Isotoxal figure 2-dimensional tessellation constructed from regular polygons is isotoxal. For instance, the truncated icosahedron (the familiar soccerball) has two types of edges: hexagon-hexagon and hexagon-pentagon, and it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge. An isotoxal polyhedron has the same dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids, 8 formed by the Kepler–Poinsot polyhedra, and six more as quasiregular (3 | p q) star polyhedra and their duals. There are at least 5 polygonal tilings of the Euclidean plane that are isotoxal,
Dehn plane is π. This is Dehn's semi-Euclidean geometry. It is discussed in . In the same paper, Dehn also constructed an example of a non-Legendrian geometry where there are infinitely many lines through a point not meeting another line, but the sum of the angles in a triangle exceeds π. Riemann's elliptic geometry over Ω("t") consists of the projective plane over Ω("t"), which can be identified with the affine plane of points ("x":"y":1) together with the "line at infinity", and has the property that the sum of the angles of any triangle is greater than π The non-Legendrian geometry consists of