The Square Root of the Golden Section
Eur Ing Panagiotis Stefanides
THE SQUARE ROOT OF THE GOLDEN SECTION
By:
Eur Ing Panagiotis Ch. Stefanides BSc(Eng)Lon(Hons),
MSc(Eng) NTUA ,TCG, CEng , MIEE
Under the title THE SQUARE ROOT OF THE GOLDEN SECTION ,
my work presented here involves the decodification from PLATO’S
TIMAEUS
Geometries of the Triangle ( Most Beautiful) and the Bond ( Most
Beautiful) of the
Unified Elements ,and the RULER AND COMPASS drawing of this
mathematical relationship, based on the original knowledge of the
GOLDEN SECTION drawing.
The mathematical equation involved in this relationship is the
following :
T^4 - T^2 - 1 = 0 , where T = SQUARE ROOT OF THE GOLDEN SECTION
.
THIS RELATIONSHIP in geometrical terms is configured as an
ORTHOGONAL
SCALENE TRIANGLE , whose Small side has length: T^1 ,
Big side
has length: T^2 , and
Its Hypotenuse has
length : T^3 .
The ratio of each side to its next one is in GEOMETRIC RATIO =T ,
and the PRODUCT of the Small side length ( T^1) by the
Hypotenuse(T^3) is equal to
the SQUARE ( T^4) of the Big side (T^2) .
By Pythagoras Theorem this triangle gives :
[T^3]^2 = [ T^2]^2 + [ T^1]^2 or,
T^4 - T^2 - 1 = 0
The implications of this Triangle ,and the Golden Section have been
elaborated, in connection with Mathematics, Arts,Science, Spirals and
Logarithms ,
Circle - Square Relationships (Quadrature), and, Engineering Design
of Model
Structures, such as the PLATONIC SOLIDS , and , The Most
Beautiful Bond, the
SOMATOID ,constituent part of a Model Great Pyramid.
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THE SQUARE ROOT OF THE GOLDEN SECTION
I was born in Athens (Aegaleo) in 1945, and I was brought up in
Piraeus .My basic high school education in Greece[ Lyceum of Plato]
was mainly on Classics including Arts and languages and my
University [ London University and N.T.U.Athens ]studies in
Engineering, included a big content of Mathematics.By my continuous
professional Engineering work and contact with Scientific
knowledge, in subjects concerning motion, forces, energy, power etc.,
I noticed, that, there exist seven (7) basic forms, which appear to
derive one from the other and thus related. These forms
(relationships) necessary for the creation of a (powerful) work
element, from its conceptual idea to materialization are : Line,
surface, volume per unit time (mass rate of unity water density),
momentum, force, work (or energy) and power. They are fractions with
numerators powers of space (length) and denominators powers of time :
L^1 /T^0 , L^2 /T^0 , L^3 /T^1 , L^4 /T^1, L^4 /T^2, L^5 /T^2, L^5
/T^3 Length (L^1 ) and surface (L^2) are timeless (T^0 =1), Encephalic
Concepts. Time , for PLATO ( TIMAEUS 37 D ) is the moving
accordingto numbers , picture of eternity and it was born
together with Heaven ,inorder to dissolve themselves simultaneously
(TIMAEUS 38 D ) .Presenting my work in this subject, I will refer to
the contents of various sources which are related to the Greek Culture
, Art, and, and Geometry, and the GOLDEN SECTION.In Encyclopaedia
Britannica (Vol. 10, 1972, page 829, Greek Architecture) it is
stated:" ...To the Greeks fell the role of inventing the grammar of
conventional forms on which all subsequent European Architecture was
based... Greek was the patient genius, with which perfected every
element, rarely deviating from the forward path to invent new forms or
new solutions of old problem... This conservative adherence to older
types led to such masterpieces as the Parthenon and Erechtheum."
According to THEOPHANIS MANIAS[3,4], the Greek Beauty and the Greek
Spirit found in many works of Antiquity, were not ruined by time,
death of people or peoples' fanatism and mania.Cities and Sacred
Temples, were founded according to plans and scientific computations.
Religion of the Ancients was the Absolute Beauty, and the Greeks
believed as God this Absolute Beauty. Aesthetic Beauty, Optical
Beauty, in forms and colours, and Acoustic Beauty in music, Ethic
Beauty, found in virtue, and SpiritualBeauty in good learning and
knowledge.
