From Moving Everything to Fixing One Figure
The last three guides built a complete toolbox of plane motions. You met the slide, the turn, and the flip; you learned that composing a flip with a slide gives the glide reflection; and you saw the punchline that every isometry of the plane is one of exactly four kinds — translation, rotation, reflection, glide reflection — by the classification of plane isometries. That was a statement about the motions themselves, asked with no figure in the picture. Now we point the same machinery at a single shape and ask a sharper question.
Pick a figure — say a square sitting in the plane. Most isometries move it somewhere else: slide it three units right and it is plainly no longer where it was. But a few special motions pick the square up and set it back down so it occupies the exact same patch of plane, edge for edge, corner landing on corner. A quarter-turn about its centre does this. So does a flip across a diagonal. We call such a motion a symmetry of the figure: an isometry that maps the figure onto itself. The figure may have its corners shuffled, but as a set of points it is unmoved.
Why the Symmetries Form a Group
Here is the first surprise, and it is the whole reason the subject has a clean theory. The symmetries of a figure are never a loose, shapeless collection — they fit together with an iron logic. Do one symmetry, then another, and the figure returns to itself twice over, so the combined motion is again a symmetry. Every symmetry can be undone, and its undoing is also a symmetry. And doing nothing — the identity — is trivially a symmetry. These three facts say the symmetries are closed under composition, closed under inverses, and contain the identity: that is exactly the definition of a group.
So to each figure we attach its symmetry group: the set of all isometries that map it onto itself, with composition as the operation. From the earlier rung on transformation groups you already know composition of isometries is associative — sliding-then-the-result behaves the same however you bracket it — so the last group axiom comes for free. This is why the transformation group viewpoint was worth building: a symmetry group is just a transformation group, narrowed to the motions that respect one chosen figure. The figure picks out its own group, and the group remembers everything rigid you can do to the figure.
One quiet but important fact pins everything down for a bounded figure — one you could fence inside some big circle. Such a figure cannot have any translation or glide reflection as a symmetry, because either of those would shift the figure permanently off in one direction, and a bounded shape has nowhere to go. So only rotations and reflections survive as possible symmetries, and what is more, every one of them must fix the figure's centre point. That single restriction is what makes the answer for bounded figures so short.
Two Families: Cyclic Wheels and Dihedral Mirrors
Now we can say exactly what is possible. Imagine a figure with only rotational symmetry and no mirrors — a child's pinwheel, or a triskelion of three bent arms all curling the same way. Its rotations form a wheel: if the smallest turn that works is 360/n degrees, then turning by that angle 0, 1, 2, up to n-1 times gives n distinct symmetries, and the n-th turn is a full circle back to the start. These n rotations are the cyclic group, written C_n. A figure with C_4 symmetry looks the same after every quarter-turn but is changed by any flip — the pinwheel's arms would reverse their swirl in a mirror.
Now add mirrors. A figure with the full symmetry of a regular polygon — a regular n-gon, or a snowflake, or a plain square — has both the n rotations and n reflection axes. For the square those axes are the two diagonals and the two lines through opposite edge-midpoints: four mirrors in all, matching its four rotations. Together that is 2n symmetries, and they form the dihedral group D_n. So the square's symmetry group is D_4, with eight elements; an equilateral triangle has D_3 with six. The two families exhaust every bounded figure: its symmetry group is C_n if it has rotations only, or D_n if it has mirrors too. There is nothing else.
Bounded figure -> symmetry group is C_n or D_n C_n (cyclic): n rotations, NO mirrors | C_n | = n D_n (dihedral): n rotations + n reflections | D_n | = 2n pinwheel (4 arms) -> C_4 (4 elements) square -> D_4 (8 elements) equilateral tri -> D_3 (6 elements) the letter S -> C_2 (180-deg turn only) the letter A -> D_1 (one mirror only) scalene triangle -> C_1 (identity only)
Reading a Figure's Group, and Why Mirrors Come in Matched Counts
Finding the group of a given figure is a short, honest procedure, and the alphabet makes a perfect practice set. The capital letter S has no mirror line at all — try to flip it and the curves go the wrong way — but a half-turn about its centre lands it perfectly on itself, so its group is C_2. The capital A has a single vertical mirror and no nontrivial rotation, giving D_1, a group of just two elements (identity and that one flip). The capital H has both a vertical and a horizontal mirror, and the 180-degree turn that their combination forces, so its group is D_2 with four elements. A plain scalene triangle has no symmetry but the identity: its group is C_1, the trivial group.
- Find the centre. For a bounded figure every symmetry fixes one common point; locate it (for a polygon, the balance point) before looking for any motion.
- Find the smallest turn that works. Rotate about the centre until the figure first lands on itself; if that angle is 360/n degrees, you have n rotations and an underlying C_n.
- Hunt for a single mirror. Test lines through the centre for a reflection that maps the figure to itself; you only need to find one.
- Decide the family. No mirror at all means the group is C_n; one mirror automatically forces n of them, and the group is D_n with 2n elements.
That last step hides a small marvel worth pausing on: why does one mirror force exactly n of them, never some odd in-between count? Because a reflection composed with a rotation is itself a reflection, by the classification you proved earlier. Take your one mirror, then compose it in turn with each of the n rotations in the group; the group's closure guarantees all n results are symmetries, and each is a reflection across a fresh axis. So the moment a single mirror exists, the n rotations breed exactly n mirrors — no more, no fewer. This is the structural reason D_n always has its rotations and reflections in equal number, and it is the group axioms doing the bookkeeping for you.
Honest Limits, and the Door to Patterns
It is worth being clear about what this clean classification does and does not claim. It is exactly right for bounded figures — single shapes you can box inside a circle — and the credit is usually given to Leonardo da Vinci, who catalogued these symmetry types while planning the chapels and niches that could ring a central church without spoiling its balance. But the theorem leans entirely on boundedness. The instant a figure runs off to infinity — a stripe of repeating footprints, an endless tiled floor — translations come back into play, and they are precisely the symmetries that a bounded figure forbids.
And once translations are allowed, the gentle two-family picture explodes into something richer. A pattern that repeats along a single line — a frieze along the top of a wall — has a symmetry group containing a smallest translation, and it turns out there are exactly seven such frieze groups, no matter how ornate the artist gets. A pattern that repeats in two independent directions to fill the whole plane — wallpaper, a tiled floor, a honeycomb — has, astonishingly, exactly seventeen possible wallpaper groups. Those counts are not loose observations; they are theorems, proven by pushing this same group idea much harder.
So this guide is the hinge of the rung. Behind us, the four kinds of motion and the proof that nothing else exists; ahead, those same motions assembling into the seven friezes and seventeen wallpapers. The bounded story you just learned — C_n and D_n, a wheel and its mirrors — is the seed of the unbounded one, with translation the single new ingredient that turns a finite group into an infinite, repeating one. Keep the group in mind, not just the picture: it is the object that will let us count patterns that the eye alone could never sort.