1. The problem of extreme distribution of points on the surface of a sphere.
- Author:
- Roman Chijner
- Topic:
- Vectors 2D (Two-Dimensional), Vectors 3D (Three-Dimensional), Centroid or Barycenter, Equilateral Triangles, Optimization Problems, Geometry, Intersection, Linear Programming or Linear Optimization, Mathematics, Matrices, Means, Median Value, Plane Figures or Shapes, Planes, Polygons, Scalene Triangles, Solids or 3D Shapes, Special Points, Sphere, Surface, Geometric Transformations, Triangles, Vectors, Volume
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(pn). 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) |