Symmetry Group Visualizer

Turn a picture of a snow crystal by a sixth of a turn and it looks exactly as it did. Flip it across a line through two opposite arms and again nothing seems to happen. That nothing is the subject.

A symmetry of a figure is a rigid motion that carries the figure onto itself, shuffling its parts while leaving the whole unchanged. The capital letter A has one mirror line. The letter N has none, yet a half turn about its center puts it back on itself. The letter H has two mirrors and a half turn. By convention the motion that does nothing is counted too, so every figure, however lopsided, has at least one symmetry.

Seen this way, symmetry is not a look. It is a list of moves, and moves can be done one after another. Do one symmetry and then a second, and the figure is still sitting on itself, so the combination is on the list as well. Every move can also be undone. A list that is closed under combining and undoing is what mathematicians call a group, and the surprise is how few such lists exist when one point has to stay put.

Turns only, or turns and mirrors

The simplest groups in the plane contain turns and nothing else. Pick a whole number n and let r be a turn by one n-th of a circle, counterclockwise. Repeating r gives n different turns before everything is back where it began. This is the cyclic group Cn, and the letter N lives in C2. The second family adds mirrors. The dihedral group Dn holds the n turns of Cn together with reflections in n lines through the center, lines that cut the plane into 2n equal wedges. It therefore holds 2n elements, and for n of 3 or more it is the full symmetry group of a regular polygon with n sides.

The letters fit here as well. A has the group D1, a single mirror and the identity. H has D2, with four elements. A square has D4, with eight. Some books call that last group D8, after its size.

Nothing else is possible. The result is known as Leonardo’s theorem: a finite group of rigid motions of the plane is either a cyclic group or a dihedral group. The name honors Leonardo da Vinci, who wanted to know every way of attaching chapels and niches to a central building without spoiling the symmetry of its core.

The argument is short. A finite group cannot contain a slide, because repeating a slide never returns to the start, and it turns out that all its motions share one fixed point. Among the turns about that point, pick the one with the smallest angle. Every other turn must be a multiple of it, since otherwise the leftover would be a still smaller turn. So the turns alone form some Cn. If there is also a mirror, combining it with each of the n turns produces n mirrors, and a matching count the other way shows there can be no more. Turns only, or as many mirrors as turns: those are the two families.

This is why a drawing with such symmetry needs so little ink. One wedge, a 2n-th of the plane for Dn and an n-th for Cn, determines everything. Draw in it once and the group does the rest.

Why the order of two moves matters

Turns get along with each other. Two turns about the same center can be done in either order with the same result, so every cyclic group is commutative, or abelian. Mirrors are less agreeable. In a dihedral group with n of 3 or more, a turn r and a mirror s do not commute, and the rule that governs them is strict: turning and then mirroring equals mirroring and then turning the opposite way.

The tool above makes this concrete. Choose Dihedral and set n to 6. The group has 12 elements, named e, r up to r^5, and s, s r up to s r^5, where s is the mirror in the horizontal line and s r^k means turn k steps, then mirror. In the table, tap the row r and the column s. The column acts first, so this is mirror, then turn, and the answer is s r^5, the mirror at 30 degrees. Now tap the row s and the column r. The answer is s r, the mirror at 150 degrees. Same two moves, opposite order, different result.

Try one more cell: row s, column s r. Two mirrors whose lines stand 30 degrees apart combine into r, a turn of 60 degrees. A composition of two reflections of the plane is always a translation or a rotation, never a third mirror. When both mirror lines pass through the same point, that point stays fixed, so the result is a turn.

That fact is the working principle of a kaleidoscope. David Brewster wrote that the first idea of the instrument came to him in 1814, during experiments on the polarisation of light by reflection between plates of glass. His treatise of 1819 lays down a condition for perfectly symmetrical figures: for an object in any position, the two reflectors must be inclined at an even aliquot part of a circle, and he names 45, 36 and 30 degrees. Those are a full circle divided by 8, 10 and 12, exactly the wedges of D4, D5 and D6. Set the tool to D6 with the guides on, and the shaded wedge is Brewster’s 30 degrees.

Reading the table

A Cayley table records every product at once. The entry in a given row and column is the row’s element combined with the column’s element. In a group every element appears exactly once in each row and once in each column, which makes the table a Latin square. The colors in the tool’s table show it at a glance: no color repeats along any line.

