From one figure to an endless pattern
In the previous guide you watched the symmetry group of a single bounded figure take shape: gather every isometry that maps the figure onto itself, and those motions form a group — closed under composition, every move undoable. For a square that group had eight elements; for a snowflake, twelve. Now we make one decisive change: instead of a single tile, picture a pattern that goes on forever in one or two directions. A bounded figure cannot move by a translation and land on itself, but an infinite stripe of identical footprints can. That single new ingredient — a genuine translation that is a symmetry — changes the whole story.
The catalogue of possible symmetries has not grown, though. The classification you met two guides back still holds: every plane isometry is a translation, a rotation, a reflection, or a glide reflection — there is nothing else. So a repeating pattern's symmetry group is assembled entirely from those four kinds of move. What makes the subject sharp and finite is that translations and the other symmetries cannot be combined freely. They constrain one another, and out of those constraints fall the short, complete lists that this guide is about.
Friezes: patterns with one direction of repeat
Start with the simplest infinite case — a frieze, the kind of decorative border that runs along the top of a wall or around a vase, repeating in exactly one direction. Its symmetry group is a frieze group. By definition it always contains translations along the strip, all of them whole-number multiples of one shortest translation; that shortest step is the pattern's repeat unit. The question is what else can be a symmetry without breaking the strip. The honest constraint is geometric: any extra symmetry must map the infinite horizontal strip to itself, so it can only be a horizontal reflection across the strip's centre line, a vertical reflection across a line cutting straight up through the strip, a half-turn (a 180-degree rotation), or a glide reflection along the strip.
Working through which of those can coexist gives a clean result: there are exactly seven frieze groups, no more and no fewer. A footprint trail that just steps forward, left-right-left-right, realises one of them — it has translation and a glide reflection but no mirror. A row of capital letter A's repeated, AAAA, realises another — each A has a vertical mirror, so the whole strip does too. A row of H's is the richest, carrying horizontal and vertical mirrors and half-turns at once. The point is not to memorise all seven but to feel why the list closes: only four kinds of extra move are allowed, and they can be switched on or off only in combinations that are mutually consistent.
Wallpapers: two directions, seventeen groups
Now let the pattern repeat in two independent directions, filling the whole plane like wallpaper or tiled floor. Its symmetry group is a wallpaper group, and it must contain translations along two directions that are not parallel — two shortest steps whose whole-number combinations carry every cell of the pattern onto another. On top of those translations you may stack rotations, reflections, and glide reflections, exactly as before, but now in a two-dimensional grid the constraints are tighter and richer. The remarkable theorem, proved in the nineteenth century and central to crystallography, is that there are exactly seventeen wallpaper groups. Every periodic plane pattern that has ever been drawn — Islamic tilework, Escher's interlocking lizards, a brick wall, a honeycomb — has a symmetry group that is one of those seventeen.
Seventeen is small for a reason worth stating plainly: the two-dimensional translation lattice severely limits which rotations can appear. The deepest single fact behind the whole classification is the crystallographic restriction — in a pattern that repeats in two directions, the only rotational symmetries possible are by 360, 180, 120, 90, or 60 degrees, that is, rotations of order 1, 2, 3, 4, or 6. Fivefold rotational symmetry, and sevenfold, and every order above six, are flatly forbidden for a repeating pattern. You will never wallpaper a floor with regular pentagons whose pattern repeats; the arithmetic will not allow it.
Why fivefold symmetry is impossible
This restriction is one of the rare results in this whole ladder whose proof really is elementary, so let us actually see it rather than just assert it. The argument is a short impossibility proof by contradiction, the same flavour you met when squaring the circle was ruled out: assume the forbidden symmetry exists and watch it generate something smaller than the smallest allowed thing. Here the smallest allowed thing is the shortest translation, the minimum repeat distance of the pattern.
- Suppose the pattern has a rotation of some order n about a centre, and let d be the shortest translation distance in the pattern. Pick two rotation centres A and B that are exactly d apart — the translation symmetry guarantees a whole grid of equivalent centres, so two of them sit at this minimum distance.
- Rotate B about A by the rotation angle to get a new centre B', and rotate A about B by the same angle to get A'. Because A and B are symmetry centres of the same order, B' and A' are again genuine rotation centres of the pattern, so the distance |A'B'| is some whole-number multiple of d — it cannot be a non-multiple, or it would beat d as the shortest translation.
- Now do the trigonometry. For a fivefold rotation (72 degrees), the construction makes |A'B'| come out shorter than d but not zero — a translation strictly between 0 and the supposed minimum. That is a flat contradiction, so order 5 is impossible. Running the same computation, only n = 1, 2, 3, 4, 6 survive; n = 5 and every n above 6 fail the same way.
That is the entire heart of why there are seventeen wallpaper groups and not more: the allowed rotation orders are pinned down to five values, and from those few choices, combined with the possible mirrors and glides, the complete count follows. The argument is honest and finite, and it is one of the cleanest payoffs of treating symmetry as a classification of isometries rather than a vague aesthetic.
Tilings: covering the plane with tiles
A close cousin of wallpaper is the tiling, or tessellation: a covering of the whole plane by tiles with no gaps and no overlaps. The cleanest cases use a single regular polygon repeated edge to edge — a regular tiling. Here a small piece of the angle arithmetic you already own settles everything. At every vertex the tile angles must sum to exactly 360 degrees, and a regular polygon with more sides has a larger interior angle. An equilateral triangle's angle is 60 (six fit around a point), a square's is 90 (four fit), a regular hexagon's is 120 (three fit). The regular pentagon's interior angle is 108 degrees, which divides into 360 only three-and-a-third times — it does not fit a whole number of copies around a vertex, so regular pentagons cannot tile the plane.
So there are exactly three regular tilings of the plane: by triangles, by squares, and by hexagons — the honeycomb being nature's choice. That is a genuinely complete list, forced entirely by the divisor condition on 360. But be careful not to overstate it. The claim 'only three tilings' applies strictly to regular polygons, all of one kind, meeting edge to edge. Drop any of those conditions and the world opens up: mix two shapes and you get the eight semiregular tilings; allow irregular tiles and there are infinitely many. Famously, every triangle tiles the plane, and so does every quadrilateral, convex or not. The restriction lives in the word 'regular', and stating that honestly is the whole point of guide-by-guide care.
When repeating fails: aperiodic tilings
There is one more honest twist, and it is the most surprising. Everything above assumed the pattern eventually repeats — that it has a translation symmetry. But a set of tiles can be forced to fill the plane while never repeating at all. The celebrated example is the Penrose tiling, built from just two rhombus shapes (or, in another version, a kite and a dart). With the right edge-matching rules these tiles cover the whole plane, yet no translation maps the finished pattern onto itself — there is no repeat unit anywhere. Such a tiling is called aperiodic.
Here is the beautiful sting in the tail. Because a Penrose tiling does not repeat, the crystallographic restriction does not apply to it — and it can, and does, display fivefold rotational symmetry, the very thing forbidden to ordinary wallpaper. This is not a contradiction; it is the restriction's fine print made visible. Fivefold symmetry is impossible only for patterns with translation symmetry. Remove the demand to repeat, and the door that the impossibility proof bolted shut quietly reopens. In the 1980s, physical materials called quasicrystals were discovered that arrange their atoms in just this aperiodic, fivefold way, turning a piece of pure tiling geometry into a real chapter of chemistry.