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,
Skew apeirogon they can be called quasiregular. Example quasiregular skew apeirogons can be seen in the Euclidean tilings as truncated Petrie polygons in truncated regular tilings: In hyperbolic geometry, regular skew apeirogons are similarly found like in the Euclidean plane. Hyperbolic regular skew apeirogons also exist as Petrie polygon zig-zagging edge paths on all of the regular tilings of the hyperbolic plane. And again like Euclidean space, hyperbolic quasiregular skew apeirogons can be constructed as truncated petrie polygons within the edges of a truncated regular tiling. A helical skew apeirogon can exist in three dimensions, where the vertices can be seen as