[i][b][size=85] Generating a uniform distribution of points on a sphere.[/size][/b][size=85][color=#333333] Let S be a sphere of radius R around the point O: S:={x∈[/color][size=100]ℝ[/size][sup]³[/sup][color=#333333]: ||x||=R}. [/color][i]There is a set lP={A1, A2,...,An} [/i]of [b][u]n movable free points on a sphere[/u][/b].[br][b] Problem[/b]: use the [b][i]method of Lagrange multipliers[/i][/b] find such their distribution corresponding to the [i][b][color=#ff0000]maximum[/color][/b][/i] [b]sum of all their mutual distances[/b]. [br][i] This means need to find out such [i]mutual arrangement[/i] of "[i][i][color=#ff0066]repulsive" set of [/color][u][color=#0000ff]particles on a sphere[/color][/u][color=#ff0066],[/color][/i] [/i]when each point of this set is [i][b][color=#ff7700]Geometric median[/color][/b] [color=#333333]([/color][b][color=#ff7700]GM[/color][color=#333333])[/color][/b][/i] of the remaining n-1 points. [i]We assume that the equilibrium - [/i][i]stationary state [/i][i][color=#333333]in system of "charges" is reached if the sum of their mutual distances is [/color][color=#ff7700]maximal[/color][color=#333333]. [/color][/i][i][i]Iterative approach[/i][/i] of [i][i]particle placement[/i][/i] is applied for [i]achieving [/i][i]a stationary state.[/i][/i][/size][/i][size=85][i][br] An [/i][url=https://www.geogebra.org/m/xeangb8c]iterative procedure[/url][i] for calculating a polyhedron with an extreme vertex arrangement allows one point to be fixed at any point on the sphere. In the applet, point A can be set at any point in the sphere. For orientation, you can set Check box "[/i][color=#b45f06]sphere[/color][i]" - true. Click on the "[/i][b]Click→Input,Initial settings[/b][i]"-button and other starting points will be randomly set. Click the "[/i][color=#ff7700]Start[/color][i]"-button and wait for the iteration process to finish! [br]Note: [br]1.Theoretical instructions can be found in [url=https://www.geogebra.org/m/rcm4ayek]https://www.geogebra.org/m/rcm4ayek[/url][br]2.Images and explanations to them in [url=https://www.geogebra.org/m/uegb5ym3]https://www.geogebra.org/m/uegb5ym3[/url][br]3.This applet might not work properly in the online worksheet, but it works well in a ggb file. [br][br]*old [/i][url=https://www.geogebra.org/m/caqpbbrf]version[/url][/size]