[size=85]For a given number of n particles on a sphere, the coordinates of convex polyhedra whose vertices lie on the same sphere and have an extremum are calculated .[br] By extreme distributions, we mean the distribution of points on the sphere that correspond to the local extrema (maxima) of [i]Distance Sum[/i]. 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 [color=#1e84cc][i]average distance between particles on the unit sphere[/i][/color] -p[sub]n[/sub]. Case of optimal-extreme distribution of points on the surface of a sphere [i]comes down[/i] to the case that each of its vertices on a sphere is located at the points of the geometric medians of its other vertices ([color=#ff7700]Δφ[sub]i[/sub] [sub]GM[/sub][/color]=0 rad, i=1,..,n). The [i]method of Lagrange multipliers[/i] is used to find the extreme distributions of particles on a sphere.[/size]

[size=85] Calculations have shown that for certain numbers of vertices, there are several extreme distributions on the sphere. For example, for n=8 vertices, there are 4 well-known distributions: as [i]Gyrobifastigium[/i] (J[sub]26[/sub]) p=1.478 763 285, [i]Cube[/i] p=1.480 440 157 243 444, [i]Snub disphenoi[/i]d p=1.481 089 591 416 065, [i]Square antiprism[/i] p=1.481 181 823 885 056. For n=60, there are 5 distributions: [i]Subscribed Truncated Icosahedro[/i]n p=1.353 978 070 225 329, [i]Truncated Dodecahedron[/i] p=1.352 832 724 261 989, [i]Subscribed Snub Dodecahedron[/i] p=1.354 126 670 906 686, ([i]No name)[/i] p=1.354 164 857 048 013 -a distribution with very low symmetry. For some numbers of n: 7, 11, 13,... there are no extreme distributions(?)- I couldn't find them! The iterative procedure used finds mainly the distribution [i]corresponding[/i] to the [i]global maximum[/i], regardless of the initial particle distribution. Other solutions can be found under certain restrictions of the initial distributions.[/size]

[size=85] From [url=https://www.geogebra.org/m/e7v6eps2]Applet[/url]: Three-parameter model transformations of the Icosahedron. Extreme distributions.[br]Example 9. [color=#ff0000][b]as[/b][/color] [url=http://dmccooey.com/polyhedra/TruncatedIcosidodecahedron.html]Truncated Icosidodecahedron[/url][br][table][tr][td]Vertices:  [/td][td]120  (120[3])[/td][/tr][tr][td]Faces:[/td][td]62  (30 squares + 20 regular hexagons + 12 regular decagons)[/td][/tr][tr][td]Edges:[/td][td]180[/td][/tr][/table][/size]