Man sensed, and conquered, all kinds of Beauty, through Love, because
Love is the synectic substance of the Harmonic Universe.The Ancients
had studied this subject, with religious piety. They had observed the
existence, of another Beauty, in Nature. Beyond this Harmony, which is
Visible, and used, today, by architects, decorators, and generally all
those occupied with Aesthetics and Arts, there is another Invisible
Geometric Harmony.Circle, Square, Equilateral Triangle, Regular
Hexagone, Cube, Pyramids etc., have a Visible Beauty, that man senses
by his eyes, and he finds it in these geometric forms.Symmetries,
analogies and other mathematical relationships, were found in the
leaves of trees, the petals of flowers, the trunks and branches of
plants, the bodies of animals and most important, the human body,
which composed an Invisible Harmony, of forms and colours superior
than the Visible Harmony.This Invisible Harmony, we find, in all
expressions of the Hellenic Civilization.According to EVAGELOS
STAMATIS (Hellenic Mathematics No. 4 Sec. Ed. 1979), THEOPHANIS
MANIAS, discovered that the Ancient Sacred Temples of the Hellenic
Antiquity, were founded according to Geometrical Computations and
measurements. In the distances, between these Sacred Locations,
THEOPHANIS MANIAS, observes, application of, the Golden Section.
MATILA GHYKA[1,2], in his books, presents widely, the Golden Section
and Geometry in relation to painting, sculpture, architecture of human
faces and bodies, as well as bodies of animals, plants, and shells, in
relation to logarithmic spiral.Similarly, ROBERT LAWLOR[5], elaborates
on these subjects, and additionally, he states, that, the Egyptians,
while building the Pyramid, used the ratio 4/SQR[PHI] , for the value
of Pi (ratio of the circumference of a circle by its diameter).
MAX TOTH (Pyramid Prophesies Edition 1988), correspondingly, refers to
this ratio, as a useful, approximate form. He also states that, the
Mathematicians, from HERODOTUS, have modeled an Orthogonal Triangle,
whose small perpendicular, is equal to Unity, the bigger one is equal
to SQR (PHI), and its hypotenuse, is equal to PHI [i.e. GOLDEN
SECTION ]. Also, kEPLER refers to the same, triangle ( of MAGIRUS
) in a letter to his former professor Michael Mastlin- according to
Professor Roger Herz-Fischler[8 ].The GOLDEN SECTION is met in
certain problems involving transfer of information along a
communication line[CHANNEL CAPACITY]-Numerical Analysis, 2/ed,SCHAUM’S
outline series 1988.Further , my getting in touch with the
PYTHAGOREAN THEORY and,later searching PLATONIC TIMAEUS, brought me
in resonance with the relationship of THE GOLDEN SECTION and its
SQUARE ROOT.In section 53, of PLATO'S "TIMAEUS", PLATO speaks about
the triangular shapes of the Four Elements , of their kinds and their
combinations . Those are the Fire (Tetrahedron) the Earth (Cube), the
Water (Icosahedron), and the Air (Octahedron). When placing this
world in order , however these Four Generic Elements, possessing some
traces [Stefanides interpretation –SPECTRUM OF THE ELEMENTS]of their
kind were in such a situation ,as everything is naturally to be in
the absence of God,…..who began at first making them in shapes by the
way of forms and numbers. They are bodies and have depth. The depth
necessarily, contains the nature of the flat surface and the
perpendicular to this surface at the base is composed of triangles,
and all the triangles are generated by two kinds of orthogonal
triangles : the "ISOSCELES" Orthogonal and the "SCALÅÍÅ" [ Stefanides-
quarter SKELETON STRUCTURE of the GREAT PYRAMID – ].From the two
kinds of triangles (Section 54 ) the "Isosceles" Orthogonal has one
nature. (i.e. one rectangular angle and two acute angles of 45
degrees), whereas the "scalene" has infinite (i.e. it has one
rectangular angle and two acute angles of variable values having,
these two acute angles, the sum of 90 degrees). From these infinite
natures we choose one triangle "THE MOST BEAUTIFUL". If anyone can
claim that he has chosen one that is more beautiful he will not be
kept enemy ,but friend .Thus, from the many triangles, we accept that
there is one of them "THE MOST BEAUTIFUL", and we leave those by which
the equilateral triangle is constructed (i.e. by using six "scalene"
orthogonal triangles, having 30 and 60 degrees their acute angles).
Plato , states that if anyone examined and discovered,that,it is not
so,there will be friendship kept for his prize.
