Concept

Orbifold — where it appears

The surface a pattern folds up into when every pair of points its group identifies is glued, marked where rotations and mirrors were. Its shape is a complete name for the group, and reading the name off costs an arithmetic sum that has to come to two.

Named by 8 essays across 3 fields — 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

The (2, 3, 7) group, in the Poincaré disk. A triangle with angles π/2, π/3 and π/7, reflected in its own three sides until depth 12: 380 triangles, alternating in handedness because every generator is a reflection. The sum 1/2 + 1/3 + 1/7 is less than one, so the triangle does not fit in the flat plane and the drawing is of the hyperbolic one, with the whole plane squeezed inside a disk. Every triangle has the same hyperbolic area; the ones near the edge look small because the model shrinks distances there, and the tiling stops at the edge of the drawing rather than at the edge of anything.

Past two, the list does not stop

Conway's accounting says a wallpaper group costs exactly two dollars, and there are seventeen ways to spend it. Spend less and the answer is a finite group. Spend more and the list is infinite — but the cheapest thing past two costs two and one eighty-fourth, and nothing at all lies in between.

classification · Seventeen

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

p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

The two that fold into a surface

Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

classification · Flat space

The relations a polygon dictates

Poincaré's theorem has two halves. The walls of a fundamental domain name the generators, which is the half this collection already computes; walking round its corners names the relations, which needs a domain with corners rather than a domain made of pixels. Building the Dirichlet polygon exactly gives a presentation of each of the seventeen — and coset enumeration says every one of them is right.

operations · Fundamental domain

The seven friezes rolled into cylinders are the seven axial families. Each of the seven frieze groups drawn on a strip 3 cells long, beside the same strip rolled into a cylinder so that its ends meet. A translation by one cell becomes a rotation by a 3th of a turn about the axis, a mirror across the strip a mirror containing the axis, the centre line a mirror perpendicular to it, a half-turn in the strip a half-turn about a horizontal axis, and a glide a rotation by half a cell's angle combined with that perpendicular mirror. Each cylinder's symmetry group was built from the rolled strip and again from the family's own generators, and the two agree. At n = 3 the orders are 3, 6, 6, 6, 6, 12, 12, and the last column names the crystal class each member is, coloured by whether it is proper, contains the centre, or is neither.

Seven friezes round a cylinder

A point group with one principal axis belongs to one of seven infinite families, and there are seven frieze groups. They are the same seven. Draw a frieze on a strip, roll the strip into a cylinder, and every translation becomes a turn about the axis and every glide a rotoreflection.

restriction · Finite groups

Three mirrors of a 30–30–120 triangle, closed. A triangle with angles of thirty, thirty and a hundred and twenty degrees, shaded, and every image of it under the group its three side-mirrors generate, within a window. The images alternate between the triangle and its mirror image, and they tile the plane — so the closure is discrete. But the group it produces has a mirror along the triangle's own altitude, drawn dashed, that is none of the three given: the triangle is two copies of the thirty–sixty–ninety triangle, and the group is that triangle's, p6m. The closure found a mirror the generators did not name.

Four groups are made of mirrors

Close the plane from located mirrors instead of rotation centres and the closure is discrete only for four triangles of mirrors — and one of those four manufactures a mirror along its own altitude. Of the ten plane groups that contain a reflection, exactly four are generated by their mirrors; three hold a kaleidoscope at index two, three and four, which are the numbers in front of the star in their orbifold symbols; and three have mirrors in one direction only, which generate nothing along themselves.

operations · Composition

Named alongside it

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

The Euler characteristic Fundamental domain Glide reflection Abelianisation Cone point Crystallographic restriction Magic theorem Plane group Presentation Axial class Bad orbifold Commutator

All concepts