Keynote 20-23 June 2021
The “Generator Polyhedron”, Complying with Plato’s Solids’ Philosophy and Discrete Geometry, is Proposed as an Optically Tracked Satellite Model
Panagiotis Stefanides1 1Alumnus of the University of London [ex] R&D and Air-Engines’ Manufacturing, Hellenic Aerospace Ind. S.A. Institution of Engineering and Technology-Hellas Network Honourary Secretary Corresponding Author E-mail: panamars@otenet.gr Corresponding Author ORCID: 0000-0002-8259-6340
Keywords Abstract Satellite Optical Tracking, Generator Polyhedron, Platonic/Eucleidean Polyhedra, Geometry, Plato’s Most Beautiful Triangle, Symmetry, Harmony.
It is anticipated that the future shall need consideration of more frequent space travel and space research, for any form of scientific understanding of the universe we live in. This futuristic architectural proposal is based on a very special geometry of a recently Discovered Invention of a non-regular Icosahedron, related geometrically, as a whole, to the five Platonic/Eucleidean Polyhedra. The Model built is a non-regular Icosahedron having 12 Isosceli triangles and 8 Equilateral triangles. Mirror triangles cut to size, invested the structure for the configuration of a “Polyhedroheliotrope” Satellite Optical Tracking application [size the Polyhedron was built is 4x scale of theory drawings presented: approx. encapsulation in cube of size 16x16x16 cc]. Homemade dark room for satellite - Space simulation involved for photos and video clips taken. The “Skeleton “Structure of this Polyhedron consists of three Orthogonal Parallelogramme Planes, vertical to each other of size each 4Xπ by area [orthogonal parallelogramme sides’ ratio: 4/ = √/1. The geometry of this work is part of the published book “Treatise on Circle Generator Polyhedron Harmony and Disharmony Condition of Three Concentric Circles in Common Ratio”. ISBN 978 – 618–83169–0-4, National Library of Greece 04/05/2017 by Panagiotis Ch. Stefanides [leading to this Polyhedron]. It concerns relationships of 3 Concentric Circles in Ratio to each other of √/1 = 4/ = 4/√ = 3.14460551 ,√/1 = = 1,2701965 [Ref: Bibliography 2017: stefanides.gr]. Generator, refers to the geometric characteristics of this non-regular Icosahedron found to be roots of the Platonic/Eucleidean Polyhedra. It is related to the Icosahedron, which in turn relates itself to the Dodecahedron of involved sides’ dimensions of their orthogonal parallelogrammes skeleton planes’ structures, in the ratios: ^2 / = /1 for Dodecahedron to Icosahedron and /√ = √/1 for Icosahedron to Generator Polyhedron. Further important angular relationship between Dodecahedron and Generator Polyhedron found: [1/8] ∗ {4/[{ [(5) + 1]/2}]}^2 = [ 1/(54)] = [1/8] ∗ [ 4/ 1.27201965 1]^2 = [1/8] ∗ [3.14460551]^2 = 1.236067977 = 1/ (54). Numerical value 1.236067977 and its geometric relationships, are found in the structure of the Generator Polyhedron, in this paper.
1. Introduction
Since 1985, I posed a Conjecture to myself, that there should be at least one reliable Formula or Method [possibly a ruler and compass one] with its proof for the Value of π [as there are many of the kind, but not leading to the same Exact Value]. The concept envisaged, should involve Eucleidean Geometry [as it bears reliable consequence, and should be easily examined for its Truth or not]. Amongst the various methods that I have come to, [and delivered to various Conferences and Exhibitions, nationally and internationally], for some classical problems’ solutions, [as those additional ones presented here as contents of the book, which to me are very interesting and challenging], are problems with theoretical geometric solutions and proofs, simply, by ruler and compass with purpose of performing, as far as possible, more analytic and more simplified presentations.
The work presented [Generator Polyhedron] resulted from elaborating on my work “Treatise on Circle” which concerns 3 Concentric Circles in Ratio to each other of 4/π (or square root of the golden ratio), analyzing and comparing the results, for evident conditions for found Symmetries or Dissymmetries and consequently conditions for Harmony or Disharmony. Part of this paper is, an extract of my published book| [ISBN 978 – 618 – 83169 – 0 - 4] titled: Treatise on Circle – Generator Polyhedron Harmony and Disharmony Condition of Three Concentric Circles in Common Ratio [Ref. Bibliography 2017: www.stefanides.gr].
