Three-parameter model transformations of the Octahedron. Extreme distributions.
[size=85]Three parameters are used to transform Icosahedron faces:[br] t - for truncating, [br] q-similarity transformations,[br] α-angle rotation of faces.[br] With these parameters, using the well-known [url=https://en.wikipedia.org/wiki/Expansion_(geometry)]Expansion[/url] and [url=https://en.wikipedia.org/wiki/Snub_(geometry)]Snub[/url] operations applied to a polyhedron, you can explore various polyhedra and get a number of [url=https://www.sacred-geometry.es/?q=en/content/archimedean-solids]Archimedean Solids[/url]. [br] The p (distance sum) is defined here as the ratio of the sum of all pairwise distances between the vertices of the polyhedron to the number of these pairs and the radius of circumscribed sphere. Value p can be given the meaning of the mean distance between the vertices of a polyhedron on a unit sphere.[br] There are three parameters, that are used for three maximizations of the average distance between vertexes. Cases: 8 -corresponds to the [i]Biscribed Truncated Cuboctahedron[/i], 9 -?, 11 -corresponds to the [i]Biscribed Snub Cube[/i].[/size]
Values of parameters in the polyhedron model defining the well-known polyhedra.
Example 8. Pmax=1. 1.358 608 009 549 744 , t, q, α=0, V=48
[size=85]Case of optimal distribution of points on the surface of a sphere. This means 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]GM[/sub][/color]=1.32*10[sup]-9[/sup] rad).[/size]
Example 9. Pmax= 1.358 644 617 482 728 , t, q, α≠0, V=120
[size=85]The case of an nearly optimal distribution of points on the surface of a sphere (with accuracy [color=#ff7700]Δφ[sub]GM[/sub][/color]=2.40*10[sup]-3[/sup] rad).[/size]
Example 11. pmax =1.384 075 508 822 845; t=0.5, q, α, Biscribed Snub Cube(laevo), V=24.
[size=85] Case of optimal distribution of points on the surface of a sphere. This means 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]GM[/sub][/color]=0*10[sup]0[/sup] rad).[br]This is a well-known [url=http://dmccooey.com/polyhedra/BiscribedLsnubCube.html]polyhedron[/url]- Biscribed Snub Cube (laevo): [br]Vertices: 24 (24[5])[br]Faces: 38 (8 equilateral triangles + 24 acute triangles + 6 squares)[br]Edges: 60 (24 short + 24 medium + 12 long)[/size]
