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(pn). 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=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.