Seven friezes round a cylinder

The finite groups of rotations of space come in two infinite families and three exceptions. Adding the operations that reverse handedness does not produce anything new in kind — every improper group is a rotation group with the centre added, or one with half of itself negated — but it does multiply the families. The groups with a single principal axis, the ones that single out a line through the point and treat its two ends as the only special directions, fall into seven infinite families. Schoenflies named them Cₙ, S₂ₙ, Cₙₕ, Cₙᵥ, Dₙ, Dₙd and Dₙₕ in 1891, and every crystal class that is not cubic is a member of one of them.

A strip has seven symmetry groups too. The seven friezes are the ways a motif can repeat along a line, and the sixteen candidates collapse to seven by an argument short enough to do by hand. Two classifications, each of exactly seven, one about a point in space and one about a line in the plane.

They are the same seven, and the reason is a construction rather than a coincidence of counting. Roll the strip into a cylinder.

Rolling a strip turns each motion into a motion of space #

Take a frieze drawn on a strip n cells long, and roll the strip about a vertical axis until its two ends meet. A point at position x along the strip and height y lands at angle 2πx/n round the axis and at height y up it. Each motion of the strip that respects the rolling becomes a motion of space:

• a translation by one cell becomes a rotation by 2π/n about the axis;
• a mirror across the strip, perpendicular to its length, becomes a mirror plane containing the axis;
• the mirror along the strip’s centre line becomes the mirror plane perpendicular to the axis, halfway up the cylinder;
• a half-turn about a point of the centre line becomes a half-turn about a horizontal axis through that point of the cylinder;
• a glide — reflect in the centre line and slide half a cell — becomes a rotation by π/n combined with the horizontal mirror, which is a rotoreflection.

The rolling respects composition. Doing two motions of the strip and then rolling gives the same motion of space as rolling each and composing them, because the angle round the cylinder is proportional to the distance along the strip and the height is untouched. So rolling is a homomorphism from the frieze group into the finite group of motions of the cylinder. The only strip motions it sends to the identity are the translations through a whole number of turns of the cylinder, n cells at a time, and those are exactly the motions that carry each point of the strip to a point the rolling has already glued to it.

That settles the size of every rolled group. A frieze group has a fixed handful of motions per cell — one for the hop, two for the step, four for the spinning jump — and rolling onto n cells keeps n cells’ worth of them. The seven families have orders n, 2n, 2n, 2n, 2n, 4n and 4n for that reason: the number of motions of one cell of the frieze, times the number of cells.

The step makes the construction concrete. Its one symmetry beyond translation is a glide, and on three cells the glide rolls into a turn of sixty degrees combined with the horizontal mirror. Perform that three times and the turns add up to a half-turn while the mirror, applied three times, is still a mirror; a half-turn combined with the horizontal mirror is the inversion. A glide rolled round three cells contains the centre of the cylinder, which is why its class is 3̅ — a three-fold rotoinversion, which contains the inversion among its powers.

Seven names for one list #

The correspondence the rolling produces is one to one, and the letters Schoenflies chose read as descriptions of the frieze.

The hop has translations only and rolls into Cₙ, the rotations about one axis. The step, the glide, rolls into S₂ₙ, where S is for Spiegel, the mirror a rotoreflection carries. The jump, with a mirror along the centre line, rolls into Cₙₕ, where h is the horizontal mirror. The sidle, with mirrors across the strip, rolls into Cₙᵥ, where v is for the vertical mirrors that contain the axis. The spinning hop, with half-turns, rolls into Dₙ, the dihedral rotations. The spinning jump, with every feature at once, rolls into Dₙₕ. And the spinning sidle rolls into Dₙd, whose d stands for the diagonal mirrors — which are the frieze’s mirrors, lying halfway between its half-turn centres, a quarter of a cell from each.

Every one of those identifications was made twice rather than assumed. Each family was built from its own generators — a rotation by 2π/n, a horizontal or vertical mirror, a half-turn, a rotoreflection — and closed under composition; each frieze was rolled onto the same number of cells; and the two sets of matrices were required to be identical, member by member, for every family and every n from one to eight.

There is a second notation in which the correspondence costs nothing to state. Conway’s orbifold symbols name a frieze by what folding the strip up along its symmetries leaves, and a point group by what folding the sphere up leaves. The hop is ∞∞ and Cₙ is nn; the step is ∞× and S₂ₙ is n×; the spinning sidle is 2*∞ and Dₙd is 2*n. Replace ∞ by n and each frieze’s symbol becomes its family’s, which is the rolling, written down: a strip is the case where the cylinder has infinitely many cells.

Thirty-five members, twenty-seven classes #

A lattice permits a rotation of order 1, 2, 3, 4 or 6 and nothing else, so the families meet crystallography at those five values of n. Seven families at five orders is thirty-five members.

