Crystallography & Symmetry
patterns derived from their groups
Concept

Perpendicular space — where it appears

The complement of physical space in a cut-and-project construction, where the window that selects the kept points lives. Two points close together in physical space can be far apart in it, and that separation is what prevents the pattern from repeating.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Sliding the window catches different points. The periodic lattice that cut-and-project starts from, with the strip drawn at two positions 0.21 apart, which is 15 per cent of the window's width. Most lattice points are caught by both; a few are caught by one and not the other, and those are the whole difference between two quasicrystals. The slope has not changed, so the density, the two tile lengths and the ratio of their frequencies are identical — the offset is a parameter with no energy attached to it, which is what makes a phason a degree of freedom rather than a defect.

The freedom a crystal has not

Slide the window of a cut-and-project construction and the tiling changes — different tiles in different places — while its density, its tile ratio and its diffraction pattern do not. That parameter is a phason, it costs nothing, and no local measurement whatever can determine where it sits.

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 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

Named alongside it

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

Cut-and-projectHigher-dimensional latticeQuasicrystalWindowAmmann beenkerAperiodicityCrystallographic restrictionDiffuse scatteringGolden ratioIcosahedral groupIndexingInflation

All concepts