2. Polyhedron Geometry and Construction
CONDITION THAT = 4/ [ ] = 3.14460551..
Figure 1. Skeleton Paper Structure
Figure 2. X-Y-Z Co-ordinates’ Skeleton
Figure 3. Paper Structure
Figure 4. X-Y-Z Structure
Figure 4. Mirrors’ Invested Polyhedron
Satellite Optical Tracking application [size the Polyhedron was built is 4x scale of theory drawings presented: approx. encapsulation in cube of size 16x16x16 cc]
Figure 6. Generator Polyhedron AutoCad
Satellite Rotating in Space around itself], Reflects [Sun] Rays in a Codified [Dual] Form due the different amounts of light energy leaving from the dissimilar kinds of Triangular Mirrors.
Figure 7. Polyhedron Skeleton Structure AutoCad
Three Orthogonal Parallelogramme Planes, Vertical to each other [ F7] of Size each 4 by area [orthogonal parallelogramme sides’ ratio: 4/ = √/1].
CONDITION THAT = 4/ [ ] = 3.14460551..
2.2. Polyhedron Theory Based on Plato’s Timaeus “Most Beautiful Triangle Proposed by Panagiotis Stefanides
Figure 8.
PACE 2021- Ataturk University, Engineering Faculty, Department of Civil Engineering, Erzurum, 25030, TURKEY 20-23 June 2021 3
“Ruler and Compass” Quadrature of Circle By application of Plato’s Most Beautiful Triangle Theory [ Kepler/Magirus Triangle (ABC) is Similar to and Part of Plato’s].
Geometry,Vector and Co-ordinates Definition By Panagiotis Stefanides. AutoCad Computations by Dr. Giannis Kandylas [ F6, F7, F8].
3. Planes’ Vertices’ Combinations of Connections
AE, AF, AI, AL
BF, BK, BE, BJ
CK, CG, CH, CJ
DI, DL, DG, DH
IA, IE, ID, IH
JE, JH, JB, JC
KB, KC, KF, KG
LA, LD, LF, LG
EA, EB, EI, EJ
FA, FB, FK, FL
GK, GL, GC, GD
HC, HD, HI, HJ
4. Geometry, Sections, Planes and Stereometric Measurements of
the Solid Polyhedron
[ 3.14460551/2]/ 0.336234778 = [ /2] / [/2 – [^2]/8] = 4.67620501
= 4/ [ ] = 3.14460551
Stefanides
Figure 10.
= 4/1.27201965 ] = 3.14460551
= √{ [ 4 − ]/2]^2 + 2^2} = 2.045220021 = [ 4 − ]/2 = 0.427697244
= √ [ ^2 + [ (/2)^2 ] = 2.579740469
[ ] = /(/2) = (2.045220021)/(1.572302757) = 1.300780027
= [1.300780027] = 52.44801593 2 = 104.8960319
[180 – 2] = 75.10396814
= [4.676205016] = 77.92918912
[ 90 − ] = [ 90 – 77.92918912] = 12.07081088
2 = 155.8583782
[ 180 − 2] = 24.14162176
5. Important Discovery [2017]
From the geometry of the “GENERATOR POLYHEDRON”, we find relationships: 3 parallelegrammes vertical to each-other. Sides’ lengths,of each parallelogramme, are in ratio of 4/ = 1.27201965 [ = 3.14460551 . . 4/ ( )].
[4/2]/ [/2] = [/2]/ , = [/2]^2}/[4/2] = 2.472135953/2 = 1.236067977
Wolfram Alpha Checking Solutions:
= [ ½]
(54) = 0.850650808 ,
= 3.14460551102969314427823434337183571809248823135
0892950659
[1/8] ∗ {4/[{ [(5) + 1]/2}]}^2 = [3 + (5)] / [ 2 + (5)]
Relationships with the DODECAHEDRON PENTAGON Angular Structuring:
[1/8] ∗ {4/[{ [(5) + 1]/2}]}^2 = [ 1/(54)]
[1/8] ∗ {4/[{ [(5) + 1]/2}]}^2 = [3 + (5)] / [ 2 + (5)]
1.236067977 = 1/ [ (54)] =
5.1. Dodecahedron Pentagon Relationship with the Generator Polyhedron
Figure 11.
