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Tiling the floor: which shapes cover a flat surface with no gaps

Geometry · level 2, lesson 5 of 6

A question to start

Look at a bathroom floor, a kitchen wall or a sidewalk. The tiles are usually squares or rectangles. Sometimes you see triangles, and sometimes hexagons, like a honeycomb.

But have you ever seen a floor covered with regular pentagons, the five-sided shapes with all sides and corners equal? Or with regular octagons, all by themselves?

You will not find one, anywhere in the world. It is not that nobody thought of it. It is impossible. In this lesson you will find out exactly why, using nothing but the corners of the shapes.

The idea

What a tiling is

A tiling is a way of covering a flat surface with shapes so that there are:

  1. no gaps between the shapes, and
  2. no overlaps (no shape lies on top of another).

The pattern could go on forever in every direction. Another name for a tiling is a tessellation (say "tess-uh-LAY-shun").

Three regular shapes that tile

squares triangles hexagons
  • Squares tile, like a chessboard.
  • Equilateral triangles (all sides equal) tile. Half of them point up and half point down.
  • Regular hexagons tile, like the cells of a honeycomb.

These are the only three regular polygons that can tile a floor all by themselves. Let us find out why.

The secret is at the corners

Look at any point in a tiling where corners of tiles meet. The corners have to fit all the way round the point, with no gap and no overlap. All the way round a point is one full turn, like turning yourself all the way round to face the front again.

  • In the square tiling, 4 square corners meet at each point. A square corner is a quarter of a full turn, and 4 quarters make a whole. They fit exactly.
  • In the triangle tiling, 6 triangle corners meet at each point. Each equilateral triangle corner is a sixth of a full turn, and 6 sixths make a whole.
  • In the hexagon tiling, 3 hexagon corners meet at each point. Each hexagon corner is a third of a full turn.

Here is a way to see the hexagon fact without measuring anything. A regular hexagon can be cut into 6 equilateral triangles, all meeting at its centre. Each hexagon corner is made of 2 triangle corners. So 3 hexagon corners are the same as 6 triangle corners, and we already know 6 triangle corners fill a full turn.

Why regular pentagons fail

Now the pentagon. How big is a pentagon's corner? We can compare it without any measuring tool.

  • A regular pentagon's corner is wider than a square corner. (Trace one and lay it on a square corner: it sticks out.)
  • It is narrower than a regular hexagon's corner.

Now try to fit regular pentagons round a point.

gap 3 pentagons leave a gap; a 4th one will not fit
  • 3 pentagon corners: each one is narrower than a hexagon corner, and 3 hexagon corners make exactly a full turn. So 3 pentagon corners make less than a full turn. There is a gap.
  • 4 pentagon corners: each one is wider than a square corner, and 4 square corners make exactly a full turn. So 4 pentagon corners make more than a full turn. They overlap.

Three is too few and four is too many. There is no number in between! So regular pentagons can never tile. That is a real proof: a reason that works every time, not just a guess from one drawing.

What about bigger regular polygons?

The more sides a regular polygon has, the wider its corners get. (A regular octagon is rounder than a square, and its corners are flatter.)

  • 3 corners of a regular octagon: each is wider than a hexagon corner, so 3 of them make more than a full turn. Overlap.
  • 2 corners: each corner of any polygon like this is less than half a turn (less than a straight line), so 2 of them make less than a full turn. Gap.

So octagons cannot tile alone, and neither can any regular polygon with more than 6 sides. With 7 or more sides, 3 corners are already too many, and 2 are too few. That is why the list stops at triangles, squares and hexagons.

Mixing shapes

Octagons cannot tile alone, but they can tile with a helper.

octagons and squares together circles leave gaps

At each point, 2 octagon corners and 1 square corner meet. The gap that two octagons leave is exactly the size of a square corner, so little squares fill the gaps perfectly. You may have seen this pattern on a kitchen floor.

Circles, on the other hand, can never tile. However you pack them, they leave curved gaps (on the right), because a circle has no corners and no straight sides to press against its neighbours.

Any triangle tiles, and any four-sided shape tiles

Here is something surprising. You do not need a regular shape to tile. Every triangle tiles, even a lopsided one. And every four-sided shape tiles, even a strange one (as long as its sides do not cross).

any triangle any four-sided shape

Why? Try this experiment. Cut out a paper triangle, any triangle. Tear off its three corners and put them side by side, points together. They always make a straight line, which is half a turn. So if you put the triangle and a copy turned upside down together, around each meeting point you get each corner twice: two half turns, which make a whole turn. No gap, no overlap.

