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
Computational geometry main branches of computational geometry are: The primary goal of research in combinatorial computational geometry is to develop efficient algorithms and data structures for solving problems stated in terms of basic geometrical objects: points, line segments, polygons, polyhedra, etc. Some of these problems seem so simple that they were not regarded as problems at all until the advent of computers. Consider, for example, the "Closest pair problem": One could compute the distances between all the pairs of points, of which there are "n(n-1)/2", then pick the pair with the smallest distance. This brute-force algorithm takes O("n") time; i.e. its execution