Twenty-seven classes appear, and they are exactly the classes that keep one direction to themselves — the ones a rod or a layer can have, found there by a different test that asks which of the thirty-two classes carry a line onto itself. The other five of the thirty-two are the cubic classes, which have four three-fold axes and no principal one, and no strip rolls into them.

The eight members that do not add a class split evenly, and each half has a single cause.

Every repeat has one cell in it. A strip one cell long rolls into a cylinder whose principal axis is a rotation by a full turn, which is the identity, so the axis does nothing and the real symmetry lies across it. D₁, the spinning hop on one cell, is a single half-turn about a horizontal axis — which is C₂ lying on its side. D₁d is C₂ₕ lying on its side, and D₁ₕ is C₂ᵥ. C₁ᵥ and C₁ₕ are each a single mirror, one containing the idle axis and one perpendicular to it, and a single mirror is the class m whichever way it faces. Nothing at two or more cells repeats.

Every exclusion is a rotoreflection of order eight or twelve. S₈, D₄d, S₁₂ and D₆d all contain a rotation through a quarter or a sixth of a turn combined with the horizontal mirror applied at the half-cell offset, and the operation that results has order eight or twelve. Its trace is √2 − 1 or √3 − 1, which is not an integer, so no lattice holds it. That is the crystallographic restriction biting an improper operation whose rotations are all permitted: the square of the eight-fold rotoreflection in S₈ is an ordinary four-fold rotation, and S₈ is forbidden anyway. A four-fold axis may carry a horizontal mirror, as in 4/m, and may not carry a glide rolled round four cells.

The census also shows the one place where the two notations for point groups cross. At n = 3 the step rolls into S₆ and the jump into C₃ₕ, and in Hermann–Mauguin’s symbols those are 3̅ and 6̅ — the family indexed by three becomes the class named by six, and the other way about. The two notations are derived from different constructions: Schoenflies counts the rotoreflection, whose order is six for S₆, and Hermann–Mauguin counts the rotoinversion, whose order for the same group is three. The frieze decides which is which without either convention: the glide is the S, the centre-line mirror is the h.

Where the centre falls #

Every improper group either contains the inversion or is a rotation group with half of itself negated, and the rolled friezes sort into those two kinds by a rule that involves nothing but the parity of n.

The inversion through the centre of the cylinder sends a point at angle θ and height y to the point at angle θ + π and height −y. On the strip, that is a reflection in the centre line combined with a slide of half the circumference — n/2 cells. So a rolled frieze contains the centre exactly when it contains a reflection in the centre line with a slide of n/2 cells.

The jump’s reflections in the centre line slide by whole numbers of cells, so it contains one sliding by n/2 exactly when n is even. The step’s reflections slide by half-odd numbers of cells, so it contains one exactly when n is odd. The sidle has no reflection in the centre line at all, and the hop and the spinning hop have nothing that reverses handedness. The spinning jump behaves as the jump does, and the spinning sidle as the step.

So the kinds alternate along the families, and in opposite phase. C₄ₕ is 4/m and holds the centre; C₃ₕ is 6̅ and does not. S₆ is 3̅ and holds it; S₄ is 4̅ and does not. D₃d is 3̅m, centred, and D₂d is 4̅2m, twisted. A crystallographer meets each of those as a separate fact about a separate class, and the frieze makes them one fact about which slides are available: a mirror across the centre line comes with whole cells and a glide with half cells, and the centre needs n/2 of them.

The same rule sorts the census without anything further being counted. At the even orders the centred classes are 2/m, mmm, 4/m, 4/mmm, 6/m and 6/mmm, and every one is a jump or a spinning jump; the steps and spinning sidles there — 4̅ and 4̅2m — are twisted, and so is every sidle. At n = 3 it is the other way round: 3̅ and 3̅m, the step and the spinning sidle, hold the centre, and 6̅ and 6̅2m, the jump and the spinning jump, do not. A centred axial class is a mirror across the centre line at an even order or a glide at an odd one, and a crystallographer reading a table of the twenty-seven is reading that sentence thirteen times.

Sixteen candidates, seven groups, twice #

The frieze classification ends at seven because most combinations of features force others. A strip may or may not have a mirror in its centre line, a mirror across it, a half-turn and a glide: sixteen combinations, and nine of them come back from closure holding a feature they were not given — a mirror across and a half-turn force the centre-line reflection, a half-turn and a glide force a mirror across, and so on down the list.

Rolling carries that argument across whole. The four features become four kinds of operation a group with a principal axis may have: a horizontal mirror, vertical mirrors, horizontal two-fold axes and a rotoreflection. Any combination of them closes up into one of the seven families, because the rolling is a homomorphism and closure on the strip is closure on the cylinder. The seven families are not a separate classification that happens to agree with the friezes; they are that classification with a turn about an axis in place of a step along a line, and the nine forced collapses happen for the same reasons in both.

