Tessellations

Covering a floor with no gaps

Look down at a tiled bathroom floor, a brick wall, or a honeycomb. Copies of a shape fit together to cover a whole surface with no gaps and no overlaps. That is a tessellation (or tiling) — a way of filling the entire flat plane, edge to edge, forever, using copies of one or more shapes.

Some shapes do this easily: squares, triangles, hexagons. Others simply refuse — try to pave a floor with regular pentagons and you are always left with an awkward gap. The surprising thing is that a single tidy rule about angles decides exactly which regular polygons can tile the plane and which cannot. Learn that one rule and you can predict the answer for any shape.

The rule lives at each corner

Pick any point where tiles meet — a vertex. The tiles crowd around that point and, between them, they sweep out a full turn. So the interior angles meeting at every vertex must add up to exactly 360^\circ. Too little and a gap is left; too much and the tiles overlap. Exactly 360^\circ is the whole game.

\text{angles meeting at a vertex} = 360^\circ.

For a regular tessellation — one made from copies of a single regular polygon — every tile has the same interior angle, call it A. If k of them meet at each vertex then k \times A = 360^\circ. Since k must be a whole number (you cannot fit two-and-a-half tiles round a point), the interior angle A has to divide evenly into 360^\circ.

That is the key test. Recall the interior angles of the small regular polygons: triangle 60^\circ, square 90^\circ, pentagon 108^\circ, hexagon 120^\circ. Which of these go into 360 a whole number of times?

See what fits around a point

Step through the four cases. The first three close up perfectly — the tiles fan all the way round and meet exactly, so they extend to fill the whole plane. The pentagons fall short: three of them leave a 36^\circ wedge of empty space, and no whole number of pentagons ever closes the gap.

Worked example 1 — apply the vertex test

Equilateral triangle. Interior angle 60^\circ. Divide:

\frac{360^\circ}{60^\circ} = 6.

A whole number — so six triangles meet at each vertex and the triangle tiles. Likewise the square: 360^\circ / 90^\circ = 4 (four squares per corner), and the hexagon: 360^\circ / 120^\circ = 3 (three per corner).

Regular pentagon. Interior angle 108^\circ. Divide:

\frac{360^\circ}{108^\circ} = 3.33\ldots

Not a whole number. Three pentagons give 3 \times 108^\circ = 324^\circ (a gap), and four give 4 \times 108^\circ = 432^\circ (an overlap). There is no way to hit 360^\circ exactly, so the regular pentagon cannot tile the plane. The same failure hits every regular polygon with seven or more sides — their angles all sit strictly between 120^\circ and 180^\circ and none divides 360^\circ.

Only three regular polygons pass the test. That is the whole answer:

The plane can be tiled by copies of a single regular polygon in exactly three ways: with equilateral triangles, with squares, or with regular hexagons — and in no other way.
The reason is the vertex rule: a regular polygon tiles precisely when its interior angle divides 360^\circ — true only for 60^\circ (six meet), 90^\circ (four meet) and 120^\circ (three meet).

Mixing shapes: semiregular tilings

If you allow more than one kind of regular polygon, more patterns open up. A semiregular (or Archimedean) tiling uses two or more regular polygons and looks identical at every vertex — the same shapes meet in the same order all over the tiling. We name such a tiling by its vertex figure: the list of polygons circling one corner, written by their number of sides.

Take the tiling written 4.8.8 — a square and two regular octagons at each vertex (you have seen it on countless floors). Check the angles: a square contributes 90^\circ and each octagon 135^\circ, so

90^\circ + 135^\circ + 135^\circ = 360^\circ. \checkmark

It closes up, so the pattern tiles. There are exactly eight semiregular tilings in all — others include 3.3.3.3.6, 3.3.4.3.4 and 3.6.3.6 — each just a different whole-number way of hitting 360^\circ with regular polygons.

Worked example 2 — read a vertex figure

Does the vertex figure 3.12.12 — a triangle and two regular twelve-sided polygons (dodecagons) at each corner — actually tile?

Just add the interior angles meeting there. The triangle gives 60^\circ. A regular dodecagon has interior angle 180^\circ - 360^\circ/12 = 150^\circ. So the corner totals

60^\circ + 150^\circ + 150^\circ = 360^\circ. \checkmark

It closes exactly, so 3.12.12 is a genuine semiregular tiling. The recipe never changes: list the polygons at a vertex, add their interior angles, and check for 360^\circ.

These patterns repeat: periodicity

Every tiling above is periodic: there is a small block — a unit cell — that you could stamp out and slide sideways and up-down to rebuild the whole pattern, forever. Shift the square grid by one square and it lands back on itself; the honeycomb repeats the same way. In the language of translation, a periodic tiling maps onto itself under a repeating grid of slides, and it carries rich symmetry — the flat plane has exactly 17 such repeating symmetry patterns, the wallpaper groups.

This raises a deep question: must a tiling repeat? For a long time everyone assumed a set of tiles that can cover the plane can always do so periodically. Astonishingly, that is false — some tile sets cover the plane only in patterns that never repeat. Those Penrose tilings are a story for another page.

Now you know why the world is tiled the way it is. Bathroom and kitchen floors are squares because squares tile with the simplest possible grid — four to a corner, easy to cut and lay. Bees build hexagonal honeycomb because the hexagon not only tiles (three to a corner) but wraps the most space inside the least wall, so it stores the most honey for the least wax.

And you will never see a floor paved with regular-pentagon tiles, because 108^\circ stubbornly refuses to divide 360^\circ. The gap is not a manufacturing problem — it is a fact of arithmetic no tiler can defeat.

The rule that only three regular polygons tile is about regular polygons — equal sides and equal angles. Do not over-read it:

Plenty of irregular shapes tile. Every triangle tiles the plane, and so does every quadrilateral, regular or not. The real requirement is still only that the angles meeting at each vertex total 360^\circ — being regular is not needed.
Irregular pentagons CAN tile even though the regular one cannot. Mathematicians have found 15 distinct families of convex pentagons that pave the plane. It is only the perfectly even, regular pentagon that fails.
So "pentagons can't tile" is wrong; "the regular pentagon can't tile" is right. Always check the 360^\circ condition at the vertex — that, not regularity, is what actually decides.