Page 5One of these two is the "ISOSCELES" orthogonal triangle, the
other is the "SCALENE" orthogonal triangle, its hypotenuse having a
value [Stefanides Interpretation] equal to the "CUBE" [ T^3 ] ,of the
value of its horizontal smaller side[T^1] and having its vertical
bigger side the value of the "SQUARE" [T^2] of its smaller horizontal
side.Thus , by Pythagoras : [T^3]^2 =[T^2]^2 + [T^1]^2 or
T^6=T^4+T^2 or, T^4 - T^2 - 1 =
0 The value of the smaller horizontal side( T ) is equal
to the SQUARE ROOT of the GOLDEN NUMBER, the ratio of the consecutive
sides is equal, again, to the SQUARE ROOT of the GOLDEN NUMBER
(geometrical ratio) and the Tangent of the angle between the
hypotenuse and the smaller horizontal side is also equal to the SQUARE
ROOT of the GOLDEN NUMBER (ANGLE THETA=51deg, 49 first , 38 , sec,
15 third …….). The product of the smaller horizontal side and that of
the hypotenuse is equal to the "SQUARE" of the bigger vertical side,
of this triangle, and at the same time the "PYTHAGOREAN THEOREM" is
valid. The values of the sides of this triangle are given by surd
numbers, (solution of a fourth degree equation). Reorganizing this
triangle, we get another one with the same angle values, which has its
bigger vertical side equal to FOUR (4), its smaller horizontal side
equal to FOUR divided by the SQUARE ROOT of the GOLDEN NUMBER, and its
hypotenuse equal to FOUR multiplied by the SQUARE ROOT of the GOLDEN
NUMBER. Further relating PLATOS, TIMAEUS (section 54) where Plato
refers to the "MOST BEAUTIFUL TRIANGLE" and that of the [SOMATOIDES]
STEREOID (section 31 and 32) where the 4 elements are bound TOGETHER
to become unity by the MOST BEAUTIFUL BOND (....which most perfectly
unites into one both itself and the things which it binds together;
and to effect this in the Most BEAUTIFUL manner is the natural
property of PROPORTIONS),and also, analysing these ratios:
FIRE : AIR = AIR : WATER and AIR : WATER = WATER : EARTH and thus
FIRE : AIR = AIR : WATER = WATER : EARTH , and,
Interpreting [ Stefanides] them as two pairs of orthogonal scalene
triangular surfaces,one pair of same orthogonal triangles with
sides T^3 , T^2 , T^1 , and the otherwith sides T^2 , T^1 , 1
,
Page 6
i.e. four triangular surfaces bound together on a system of three
Orthogonal Cartesian axes of reference (x,y,z), we construct a SOLID
, i.e. this [SOMATOIDES] STEREOID , with coordinates : (O,O,O),
(O,O,T^2 ), (T,O,O) , (0,1/T, 1/T^2 ) .It is found that this SOLID
is a constituent part of a Model Great Pyramid,and also ,a Quarter
Piece,of this Model has a SKELETON STRUCTURE two PERPENDICULAR
TRIANGLES , the ISOSCELES ,and the MOST BEAUTIFUL [SCALENE] ,whose
small horizontal side is the HEIGHT of the ISOSCELES, and the bigger
side of the SCALENE is,VERTICAL on the section of the two ISOSCELES’S
sides.On the basis of the fourth order equation ,above[T^4 –T^2 – 1 =
0 ] , the COMPASS and RULER SQUARE ROOT OF THE GOLDEN SECTION is
drawn here ,on the assumption of the knowledge of drawing, as such,
the GOLDEN SECTION [Phi] = [T^2] ,
http://www.stefanides.gr/gmr.htm [We note that a solid one
eightth (1/8) of a Model Great Pyramid , is formed by unfolding the
two triangles ,so that their surfaces are perprndicular to each
other,along their common hypotenuse].
