Of all the regular polygons, only three can tile the plane on their own: the equilateral triangle, the square and the regular hexagon. The condition is bluntly arithmetic — the angles meeting at a vertex must total 360 degrees — and 360 divided by the pentagon’s 108 degrees gives 3.33, so the regular pentagon is out. A periodic pattern is likewise restricted to rotation centres of order 2, 3, 4 or 6. Penrose tilings slip past that law by giving up periodicity: fat and thin rhombs cover the plane with five-fold symmetry, and the ratio of fat tiles to thin ones tends to φ.
Of all the regular polygons, only three can tile the plane on their own: the equilateral triangle, the square and the regular hexagon. The condition is bluntly arithmetic — the angles meeting at a vertex must total 360 degrees — and 360 divided by the pentagon’s 108 degrees gives 3.33, so the regular pentagon is out. A periodic pattern is likewise restricted to rotation centres of order 2, 3, 4 or 6. Penrose tilings slip past that law by giving up periodicity: fat and thin rhombs cover the plane with five-fold symmetry, and the ratio of fat tiles to thin ones tends to φ.
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