Regular polytope impossible ("not constructible"). The simplest few regular polygons that are impossible to construct are the "n"-sided polygons with "n" equal to 7, 9, 11, 13, 14, 18, 19, 21... Constructibility in this sense refers only to ideal constructions with ideal tools. Of course reasonably accurate approximations can be constructed by a range of methods; while theoretically possible constructions may be impractical. Euclid's "Elements" gave what amount to ruler-and-compass constructions for the five Platonic solids. (See, for example, Euclid's Elements.) However, the merely practical question of how one might draw a straight line in space, even with a ruler, might lead
Regular polytope one to question what exactly it means to "construct" a regular polyhedron. (One could ask the same question about the polygons, of course.) The English word "construct" has the connotation of systematically building the thing constructed. The most common way presented to construct a regular polyhedron is via a fold-out net. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along just enough edges so that the surface may be laid out flat. This gives a plan for the net of the unfolded polyhedron. Since the Platonic solids have only triangles,