[size=85] [b]The Problem:[/b] [b] Find a uniform distribution of particles on the surface of the sphere of radius R.[/b][br][i][color=#333333] It is assumed[/color][/i] that this distribution must satisfy the [i]Principle[/i] of [color=#ff0000]Maximum[/color] [b]Distance Sum[/b]. The sum of distances([b]Distance Sum[/b]) is measured by summing all the segments connecting each possible combination of 2 points. In this case, the "measure" of the distribution is the [color=#1e84cc][i]average[/i] distance between the particles on the unit sphere([/color][color=#9900ff]p[sub]n[/sub][/color][color=#1e84cc]).[/color] The problem comes down to finding distributions that [color=#ff0000]maximize[/color] 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,...?[br][b]a[/b]. My solution to this problem started with directly maximizing the Distance Sum. The disadvantages of the proposed algorithm are slow convergence [b]([[url=https://www.geogebra.org/m/dVXk2Gbq]1[/url]],[[url=https://www.geogebra.org/m/aeqJmSdH]2[/url]][/b]).[br][b]b[/b]. Then, using the [b]Lagrange method ([[url=https://www.geogebra.org/m/stfexhyj ]3[/url]],[[url=https://www.geogebra.org/m/d9ytv4wg]4[/url]] or [[url=https://www.geogebra.org/m/tnut2f9y ]5[/url]])[/b] it turned out that the problem [i]has a simple geometric solution[/i]. The iterative procedure does not require calculating all the constantly changing distances between particles. Using GeoGebra Script, the programmed iterative [url=https://help.geogebra.org/topic/geogebra-windows-portable-zip-for-december]procedure[/url] could not cope with a large array of numbers: [url=https://www.geogebra.org/m/b5zcy52h]Generating an extreme arrangements of points on a sphere[/url].[br][b]c. [/b]Finally[b],[/b] 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 .[br] *[size=85]You can find simultaneous worksheet of this and other applets in [url=https://www.geogebra.org/m/wsj6hdrs]https://www.geogebra.org/m/wsj6hdrs[/url] .[/size][/size]

[size=85]*[url=http://dmccooey.com/polyhedra/BiscribedLsnubCube.html]Biscribed Snub Cube (laevo)[/url][table][tr][td][url=http://dmccooey.com/polyhedra/BiscribedLsnubCube.html][/url][/td][/tr][/table][table][tr][td][/td][td]biscribed form[/td][/tr][/table][table][tr][td]Vertices:  [/td][td]24  (24[5])[/td][/tr][tr][td]Faces:[/td][td]38  (8 equilateral triangles + 24 acute triangles + 6 squares)[/td][/tr][tr][td]Edges:[/td][td]60  (24 short + 24 medium + 12 long)[/td][/tr][/table][/size][table][tr][td][/td][/tr][/table][table][tr][td][/td][/tr][/table]