# 1. The problem of extreme distribution of points on the surface of a sphere.

**The Problem:**

**Find a uniform distribution of particles on the surface of the sphere of radius R.**

*It is assumed*that this distribution must satisfy the

*Principle*of Maximum

**Distance Sum**. The sum of distances(

**Distance Sum**) is measured by summing all the segments connecting each possible combination of 2 points. In this case, the "measure" of the distribution is the

*average*distance between the particles on the unit sphere(p

_{n}). The problem comes down to finding distributions that maximize the selected "measure". So what is the configuration (set of locations) of n points on the sphere so that the sum of the distances is maximal for n=1,2,3,...?

**a**. My solution to this problem started with directly maximizing the Distance Sum. The disadvantages of the proposed algorithm are slow convergence

**([1],[2]**).

**b**. Then, using the

**Lagrange method ([3],[4] or [5])**it turned out that the problem

*has a simple geometric solution*. The iterative procedure does not require calculating all the constantly changing distances between particles. Using GeoGebra Script, the programmed iterative procedure could not cope with a large array of numbers: Generating an extreme arrangements of points on a sphere.

**c.**Finally

**,**I achieved my goal. A programmed task using JavaScript is fast and reliable! You can explore this with this applet. Of course, offline computing is even faster. I calculated here distributions for 72 (maybe more!) particles on a sphere . *You can find simultaneous worksheet of this and other applets in https://www.geogebra.org/m/wsj6hdrs .

*Biscribed Snub Cube (laevo)

biscribed form |

Vertices: | 24 (24[5]) |

Faces: | 38 (8 equilateral triangles + 24 acute triangles + 6 squares) |

Edges: | 60 (24 short + 24 medium + 12 long) |

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