# Polyhedra whose vertices are equivalent and have an extreme distribution on the same sphere.

- Author:
- Roman Chijner

For a given number of

**n**particles on the sphere we calculate coordinates of convex polyhedra whose**vertices**lie on the same sphere, have an extreme distribution and select those whose vertices are equivalent to each other. By*extreme distributions*, we mean the distribution of points on the sphere that correspond to the**local extrema (maxima) of Distance Sum**. The sum of distances is measured by summing all the segments connecting each possible combination of 2 points.**The "measure" of this distribution is the average distance between particles on the unit sphere(p**The method of Lagrange multipliers is used to find the extreme distributions of particles on a sphere: https://www.geogebra.org/m/pjaqednw, https://www.geogebra.org/m/puqnepmv, https://www.geogebra.org/m/rcm4ayek_{n}).*n=4: Tetrahedron; n=6: Octahedron; n=8: Cube, Square Antiprism; n=12: Icosahedron, Cuboctahedron,Truncated Tetrahedron; n=20: Dodecahedron; n=24: Biscribed Snub Cube, Truncated Cube, Biscribed Truncated Octahedron; n=30: Icosidodecahedron; n=48: Biscribed Truncated Cuboctahedron; n=60: Biscribed Snub Dodecahedron, Rhombicosidodecahedron, Biscribed Truncated Icosahedron, Truncated Dodecahedron; n=120: Biscribed Truncated Icosidodecahedron.*