Three-parameter model transformations of the Icosahedron. 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 [i]average distance[/i] between vertexes. Cases: 9 -corresponds to the [color=#ff0000]as[/color] [i]Truncated Icosahedron[/i], 11 -?, 12 -corresponds to the [i]Subscribed Snub Dodecahedron, 13 -[/i]corresponds to the[i] [color=#ff0000]as[/color] Rhombicosidodecahedron.[/i][/size]
Values of parameters in the polyhedron model defining the well-known polyhedra.
Example 9. Pmax=1.343 678 300 457 089, t, q, α=0, V=120 -as Truncated Icosidodecahedron.
[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]=3.37*10[sup]-8 [/sup][color=#333333]rad[/color]).[br][size=85]This is as a well-known polyhedron- [/size][url=http://dmccooey.com/polyhedra/TruncatedIcosidodecahedron.html]Truncated Icosidodecahedro[/url]n: [br][table][tr][td]Vertices: [/td][td]120 (120[3])[/td][/tr][tr][td]Faces:[/td][td] 62 (30 squares + 20 regular hexagons + 12 regular decagons)[/td][/tr][tr][td]Edges:[/td][td]180 (60 short + 60 medium + 60 long)[/td][/tr][/table][/size]
Example 11. Pmax=1.343 703 862 956 643, 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]=1.36*10[sup]-3 [/sup]rad).[/size]
Example 12. Pmax=1.354 126 670 906 69; t=0.5, q, α; V=60 -Biscribed Snub Dodecahedron.
[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]=3.41*10[sup]-9[/sup] rad).[br]This is a well-known [url=http://dmccooey.com/polyhedra/BiscribedLsnubDodecahedron.html]polyhedron[/url]- Biscribed Snub Dodecahedron: [br]Vertices: 60 (60[5])[br]Faces: 92 (20 equilateral triangles + 60 acute triangles + 12 regular pentagons)[br]Edges: 150 (60 short + 60 medium + 30 long)[/size]
Example 13. Pmax=1.354 119 849 172 209, t=0, q, α=0, V=60, as Rhombicosidodecahedron
[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]=0*10[sup]0[/sup]rad).[br]This is a well-known [br][table][tr][td][table][tr][td][color=#ff0000]as [/color][url=http://dmccooey.com/polyhedra/Rhombicosidodecahedron.html]Rhombicosidodecahedron[/url][/td][td][/td][/tr][/table][/td][/tr][tr][td][table][tr][td]Vertices: [/td][td]60 (60[4])[/td][/tr][tr][td]Faces:[/td][td]62 (20 equilateral triangles + 30 squares + 12 regular pentagons)[/td][/tr][tr][td]Edges:[/td][td]120[/td][/tr][/table][/td][/tr][/table][/size]
