Discovered Discrete Geometry Relationship between a Non-Regular 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.

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The Non-Regular Icosahedron, or otherwise the “Generator Polyhedron”????????????

[ PHOTO Α - ΙΝΟΧ covered Generator Polyhedron], a new Discovered Invention [2017] by Panagiotis Stefanides.

[ PHOTO B - 3D PRINT POLYHEDRAL 7 CIRCLES OF PLATOS 7 PLANETS - Proposal]?" ASYMMETROHEPTACYCLOSPHΕRA"

“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/aerospace-and-defense/8160

we find relationships:

3 parallelegrammes vertical to each-other. 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)]

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1.236067977 = 1/ [Sin (54)] = x

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This is directly Related to the Pentagon Angle of 54 Deg:

1/Sin(54) = 1.23606797

r = [?]/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

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r/h = { [ ?]/Cos(54)}/ rSin(54) = 0.850650808 / 0.68819096 = 1.23606797 = 1/Sin(54)??????????????????

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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

?" ASYMMETROHEPTACYCLOSPHΕRA"

Generator Polyhedron Skeleton Proposal[ by?Panagiotis Stefanides]?of Structure involving the 7 circles of Plato’s Planets [Plato’s Republic XIV 616 E -617A].

?( a ) ( b ) ( c ) ( d ) ( e ) ( f )

?“Generator Polyhedron” Constructions:

????????????????(a) Skeleton?PLC 3DForm?SOLIDWORKS Print Structure,

????????????????(b) Paper Structure, [Scale 4X], Invested?Mirror Triangles [Cuts?by Lousis Co?Melissia?

????????????????(c) Paper Structure [Scale 2X], Invested INOX Triangles [ Waterjet Cuts DIN Piraeus],??????

???????????????(d) PLC [Scale 2X], 3D Print [ Geometry Design and Vector Co-ordinate’sDefinition,by

???????????????????Panagiotis Stefanides, Solidworks Computations and SOLIDWORKS Print by Eng. A. Georgostathis 3DForm Maroussi Athens].

????????????????(e)?Polyhedral Skeleton AutoCad?Image?by Dr. Ginnis Kandylas,

?Geometry and Vector Co-ordinates Definition by Panagiotis Stefanides.

?????????????(f) 3DForm Image , Simulation of Interpreted form of Plato’s?Timaeus 7?????

?????????????????Circles, Soul of the World, Proposal, Geometry and Vectors’

?????????????????Definition?by Panagiotis Stefanides.?????????????????????????????????

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