Crystallography & Symmetry
patterns derived from their groups
Generator

C5: 5 operations about one point

C5: 5 operations about one point
C5: 5 operations about one point. The orbit of one asymmetric mark under C5, which has 5 operations. All of them are rotations, so every copy has the same handedness. The dot at the centre is the centroid of the orbit, computed from the 15 points drawn: every operation of the group carries it to itself, to within 4e-17. That is Leonardo's theorem in one sentence — a finite group of plane isometries fixes the centroid of any orbit, so it is a group of rotations and mirrors about that point, so it is a Cₙ or a Dₙ.

The orbit of one asymmetric mark under C5, which has 5 operations. All of them are rotations, so every copy has the same handedness. The dot at the centre is the centroid of the orbit, computed from the 15 points drawn: every operation of the group carries it to itself, to within 4e-17. That is Leonardo's theorem in one sentence — a finite group of plane isometries fixes the centroid of any orbit, so it is a group of rotations and mirrors about that point, so it is a Cₙ or a Dₙ.

11 essays call finite-groups. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 522 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

Every solution of the axis equation. The integer solutions of 2 − 2/N = Σ(1 − 1/nᵢ), which is what counting the pairs (rotation, fixed pole) two ways gives. Two classes of axis force n₁ = n₂ = N and give the cyclic groups; three classes give the dihedral family and exactly three sporadic answers — (2, 3, 3), (2, 3, 4) and (2, 3, 5), of orders 12, 24 and 60, which are the rotation groups of the tetrahedron, the octahedron and the icosahedron. Four classes are impossible, because four terms of at least a half already exceed the left-hand side. Nothing about crystals has been used. What a lattice forbids

Before the lattice has a say

Every finite group of motions of the plane is a Cₙ or a Dₙ, and every finite group of rotations of space is one of five families. Both lists come out of counting rather than out of crystallography — and then the crystallographic restriction deletes almost all of them, leaving eleven.

 Which Schläfli symbols close. Every {p, q} with p polygons round each face and q faces round each vertex, from three to six of each. A solid exists only when 2p + 2q − pq is positive, which is the same statement as 1/p + 1/q > ½; the five that qualify carry their vertex, edge and face counts, and the three on the diagonal where the expression vanishes are the three regular tilings of the plane. Past them the expression is negative and the answer is the hyperbolic plane, where the list never ends. The five, the three and the infinity are one inequality read at its three signs. What a lattice forbids

Five solids from one inequality

Five families of rotation group in space, five regular solids, three regular tilings of the plane and an endless supply of hyperbolic ones — all of it is 1/p + 1/q compared with a half, read at its three signs.

 Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds. What a lattice forbids

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

 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. What a lattice forbids

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.

 The rectangle a (4, 2) tube is rolled from. A patch of honeycomb turned so that the rolling vector C = 4a₁ + 2a₂ lies along the page. C has length √28 ≈ 5.292; the shortest lattice vector perpendicular to it, T, has length 4.583; and the rectangle on the two holds 28 hexagons and 56 atoms. Rolling the rectangle so that its left and right edges meet makes one repeat of the tube. The two lines through the corner are the sheet's mirror directions nearest C: the zigzag direction along a₁ and the armchair direction thirty degrees from it. C makes an angle of 19.11° with the first and lies on neither. What a lattice forbids

The tube has a screw no lattice allows

Roll a honeycomb along one of its lattice vectors and the tube turns and climbs with a screw of order 14, 98 or 794 — orders the flat sheet could never have. The rolling keeps the sheet's translations and spends them on turns, and it keeps the sheet's mirrors only along two directions, which is why almost every carbon nanotube comes in two hands.

 Whether a rolled sheet ever comes back round. Three plane lattices, each with the same rolling vector C = 3a₁ + a₂ drawn from the origin and the line through the origin perpendicular to it. A translation of the rolled pattern straight up the tube, with no turn, is a lattice vector on that line. The square lattice has one, marked T, and the tube repeats every 10 turns. The general rectangular lattice has none in this direction — only along its cell edges — and the general oblique lattice has none in any direction at all, so its rolled pattern climbs forever without returning to the same angle. What a lattice forbids

Most sheets roll into a tube that never repeats

Rolling the honeycomb along a lattice vector always gives a tube with a repeat, and that is a property of the honeycomb rather than of rolling. Over the seventeen plane groups, 567 of 1,008 rolling directions give a tube with no translation along its axis at all — and every direction of an oblique pattern is one of them.