Do the same with a four-sided shape: tear off the 4 corners and put them together. They always fill a full turn exactly. So at each meeting point in the tiling, one of each corner fits round the point.

(You will prove these corner facts with real reasons in level 4: angles in a triangle.)

Tilings in nature and art

Honeybees build their honeycomb from hexagons. In 1999 the mathematician Thomas Hales proved that the hexagon pattern is the best possible way to divide a flat surface into equal-sized cells while using the least total length of walls. For a bee, walls are made of wax, and wax is costly to make. So the hexagon is the bee's thrifty choice.

The artist M. C. Escher made famous tilings in which every tile is the shape of a bird, a fish or a lizard. He started with simple shapes like squares or triangles, cut a bump out of one side and stuck it onto the opposite side, so the tiles still fitted together perfectly.

And pentagons? Regular pentagons cannot tile, but some lopsided pentagons can. Mathematicians have found 15 different kinds, and the 15th was only discovered in 2015. Geometry still has new things to discover!

Worked examples

Example 1. How many corners meet? In a tiling of regular hexagons, how many hexagons meet at each corner point?

  1. Each hexagon corner is a third of a full turn (2 triangle corners, and 6 triangle corners make a full turn).
  2. Three thirds make a whole.
  3. So 3 hexagons meet at each point. Look at a honeycomb: every corner is where 3 cells meet, like the letter Y.

Example 2. Does a rectangle tile? Does a rectangle (like a brick) tile a floor?

  1. All 4 corners of a rectangle are square corners.
  2. 4 square corners fill a full turn.
  3. So rectangles fit 4 to a point, in rows, like a brick wall or a sheet of stamps. Yes, they tile. (They can also tile in a brick pattern, where the rows are shifted.)

Example 3. Mixing triangles and squares. Can equilateral triangles and squares (with the same side length) fit together round one point?

  1. A triangle corner is a sixth of a turn. A square corner is a quarter of a turn.
  2. Try 3 triangles and 2 squares: 3 sixths is a half, and 2 quarters is a half. A half and a half make a whole!
  3. So yes: 3 triangles and 2 squares fit exactly round a point. Floors are really made like this: a row of squares, then a row of triangles, then a row of squares, and so on.

Try it

  1. How many squares meet at each corner point of a square tiling?
  2. How many equilateral triangles meet at each corner point?
  3. Does a regular pentagon tile? Give the reason in one sentence.
  4. Can a regular octagon tile all by itself? What helper shape makes it work?
  5. Could a floor be tiled with copies of one lopsided triangle? How?
  6. 2 regular hexagons meet at a point. Two of their corners fill two thirds of a full turn. Which one shape's corner fills the last third? (Hint: a hexagon corner is a third of a turn.)
  7. Would circles make a good floor tile? Why not?
  8. A floor uses equilateral triangles and regular hexagons. At one point, 2 hexagon corners meet. How many triangle corners fill the rest?
Answers
  1. 4. Each square corner is a quarter of a full turn, and 4 quarters make a whole turn.
  2. 6. Each corner of an equilateral triangle is a sixth of a full turn.
  3. No. Three pentagon corners leave a gap (each is narrower than a hexagon corner) and four overlap (each is wider than a square corner), and there is no number between 3 and 4.
  4. No, not by itself: 2 octagon corners leave a gap and 3 overlap. With squares as helpers it works: 2 octagons and 1 square meet at each point.
  5. Yes. Put the triangle next to a copy turned upside down (half a turn): together they make a slanted four-sided shape. Rows of those fill the floor. At each meeting point the corners fit exactly, because the three corners of a triangle make a straight line, and each appears twice.
  6. Another hexagon. Its corner is a third of a turn, which is exactly what is left. (That is why 3 hexagons meet at every point.) Two triangle corners also fill a third, so 2 triangles work too.
  7. No. Circles leave curved gaps between them, however you pack them, because they have no straight sides to fit against each other.
  8. 2 hexagon corners fill two thirds of a turn. The rest is one third. A triangle corner is a sixth, and 2 sixths make a third. So 2 triangle corners. (This makes a real tiling: around every point, hexagon, triangle, hexagon, triangle, taking turns.)

Common mistake

Thinking any regular shape will tile if you try hard enough. Regular pentagons, heptagons and octagons never tile on their own. It is not a matter of trying harder: the corners simply do not add up to a full turn.

Thinking only regular shapes tile. The opposite surprise: every triangle and every four-sided shape tiles, even lopsided ones. Regular is not needed. What is needed is that the corners meeting at each point add up to exactly one full turn.

Where it leads