Discovered Discrete Geometry Relationship between a NonRegular Icosahedron and the other Five Regular Solids, of possible interest to Scientists.
Under the current exposure to the virus [corona] conditions one should not hesitate learning possibilities of accepting and use of new discoveries and particularly those bearing “change of paradigm,” with the involvement of discrete geometry.
The NonRegular Icosahedron, or otherwise the “Generator Polyhedron” [ PHOTO 1], a new Discovered Invention [2017] by Panagiotis Stefanides.
“Generator ” refers to the geometric characteristic of this Solid found to be the root upon which other Solid Polyhedra are based i.e. the Platonic/Eucleidean Solids [Icosahedron Dodecahedron etc.] The Geometry of this work is part of book: [ISBN 978 – 618 – 83169 – 0  4], National Library of Greece , 04/05/2017, by Panagiotis Ch. Stefanides.
From the geometry of the “Generator Polyhedron” [ Photo 1A, 1B, 2B, 3]
https://contest.techbriefs.com/2017/entries/aerospaceanddefense/8160
/>we find relationships:
3 parallelegrammes vertical to eachother. Sides’ lengths, of each parallelogramme, are in ratio of 4/π = 1.27201965 [for π = 3.14460551 i.e. 4/ SQRT(Golden Ratio)].
[4/2]/ [π/2] =[π/2]/ x, x = {[π/2]^2}/[4/2] = 2.472135953/2 = 1.23606797
Similarly the Icosahedron Parallelogramme ones are in ratio of 1.618033989 and those for the Dodecahedron are in ratio of 2.618033989.
Relationship with the Dodecahedron Pentagon [Photo 2A].
Considering:
[4/2]/ [π/2] = [π/2]/ x, x = {[π/2]^2}/[4/2] = 2.472135953 / 2 = 1.23606797
{ [ π/2] / χ is the the dimenstional ratio of each of the 3 plains of the next smaller polyhedron skeleton structure – quantized reduction copy of the model}
1/8]*{4/[sqrt{ [sqrt(5) +1]/2}]}^2 = [ 1/sin(54)]
[1/8]*{4/[sqrt{ [sqrt(5) +1]/2}]}^2 = [3+sqrt(5)] / [ 2+ sqrt(5)]
1.236067977 = 1/ [Sin (54)] = x
This is directly Related to the Pentagon Angle of 54 Deg:
1/Sin(54) = 1.23606797
r = [½]/Cos(54) = 0.850650808, h = rSin(54) =0.68819096 , r/h =1/Sin(54) = x
H = r + h = 1.538841768, h/r = 1/x = Sin(54) = 0.809016994
r/h = { [ ½]/Cos(54)}/ rSin(54) = 0.850650808 / 0.68819096 = 1.23606797 = 1/Sin(54)
Photo 1: Generator Polyhedron and Skeleton Structures.
Photo 2: Pentagon of Dodecahedron and Section of the “Generator Polyhedron”.
Photo 3: “Generator Polyhedron” Stereometric Calculations.
You Tube Video Clip :
https://youtu.be/NuNuNySN2sU?t=9
Generator Polyhedron Skeleton Proposal[ by Panagiotis Stefanides] of Structure involving the 7 circles of Plato’s Planets [Plato’s Republic XIV 616 E 617A].
Video
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ABOUT THE ENTRANT
 Name:Panagiotis Stefanides
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