Three things can be read straight off the grid. If the table is its own mirror image across the main diagonal, the group is abelian; for D6 it is not. Wherever the identity sits on the diagonal, that element undoes itself, and in D6 there are eight such elements: the identity, the half turn r^3 and all six mirrors. Finally, the table falls into four blocks. A turn times a turn is a turn, a mirror times a mirror is a turn, and a turn times a mirror, in either order, is a mirror.

The table also hides smaller groups inside it. A cyclic group of n elements has exactly one subgroup for each divisor of n, and no others. For n equal to 12 the divisors are 1, 2, 3, 4, 6 and 12, so there are six, and choosing C4 in the tool keeps only the quarter turns: a drawing under C12 drops from twelve copies to four. Dihedral groups are richer. Every subgroup of Dn is itself cyclic or dihedral, and the tool counts 16 subgroups in D6.

Three exceptions in space

Step off the page and the same two families reappear as groups of rotations. Cn is what a pyramid over a regular base allows: turns about one axis. Dn belongs to a prism. It has the n-fold axis through the two ends and n two-fold axes through the sides, at right angles to the main one. A mirror in the plane has become a half turn in space, because flipping a flat object over does the work of reflecting it. That is why the tool shows Dn as a prism and Cn as a prism tapered at one end, which can no longer be flipped.

In the plane the list ended there. In space it has exactly three more entries. The finite rotation groups are the cyclic groups, the dihedral groups, and the rotation groups of the regular solids: the tetrahedral group T, the octahedral group O and the icosahedral group I. The classification goes back to Felix Klein’s lectures on the icosahedron of 1884. The three groups contain 12, 24 and 60 rotations.

Five regular solids share those three groups. The cube and the octahedron have the same rotations, and so do the icosahedron and the dodecahedron; the tetrahedron stands alone. The tool swaps one solid for its partner while the axes stay put.

The axes tell the groups apart. The tetrahedron has four three-fold axes and three two-fold ones. The cube has three four-fold axes through its faces, four three-fold axes through its corners and six two-fold axes through the middles of its edges. The icosahedron has six five-fold, ten three-fold and fifteen two-fold axes, 31 axes in all. The counts check themselves. Each five-fold axis carries four turns besides the identity, each three-fold axis two, each two-fold axis one, and 24 plus 20 plus 15 plus the identity makes 60.

A small marker placed on a sphere around the solid shows what the count means. Off every axis, the icosahedral group copies it to 60 different places, one for each rotation. Slide it onto a five-fold axis and the copies pile up five deep on 12 points. Add reflections and the three groups double to 24, 48 and 120 elements, with mirror-image copies of the marker appearing in between.

Where the groups turn up

Snow crystals are the familiar case. The United States oceans and atmosphere agency explains that an ice crystal is symmetrical because it reflects the internal order of its water molecules, and that the six arms look alike because each one meets the same conditions on the way down. The hexagonal structure of ice is what makes the symmetry six-fold. Most flowers are radial, built of three or more identical sectors related by rotation about the center, while a minority are bilateral, split into mirror halves by a single plane. Adult echinoderms, the starfish and their relatives, have five-fold radial symmetry.

Chemists sort molecules by these same groups. A point group collects the symmetry operations that leave a molecule indistinguishable from how it started, and benzene, for one, belongs to D6 and, with its mirror planes counted, to D6h. The exceptional groups turn up too. The 60 carbon atoms of buckminsterfullerene sit in a truncated icosahedron cage of 20 hexagons and 12 pentagons, the pattern of a football. Small viruses build their shells from identical protein subunits gathered in fives at the corners of an icosahedron. What icosahedral symmetry cannot do is repeat by translation, so no crystal has it.

It is worth being plain about the limits of all this. Everything here concerns symmetry about a single fixed point. Patterns that repeat by sliding, such as friezes and wallpaper, belong to other groups and to a different tool. Natural objects are only approximately symmetric: snowflakes are never perfectly symmetrical, and a group describes the ideal. The tool builds each group by multiplying matrices until no new ones appear, comparing them within one part in a billion. Its table is drawn only up to n equal to 12, and in space it adds reflections to the three solids only, not to the prisms.

Still, the short list is the point. Turns, or turns with mirrors, and in space three groups more. Once symmetry is treated as something done to a shape, the possibilities can be counted, and a colored table holds them all.