= (54) = , = [ ½]/(54) = = = = = = 1, = [1/2]
/ = /(54) = 1/(54) = 1.236067978 = + = 1.538841768
= [2.618033989] = 1.61803398
6. Building Blocks of Solids
Figure 12.
Representation of the 4 elements [ Spectrum ] Elemental Lines [ Plato’s Timaeus “TRACES”- i.e. Matter] Structuring Orthogonal Triangles which joined together build [Tetrahedra]Volumes, which in turn precede building Solid Polyhedra [ ref: F14].
Figure 13.
Somatoides
First Built Tetrahedron from two pairs of orthogonal triangles, proceeding in structuring a Great Pyramid Model, from which the Skeleton of the Icosahedron is Structured [and so on further ref: F14] https://youtu.be/RBO2zbX8IzM?t=1
Figure 14.
7. Unified Solids
Icosahrdron Octahedron and Tetrahedron [Contained within a Cube- not shown] Joined together, by fitting equal sized equilateral triangles of the solids’´bases.
Figure 15.
The open gap is a Tetrahedral Wedge.
Dihedral angle ε = 41.81031479.. deg found, also, in the Cross-Section of the Dodecahedron
Figure 16.
(1) Unified Form of 3 Platonic Solids, Icosahedron, Octahedron and Tetrahedron [ Within Cube-not shown].
(2) Empty Wedge Gap [ of sectional angle ε - acute], relates to the Dodecahedron [ symbolized by a Hexagonal - shape envelope].
(3) Generator Polyhedron, Related to the 5 Platonic Solids.
8. The Triangle Theory
Theory is based on Plato’s Timaeus “Most Beautiful Triangle [Interpreted by Panagiotis Stefanides- being similar but not the same with the Kepler/Magirus Triangle], upon which the building blocks of the Solids is based.
The Generator polyhedron is based on this Triangle and particularly on the ratio 4/ = 4/ { []} = and on the number 4.
π being a real positive root of:
^4 + [4^2] ∗ [^2] – 4^4 = 0
= = 4/ √ = 4/ [ ] = 3.14460551
Figure 17.
[] = ^3 , [] = ^2 , [] = ^1
[ ^3]^2 – [ ^2]^2 – [^1]^2 = 0 ^6 –^4 – ^2 = 0
^4 –^2 – 1 = 0
[] = = [ ]/ [] = [^2]/[^1] = = [ ] = √ = √
9. Ontology
The Generator Polyhedron is Proposed as the “other genus - γ ν ο ς” referred to Plato’s Epinomis 981b
……Σ τ ε ρ ε δ σ μ α τ α λ γ ε σ θ α ι χ ρ κ α τ τ ν ε κ τ α λ γ ο ν π ν τ ε….“τ δ λ λ ο γ ν ο ς πα ν χ ε ι μ ο ρ φ ν μ α ν” ……. τ θ ε ι τ α τ ο ν ν τ ω ς ψ υ χ ς γ ν ο ς - …the genus of the soul.
10. Conclusions
The Realization of the Searched Discrete Geometries led to the Construction of the Generator Polyhedron, which, in turn as a Feedback, serves to demonstrate anticipated Compliances.
Declaration of Conflict of Interests
The author declares that there is no conflict of interest. They have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References
[1.] Treatise on Circle – Generator Polyhedron Harmony and Disharmony Condition of Three Concentric Circles ISBN 978 – 618 – 83169 – 0 – 4
[2.] GOLDEN ROOT SYMMETRIES OF GEOMETRIC FORMS ISBN 978 –
960 – 93 – 2219 – 5
[3.] GEOMETRIC CONCEPTS IN PLATO REVIEW PUBLICATION P.C.S. National Library of Athens No: 5659 – 29 December 1997