Schwarz triangle

Consider a regular convex polyhedron projected onto a sphere, and consider the three sets of lines, viz, the edges, lines connecting the face centres to the vertices of the said face, and lines connecting the face centres to the middles of the surrounding edges. These lines form a number of great circles on the sphere, and separate the spherical polyhedron into a number of triangles. These fundamental triangles are called M&ouml;bius triangles.

If, instead of restricting oneself to fundamental triangles, larger triangles can be created. These are called Schwarz triangles.

Each Schwarz triangle represents up to eight different polyhedra (seven simple and one snub form).

The density of a polyhedron is generally equal to the number of fundamental triangles enclosed. This number is easy to compute, by adding up the three angles for the Schwarz triangle.

The generalisation to a 3D simplex of a polychoron is called a Goursat tetrahedron.