Shapes that tile
Tonight's idea
A tiling covers a flat floor with shapes, leaving no gaps and no overlaps. A regular shape has all its sides equal and all its corners equal.
Look at one point where corners meet. The corners have to go all the way round it. A full turn is 360 degrees, written 360°. So the corner angles at that point must add up to exactly 360°.
Corner angles of regular shapes: triangle 60°, square 90°, pentagon 108°, hexagon 120°.
Six triangles fit, because \(6 \times 60 = 360\). So do four squares and three hexagons.
Three pentagons use \(3 \times 108 = 324\) degrees. That leaves a gap of 36°, far too small for another pentagon. Four would need 432°, so they would overlap. A corner pushed against a straight side leaves 72°, too small as well. Regular pentagons cannot tile.
Watch one: the regular octagon
A regular octagon has eight sides and corners of 135°. Two corners use 270°. Three would need 405°, which is too much. Octagons on their own fail.
But look at the gap after two: \(360 - 270 = 90\) degrees. That is a square corner. Two octagons and one square fit round every point, and together they do tile.
Try each shape round a point and watch the gap open or close: mathdetours.com/demos/tiling.html.
Your turn
- Two squares and one triangle (all regular) meet at a point. (a) How many degrees have they used? (b) How many are left? (c) Name one regular shape that fills the rest exactly.
- A regular shape with ten sides has corners of 144°. Can it tile on its own? Show the sums for two corners and for three.
- Bricks in a wall are rectangles, with each row slid half a brick along. Draw a small patch. Pick a point where bricks meet. Two corners meet there, and one long straight side. What angles add to 360° at that point?
- A regular shape with twelve sides has corners of 150°. (a) Two of them meet at a point. Which regular shape fills the gap? (b) One of them and one square meet at a point. Which regular shape fills that gap?
Stretch: nothing past six
A regular shape with more than six sides has corners bigger than 120° and smaller than 180°. Explain why no such shape can tile on its own, however many sides it has.
Answers
- (a) \(90 + 90 + 60 = 240\) degrees. (b) \(360 - 240 = 120\) degrees. (c) One hexagon. (Two more triangles also work.)
- No. Two corners use \(2 \times 144 = 288\) degrees, leaving a gap of 72°, too small for a third. Three would need \(3 \times 144 = 432\) degrees, which overlaps.
- Two brick corners of 90° each, and the straight side of the brick above or below, which is a half turn of 180°. \(90 + 90 + 180 = 360\).
- (a) \(360 - 150 - 150 = 60\) degrees: a triangle. (b) \(360 - 150 - 90 = 120\) degrees: a hexagon.
Stretch. Three corners, each bigger than 120°, add to more than 360°, so three overlap. Two corners, each smaller than 180°, add to less than 360°, so two leave a gap. The gap is smaller than one corner, so nothing fits in it. Sliding the rows like bricks does not help: a straight side uses 180°, one corner leaves a gap, and two are too many.