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
Skew polygon petrie polygon has. This is 5 sides for a 5-cell, 8 sides for a tesseract and 16-cell, 12 sides for a 24-cell, and 30 sides for a 120-cell and 600-cell. When orthogonally projected onto the Coxeter plane these regular skew polygons appear as regular polygon envelopes in the plane. The "n"-"n" duoprism and dual duopyramids also have 2"n"-gonal petrie polygons. (The tesseract is a 4-4 duoprism, and the 16-cell is a 4-4 duopyramid.) Skew polygon In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The "interior" surface