Tiling with any 4-gon

Why any quadrilateral can tile the floor. Around any corner of a tiling the angles add up to 360°. The four angles of any quadrilateral add up to 360° too, so four copies can meet at a point, each using a different angle once.

Method 1: turn a copy through 180° about the midpoint of one side: it fits that side exactly. Repeat at every side. This tiles the whole plane, even with a dented 4-gon.

Method 2: draw the grid made by the Varignon parallelogram, place copies on it, and the gaps turn out to be copies too.

Not every shape tiles. A regular pentagon has angles of 108°, and no whole number of 108°s (or 108°s plus a straight angle) makes 360°.

Tiles that never repeat. Some shapes tile the plane without the pattern ever repeating: the 'hat', found in 2023. Such patterns describe quasicrystals (Dan Shechtman, Nobel Prize in Chemistry 2011).

Useful polygon facts:

• angle sum of an n-gon: (n – 2) × 180°;
• number of diagonals: n(n – 3)/2;
• four lengths make a 4-gon exactly when the longest is less than the other three added together.