One thing the cylinder adds that the strip does not have is a choice of n, and with it the parity that decides where the centre goes. A frieze has no centre to hold, because a strip has no point that every motion could fix; a cylinder has one, the point on its axis halfway up, and whether the inversion through it is among the rolled motions depends on whether half the circumference is a whole number of cells. That is the only respect in which the two classifications differ, and it is invisible until the strip is closed.

A glide becomes a mirror when the cells shrink #

Nothing in the rolling needs n to be small, and letting it grow shows where the seven families go.

The jump’s reflections in the centre line sit at whole cells and the step’s at half cells. Round the cylinder, the first are at angles 2πk/n and the second at 2π(k + ½)/n, so the two sets interleave with a gap of π/n between neighbours — half a cell’s angle, ninety degrees at n = 2 and a little over eleven at n = 16.

As n grows without limit both combs fill the circle, and the two families approach the same group: every rotation about the axis together with the horizontal mirror. The step and the jump merge. The same happens to the spinning sidle and the spinning jump, which differ in exactly the same way, and both approach every rotation about the axis with every mirror containing it and the horizontal mirror too.

That lands on a list already derived by closure: Curie’s seven limiting groups, the symmetries a uniform field can have. Five of them have a principal axis — ∞, ∞m, ∞/m, ∞2 and ∞/mm — and the seven families reach exactly those five, the hop reaching ∞, the sidle ∞m, the step and the jump both ∞/m, the spinning hop ∞2, and the spinning sidle and spinning jump both ∞/mm. Seven friezes become five limiting groups, and the two that are lost are the two glides, because a glide is a mirror with a slide of half a cell and an infinitely fine cylinder has no half a cell to slide by. The other two limiting groups, ∞∞ and ∞∞m, have no axis at all and are the limit of the cubic and icosahedral groups instead.

That limit is argued rather than computed. What is computed is the gap, which is exactly π/n at every n tried, and which is what makes the merge visible before it happens.

What rolling cannot reach #

The groups with no principal axis. The five cubic classes and the icosahedral groups single out no line, and nothing rolled round a line produces them. Their construction is the one from the axis equation, with three kinds of axis meeting at a point, and a strip has only one direction to roll along.

Translations along the axis. Rolling keeps the height fixed, so every rolled group fixes the centre of the cylinder and none of them is a rod group. A helix — a structure that turns and climbs at once — needs a translation along the axis, which the strip’s height does not have, and its symmetry is one of the seventy-five rod groups rather than a point group.

The one-cell row is degenerate. At n = 1 the cylinder’s axis carries no rotation, and four of the seven members are other members lying on their sides. The correspondence holds there — every identification was checked at n = 1 as at every other value — but the principal axis it names is not principal.

What was computed, and how. Each frieze as a list of motions modulo one cell, checked to be closed under composition; the rolling, checked product by product to be a homomorphism with nothing but the identity in its kernel modulo n cells; the rolled group, checked against the family built from its own generators, for all seven families at every n from one to eight; the thirty-five members at the lattice’s orders, identified by counting operation types and compared with the twenty-seven axial classes found separately; which members contain the inversion; and the angles of the reflections in the centre line, with the gap between the two combs.

The first of those refusals is not hypothetical. The plate of the seven friezes drawn elsewhere on these pages had put the spinning sidle’s half-turn on its mirror, which composes the two into a reflection in the centre line the spinning sidle does not have, and drew five copies of the motif a cell where there are four. Nothing had checked that the copies formed a group. They are checked now, and the plate is corrected.

Who put the two lists side by side #

The families of point groups belong to the nineteenth century, and Schoenflies fixed the letters they are still written in in his book of 1891. The friezes were classified as ornament, and for most of the time since the two lists sat in different chapters of the same books, with the same count and no stated connection between them.

The connection is set out in The Symmetries of Things, by John Conway, Heidi Burgiel and Chaim Goodman-Strauss, from 2008. Their orbifold notation writes a frieze and a spherical group in the same alphabet, so that the substitution of n for ∞ is visible on the page, and the names hop, step, sidle and jump for the friezes are theirs. The rolling above is the geometry under that substitution.

Where this goes: rolling a lattice instead of a strip #

A strip has one direction of repeat and rolls into a point group. A plane pattern has two, and rolling it along a lattice vector — so that the vector becomes the circumference — keeps a translation along the cylinder’s length. The result is a rod group, and which one depends on the direction of the vector against the pattern’s own mirrors. That is the construction of a carbon nanotube from a sheet of graphene, where rolling along a mirror gives a tube with mirrors and rolling at any other angle gives a tube that is one hand or the other. Which of the seventy-five rod groups the seventeen plane groups can roll into, and at which vectors, is a question the rolling here does not reach.