Crystallography & Symmetry
patterns derived from their groups
Concept

Cone point — where it appears

A marked point of an orbifold where a rotation centre was, carrying the order of that rotation. A fold with no cone points and no boundary is an ordinary surface, which happens for exactly two of the seventeen plane groups.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A fundamental domain for p6m. One representative from every orbit of p6m, shaded, with the images that tile the rest of the cell. The domain was found by computing orbits rather than by drawing a region, and every sample's orbit was checked to meet it exactly once — so the region has neither a gap nor an overlap.

Orbifold notation, the shorter language

Fold a pattern up along its own symmetries and what remains is a small surface with marked points. Its shape is a complete name for the group, and reading the name off costs an arithmetic sum that has to come to two.

classification · Seventeen The seventeen signatures, and the seventeen groups. Every combination of features costing exactly two, beside the plane group each one names. The left column is produced by an accounting identity that has never heard of a lattice; the right by reading seventeen groups' own operations — their rotation centres and orders, which of those lie on mirrors, and how many closed curves the mirror lines make once equivalent lines are identified. The map between the two lists is a bijection, and the figure does not appear unless it is one — in both directions. A signature with no group and a group whose signature is not on the list are both refused, and so is the failure that actually happens: two groups deriving one signature, which costs exactly two and passes every check but injectivity.

Seventeen dollars

Conway's magic theorem prices the features a folded-up pattern can have — a handle costs two, a mirror boundary one, a cone point of order n almost one — and requires the total to come to exactly two. There are seventeen ways to pay, and the classification falls out of an accounting identity that never mentions a lattice.

classification · Seventeen

Named alongside it

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

The Euler characteristicOrbifoldBad orbifoldCross-capMagic theoremMirror boundaryOrbifold costOrbifold notationQuotient spaceSignature

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