 Fifty-four of the seventy-five rod groups are a rolled plane pattern. Every rod group, one dot each, grouped by its crystal class. A dot is filled when some plane pattern rolled along some lattice vector has exactly that group, and the 106 crystallographic rollings of the seventeen plane groups fill 54 of them. The eighteen improper classes are full: every one of the 43 achiral rod groups is reached. The nine proper classes are not, and the twenty-one groups named on the right are what is missing — the sixteen whose screw is one of a left- and right-handed pair, and the five bare axes with no climb at all, p1, p112, p3, p4 and p6. What a lattice forbids

A rolled sheet is never one of a pair

Rolled up along every lattice vector that gives a crystallographic tube, the seventeen plane groups reach fifty-four of the seventy-five rod groups: every achiral one and eleven of the chiral. Not one of the sixteen screws that come in left- and right-handed pairs is among them, and the reason is a single fact about how far a rolled lattice can climb.

 The turns a rolled sheet can make are square roots. Every primitive rolling of a square and of a hexagonal sheet with at most 66 cells in a repeat, one mark per distinct turn. Across is the number of cells n₀ in one repeat of the tube; up is the turn, as a fraction of a circle, of the screw that climbs one n₀-th of the repeat. Most columns are empty, and the occupied ones hold two or four marks rather than anything up to n₀: on the square sheet the turns are exactly the square roots of minus one modulo n₀, on the hexagonal sheet the roots of the corresponding congruence for minus three. The pattern is symmetric about half a turn because a root and its negative are the two hands of one tube. What a lattice forbids

A rolled sheet turns by a square root

Roll a square sheet into a tube of thirteen cells and its screw could, on paper, turn by any of twelve fractions of a circle. It turns by five thirteenths or eight, and nothing else, because five squared plus one is a multiple of thirteen. Every rolled tube's screw is a square root of minus the sheet's determinant — which is why handed tubes appear only at the orders Fermat allows, why they always come in both hands, and why the crystallographic ones need an accident.

 All thirteen families of line groups, reached by rolling. Each of the seventeen plane groups, on its generic lattice, rolled into a tube along every lattice vector that gives a repeat, with a mark under each of the thirteen families of line groups some rolling lands in. The band above each column is the family's translational part: a general screw for families 1 and 5, a half-step screw for 4, 8 and 13, pure translations or glide planes for the rest. Every family is reached, six of them by a single plane group, and the two oblique groups reach nothing because an oblique lattice rolled in any direction never repeats. What a lattice forbids

The centring that turns into a screw

Rolled with no limit on the order, the seventeen plane patterns reach every one of the thirteen families of line groups — six of them through a single pattern each. And the families' translations are decided by one integer: a tube that keeps any reflection holds one or two cells of its sheet in a repeat, never more, and it holds two exactly when the sheet is centred along the rolling direction. The centring vector of the flat lattice is the half-step screw of the tube.

 The eighty layers by what rolling does to their two faces. The eighty layer groups sorted by how the two tubes a sheet makes, one with each face outside, are related. Seventeen one-sided layers make two different tubes in every direction. Forty-six make the same tube in every direction, because they have a mirror or glide in the sheet or a centre of inversion. Sixteen relate the two tubes only along two, four or six special directions, and there as mirror images. One, the layer with a four-fold rotoinversion and nothing else that turns it over, exchanges its faces and still makes two different tubes in every direction. What a lattice forbids

A sheet with two faces rolls into two tubes

Roll a sheet whose two faces differ and every operation that turns it over dies, as expected: a tube's inside is not its outside. But the sheet can be rolled with either face out, and the operations that died are exactly the congruences between the two tubes that result. Over the eighty layer groups that settles, direction by direction, whether the two are one tube, a pair of mirror images or two different objects, and one layer exchanges its faces and still never rolls the same tube twice.

 One sheet with two faces rolled into two scrolls. The cross-section of a sheet with different atoms on its two faces, rolled three turns into an Archimedean spiral, once with face A outward and once with face B outward. Unlike a tube, a scroll has an inner edge and an outer edge and a radius that grows with every turn, so an isometry of space can neither turn it about its axis nor reverse the direction along the spiral: it can only slide it along the axis, reflect it across the axis, or reflect the spiral itself. What a lattice forbids

A scroll cannot be turned end for end

A sheet with two different faces rolls into two tubes, face A out and face B out, and for forty-six of the eighty layer groups the two tubes are one object whichever way the sheet is rolled. Roll the same sheets into scrolls, spirals with an inner edge and an outer one, and only seventeen keep that. A spiral cannot be turned about its axis or run backwards, so the half-turns and screws that joined the tubes have nothing to act on. What survives is a mirror lying in the sheet, and a two-fold axis lying along the roll.

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