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,
Pyramid (geometry) is C, which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of regular convex polygons, in which case they are Johnson solids. If all edges of a square pyramid (or any convex polyhedron) are tangent to a sphere so that the average position of the tangential points are at the center of the sphere, then the pyramid is said to be canonical, and it forms half of a regular octahedron. Pyramids with a hexagon or higher base must be composed of isosceles triangles. A hexagonal pyramid with equilateral