Aperiodic set of prototiles seen on the list of aperiodic sets of tiles. The underlying undecidability of the domino problem implies that there exists no systematic procedure for deciding whether a given set of tiles can tile the plane. Polygons are plane figures which are bounded by straight line segments. Regular polygons have all sides of equal length as well as all angles of equal measure. As early as AD 325, Pappus of Alexandria knew that only 3 types of regular polygons (the square, equilateral triangle, and hexagon) can fit perfectly together in repeating tessellations on a Euclidean plane. Within that plane, every triangle,
Euclidean tilings by convex regular polygons uniformity which requires edge-to-edge regular polygons. Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions. Euclidean and general tiling links: Euclidean tilings by convex regular polygons Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his "Harmonices Mundi" (Latin: "The Harmony of the World", 1619). Following Grünbaum and Shephard (section 1.3), a tiling is said to be "regular" if the symmetry group of the tiling acts transitively on the "flags" of the tiling, where a flag is a triple consisting