Tiling the whole floor

A tile shop sells squares, triangles and hexagons. Why not tiles with five equal sides and equal corners?

The rule

Five-sided tiles cannot fill a point. Their corners cannot add to exactly 360°.

Use copies of one regular shape, edge to edge. They tile when whole corners fill 360°. Divide 360° by the corner. A whole number means they fit. Only the triangle, square and hexagon do.

A shape with all sides and all corners equal is regular. Walk round it. You turn 360° in all. At each corner you turn 360° divided by the number of sides. The corner itself is 180° minus that.

Example: a regular hexagon has a corner of 120°. Check: 360 ÷ 6 = 60, and 180 − 60 = 120. 360 ÷ 120 = 3, so three fit. A regular five-sided shape has a corner of 108°. Check: 360 ÷ 5 = 72, and 180 − 72 = 108. 360 ÷ 108 is not a whole number.

With seven or more sides, each corner is over 120°. Three corners add to more than 360°. Two corners add to less, so a gap is left.