Shapes that tile
Pick a shape. Then press Add a shape again and again to fit copies around the black dot, corner to corner. A full turn around the dot is 360°. Do the corners fill it exactly, leave a gap, or overlap?
What's going on
A regular shape has all its sides the same length and all its corners the same angle. To cover a floor with one regular shape, corner to corner and with no holes, the corners that meet at a point have to add up to exactly 360°, one full turn.
Only three shapes manage it. Six triangles: 6 × 60° = 360°. Four squares: 4 × 90° = 360°. Three hexagons: 3 × 120° = 360°.
A pentagon's corner is 108°. Three of them make 324°, which leaves a gap of 36°, too thin for another pentagon. Four make 432°, which is too much.
After the hexagon it never works again. The corners keep getting bigger than 120°, so three are always too many, and two are never enough.
Octagons can still tile if you let a second shape help: two octagons and one square make 135° + 135° + 90° = 360°. Look for that pattern on a tiled floor.
Sheet: Shapes that tile.