Crystallography & Symmetry
patterns derived from their groups
Concept

Rotation centre — where it appears

The one point a rotation of the plane leaves where it was, found by solving for the place the motion fixes. Every rotation of a pattern has one and it is somewhere definite, so composing two of them is a question with a location in its answer as well as an angle.

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

Two half-turns make a translation. The half-turn about (0.25, 0.25) followed by the half-turn about (0.75, 0.5) is the translation by (1, 0.5) — twice the vector between the two centres, and not the vector itself. The open lens is a third centre, and it is not the midpoint of the two drawn: it is where the half-turn about the first lands when it is composed with one repeat vector of the lattice, which is half a repeat along. That is the step that puts two-fold centres on the half lattice and gives a p2 cell four inequivalent ones. Both the translation and the forced centre are computed from the operations and compared with the construction in exact rational arithmetic.

Where the product is

Composing two symmetries lands on a third — and the third one is somewhere. Two half-turns make a translation by twice the distance between their centres, and that single fact puts the lattice into a pattern before anybody chooses one.

operations · Composition The closure is a lattice exactly when the orders allow one. Twelve pairs of rotation orders, with a centre of each order placed one unit apart and the group they generate closed out to words of length 6. The linear parts reached are exactly the least common multiple of the two orders, every time — two rotations generate rotations, and the angles they generate are the multiples of the smaller of two fractions of a turn. A lattice admits rotations of order one, two, three, four and six and no others, so the closure can be a plane group exactly when that multiple is one of those five. The pairs where it is not are the pairs where the translations keep getting shorter.

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.

operations · Composition A screw out of two rotations that have none. Four pairs of located rotations of space, composed, with the result read back as a screw: its angle, and its pitch, which is the part of its translation lying along its own axis. Axes that meet give a rotation and no translation at all, because the point where they meet is fixed by both. Parallel half-turns give a translation. Skew axes give a screw — a motion with a translation in it, out of two motions with none — and the translation is twice the distance between the two axes. Nothing in either factor moves anything along the product's axis, and the product does.

The axis a product lies on

Two rotations of space about axes that do not meet compose to a screw — a motion with a translation in it, out of two that have none. The translation is twice the distance between the axes and the angle twice the angle between them, and the screw's own axis is not somewhere arbitrary: it lies on the two axes' common perpendicular, at a place the arithmetic gives.

operations · Composition The turns that keep the join discrete. Two copies of p4 on one square lattice, one turned against the other, with the shortest translation their union generates. At a turn whose cosine and sine are both rational the translations are a lattice, and its shortest vector is one over the square root of Σ — where Σ is the odd part of p² + q² for the rational point (p, q) — which the measurement reproduces to six places at every one tried. At a whole number of degrees other than a multiple of ninety there is no such point, and the search finds shorter translations the further it runs.

Two patterns laid over one another

Compose two plane groups rather than two operations. The group generated by both is one of the seventeen or is not discrete at all, with nothing between — and it takes two conditions, one on the rotation orders and one on the turn between the lattices. The turns that work have rational cosines, which by a theorem of Niven's means none of them is a whole number of degrees.

operations · Composition

Named alongside it

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

CompositionPlane groupSymmetry elementClosureCrystallographic restrictionDiscretenessFixed pointLattice translationCoincidence site latticeEnumerationHalf-turnMirror line

All concepts