Pick a regular polygon and gather several copies at a single vertex. If the angles add up to exactly 360°, it tiles with no gaps; if not, a gap is left over or the shapes overlap.
If a regular polygon's interior angle divides 360° evenly, several copies of that shape can be gathered perfectly at a single vertex. Only the equilateral triangle (60°), square (90°), and regular hexagon (120°) satisfy this condition, making them the only shapes that can tile a plane with no gaps using a single regular polygon (a "regular tessellation").
This principle is easy to spot all around us. Honeycombs being hexagonal, and bathroom floor tiles fitting together with no gaps as squares or triangles, are both thanks to this angle condition. The artist M.C. Escher used this same tessellation principle to create mesmerizing pictures of interlocking birds and fish.
Pick a shape and see how many copies you can gather at a single vertex.