Cover the plane with copies of one shape, and try to make it repeat
A tiling of the plane is a family of shapes that covers every point and overlaps nowhere. That is the whole definition. The shapes may be rotated, reflected, repeated or not; there is no requirement that the arrangement be tidy, and no requirement that it repeat.
In practice almost every tiling anyone draws does repeat, because repeating is how you get an infinite object out of a finite instruction. This page is about what repeating means precisely, and about a tiling that declines to.
A tiling is periodic if some translation — a slide, with no turning and no flipping — carries the whole thing exactly onto itself. The slides that work are never isolated: if u and v both work then so does any whole-number combination of them, so the periods of a tiling form a lattice.
One tile per cell, and the lattice is the obvious one. The translations that work form a lattice: all the integer combinations of (1.00, 0.00) and (0.00, 1.00). A tiling with even one non-zero period is called periodic, and every tiling most people can draw from memory is.
Try it: Slide the copy around and watch the distance to the nearest genuine period. It hits zero only at the lattice points, and between them the outlined copy is visibly out of step. Switch to the triangles: the two triangles in a cell point opposite ways, so no translation swaps them — only a half turn does, and a half turn is not a period.
Before asking whether a tiling repeats it is worth being able to check that it is a tiling at all, and by eye that is surprisingly hard — a hairline gap and a hairline overlap look identical at any zoom you are likely to try.
The patch below is checked by arithmetic instead. Every tile in this module sits on a triangular lattice with integer coordinates, so its area is an exact integer count of unit triangles. Add those up. Then, separately, trace the outside boundary by cancelling every edge that two tiles walk in opposite directions, and measure the area it encloses. An overlap or a gap would make the two disagree.
Areas are counted in unit lattice triangles, which is what makes the comparison exact: a tiling is defined by covering everything and overlapping nothing, and both halves of that definition are integer arithmetic here rather than a judgement about pixels.
Try it: Grow the patch and watch the two area columns stay locked together. They are computed by completely different routes — one per tile, one from the boundary alone — and they agree at every size, on the nose, with no tolerance permitted.
Here is the same kind of patch, and the question of the whole module: is there a translation carrying it onto itself? Slide it and count. A tile is green when it lands on a tile of the same shape in the same orientation, and red when it does not.
Search over all 2400 non-zero lattice shifts within 24 steps and the best still leaves tiles stranded. It is worth being careful about what that does and does not show: a finite patch always has a boundary, so some failures are boundary effects, and enlarging the patch raises the best score rather than lowering it. The tiling really is full of repetition. What it has no room for is a single shift that works everywhere at once.
Try it: Press find the best one. It gets a long way — this tiling is stuffed with repetition, and any finite arrangement you find in it turns up again and again. But something is always left over, and a search on a finite patch can never rule out a period that shows up further out. That gap in the argument is exactly what the hierarchy closes.
One canvas runs through this whole module, and it gains a control on every page as the ideas that justify the control arrive. Right now it does two things: it pans and it zooms. That is enough to make the point it is here to make — look as far as you like, at any scale you like, and the pattern never settles into a repeat.
Try it: Zoom into a corner and look for a patch you recognise. You will find it again elsewhere — that much is guaranteed. What you will not find is the same arrangement continuing in the same way, because the surroundings of a repeated patch differ every time.
A set of shapes is called aperiodic when it tiles the plane and every tiling it admits fails to be periodic. That is a strong condition: it is not enough to arrange the tiles badly, the tiles have to make good arrangement impossible. In 1961 Hao Wang conjectured no such set existed. Four years later his student Robert Berger found one — with 20,426 tiles. The rest of this module is the story of that number coming down, and of what happened when it reached one.