal must be unchanged by every symmetry the crystal has. That is a whole subject in one sentence, and it reduces to arithmetic: how many independent components a property may have is a character averaged over the point group, exact in integers.

point-groups · Neumann's principle Neumann's principle for elastic constants in 6/mmm. Each bar is one operation's contribution to the character of the representation the elastic constants live in — the stiffness of the crystal in every direction and shear. The identity contributes the unconstrained count of 21; every other operation of 6/mmm subtracts from it, and the average over all 24 is 5, which is how many independent components the property may have. The sum is exact in integers and is taken in the lattice basis, so no Cartesian frame is chosen anywhere in it.

A character does not know its basis

The number of independent elastic constants a hexagonal crystal has is computed here from integer matrices in a lattice basis, having never chosen a Cartesian frame. That looks wrong the first time: a physical tensor lives in an orthonormal frame and these matrices are not orthogonal.

point-groups · Neumann's principle p3m1 and p31m, told apart without a picture. The two groups this site returns to most often: same point group, same lattice, same number of operations, and distinguished in every other essay here by where their mirrors sit relative to the lattice — which is a fact about the plane. Abelianised, they are ℤ2 and ℤ6, which are not isomorphic. That difference is a fact about the groups: no change of basis, no redrawing and no relabelling can carry one to the other, and the argument never mentions a mirror line.

What is left when the order is forgotten

Abelianising a group throws away the order of the letters in every word and leaves a small abelian group behind. It is computed by a Smith normal form, it never mentions the plane, and it separates p3m1 from p31m — which a picture can only illustrate.

operations · Presentations the kagome net: 4, 8, 14, 18 at the first four shells. The vertices of the kagome net at graph distance one, two, three and four from a chosen vertex, each marked with its distance. Distance here is a number of edges and nothing else — no length enters, and the shells are drawn on the barycentric placement only so that they can be seen. The counts are 4, 8, 14, 18, 22, 28, 30, 38, 38, 48, 46, 58, which is the net's coordination sequence.

Counting outwards

How many vertices lie one step from a vertex, two steps, three? The counts settle into a straight line — but for some nets only along the even distances, with a different line along the odd ones, alternating for ever. The period is measured, and it is not always one.

applied · Nets 12 of the thirty-two classes have a free invariant ring. Every crystal class with its order, the number of its operations that are reflections, whether its ring of invariant polynomials is free, and the degrees of the generators when it is. A reflection here is an operation of determinant minus one whose fixed set is a plane; an inversion centre has determinant minus one and fixes only the origin and is not one. The classes with a free ring are exactly the classes generated by their reflections, which is Chevalley's theorem checked rather than quoted.

Twelve of the thirty-two are free

A crystal class leaves some polynomials alone, and the ones it leaves alone form a ring. For twelve of the thirty-two classes that ring is generated by three polynomials with no relation between them, and for the other twenty it is not — and the twelve are exactly the classes generated by their mirror planes. The two verdicts are computed by routes sharing no code, and an inversion centre is not a mirror.

point-groups · Invariants One invariant for every invertible chain. The quantity x² + y² + z² − 2xyz − 1, built from the half-traces of the transfer matrices of the two words a substitution produces at each level and of their product, for six substitutions at one energy, logarithmic. For the golden, silver and bronze chains and for the golden rule reversed it is V²/4 = 0.09 at every level — the same number for all four, computed in exact integers. For period-doubling and Thue–Morse it wanders by orders of magnitude from one level to the next.

One invariant for every chain that can be undone

The Fibonacci chain's transfer matrices obey a recursion with a quantity it cannot change, and the natural guess is that every quasiperiodic chain has its own. It does not. The conserved quantity is the trace of the commutator of the two tiles' matrices, so every chain a reversible substitution builds from the same two tiles — golden, silver, bronze — conserves the same number, exactly, at every length and every energy. Period-doubling and Thue–Morse, whose rules cannot be undone, conserve nothing.

aperiodic · Quasicrystals Tribonacci keeps one number at every length, and its neighbours do not. The size of tr(ABC) − tr(ACB), computed in exact integers on three two-site tiles, at each application of four rules, on a logarithmic scale. Under tribonacci it stays 24 while the traces themselves reach dozens of digits. Under the plastic rule and the rule x³ = x² + 1, both of which can be undone, it wanders and grows. Under a rule that cannot be undone it passes through zero and runs off the scale.

Three tiles keep a number that asks which way they face

Two tiles always conserve something: every reversible rule keeps the trace of their commutator. Three letters were expected to conserve nothing, and most rules do conserve nothing. The tribonacci rule keeps exactly one number, the difference between a product of three matrices and the same product reversed. It is zero for any tile that reads the same both ways, so it measures direction rather than difference, and one rule in fifty keeps it, for a reason that can be read off the words.

aperiodic · Quasicrystals

Named alongside it

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

TraceCharacterPoint groupCommutatorRepresentationAutomorphismCrystal classInflation factorNeumann principleSubstitutionTensorTransfer matrix

All concepts