Relating this triangle with circles squares and parallelograms we
get mathematical relationships ,
http://www.stefanides.gr/quadrature.htm and
http://www.stefanides.gr/corollary.htm , and
http://www.stefanides.gr/piquad.htm ,
http://www.stefanides.gr/quadcirc.htm ,involving quadratures. Also
relationships have been obtained with logarithms and spirals
http://www.stefanides.gr/nautilus.htm
http://www.stefanides.gr/why_logarithm.htm
By using the sections of the four solids, we find the relationships,
between them i.e. the Icosahedron with the Octahedron, the Tetrahedron
and the Cube. In addition, if we add one next to the other the three
solids, Icosahedron, Octahedron and Tetrahedron, there remains an
empty space ,a wedge [Stefanides-Pyramid of Light ],angle of
isosceles triangle , section of the wedge =2*Arctan(1/[ T^4] ) =
41.810314894 Deg.
Page 7Finally, in the section of the Dodecahedron we find a
triangle, fitting this empty triangular space, with the same angle, [
half this angle ,also , we find in the section of the Icosahedron]
.So, in this manner we obtain an additional relationship, of the
Dodecahedron with the other Four Platonic Solids.Dodecahedron was
the FIFTH COMPOSITION, mentioned as such by PLATO [and was given the
name of AETHER, by the philosophers], which , God used , to PAINT IN
DIVERSE COLOURS, EVERYTHING [Stefanides interpretation –LIGHT ].THE
GOLDEN RATIO is found in its Cross Section ,and so with the
ICOSAHEDRON.the section of Icosahedron :
http://www.stefanides.gr/icosohadron.htm , Decorated Platonic Solids
:http://www.stefanides.gr/dec_plastic.htmQuadrature Presupposition
,and Quantization of Circle
Duality:http://www.stefanides.gr/theo_circle.htm ,
http://www.stefanides.gr/quad.htm .Conclusions It is concluded here
that Plato in his Timaeic Geometry , CORRELATES the FOUR
ELEMENTS, through the general analogies of their sides Fire, Air,
Earth and Water, i.e. Fire/Air is equal to Air/Water is equal to
Water/Earth , in a codified form , and describes the use ,of the
existed FIFTH COMPOSITIONfor painting by diverse colours of
EVERYTHING ,by God.This implies a UNIFIED THEORY for the COSMOGONY of
the Classical GreekPhilosophy , delivered in a codified form, and
the SQUARE ROOT of the GOLDENSECTION, serves as a Key to its
decodification , with truth as probable , as usedMathematics’ logic
permits.Panagiotis Stefanides
REFERENCES
1. THE GEOMETRY OF ART AND LIFEMATILA GHYKA DOVER PUBLICATION, INC.
NEW YORK 1977 .2. LE NOMBRE D'ORMATILA C. GHYKAGALLIMARD 1959 .3.
PARTHENONTHEOPHANIS MANIASPYRINOS COSMOS 1987 .4. TA AGNOSTA
MEGALOURGIMATA TON ARCHAEON ELLINON THEOPHANIS MANIAS PYRINOS COSMOS
1981 .5. SACRED GEOMETRYROBERT LAWLOR THAMES AND HUDSON 1982 .6. THE
MOST BEAUTIFUL TRIANGLE - PLATO'S TIMAEUS P.C. STEFANIDES FIRST
CONFERENCE OF HISTROY AND PHILOSOPHY OF ANCIENT GREEK MATHEMATICS
MATHEMATICAL SOCIETY RESEARCH INSITUTE 2,3 AND 4 MARCH 1989.7.
GEOMETRIC CONCEPTS IN PLATO –RELATED TO ARTP.C.STEFANIDES ,
DEMOKRITEIO UNIVERSITY, 22-23 and 24 Nov. 1991. CONFERENCE OF THE
CHAMBER OF ECASTIC ARTS OF GREECE- XANTHE.8. ROGER HERZ-FISCHLER- A
MATHEMATICAL HISTORY OF DIVISION IN EXTREME AND MEAN RATIO- WLV ,
ONTARIO 1987 .WEB LINK REFERENCES :
http://curvebank.calstatela.edu/log/log.htm ,
http://www.antigona.web.pt/ [ VOL II ] ,
Hoping for comments,
Regards,
Panagiotis Stefanides
http://www.stefanides.gr
Ion Vulcanescu
I have took a look into your presentation of the research tied in the
Golden Section, and I have presented a short history of my research
to this subject.
I hope will appear.
Sorry, I can not make available scientific data: my presentation of
Sept 11 2003 was stollen by someone who entered either into my
computer or took over Math Forum Web page.
Sincerely,
Ion Vulcanescu - Philosopher
Eur Ing Panagiotis Stefanides
I, thank You ,for Your message.
Related work You may be interested in is under :
http://curvebank.calstatela.edu/log/log.htm
and, http://www.antigona.web.pt
I may inform You that next week (22-25 Oct),
there is an International Conference in Vinnitsa Ukraine
(The Agricultural State University of Vinnitsa)
on problems of Harmony ,Symmetry and Golden Section, ref:
http://www.goldenmuseum.com
Ion Vulcanescu
Research further the Evolution of all Spirale of Nautilius Shell Curve
by the bottom right section of your research where appears a
continuity of angles of 22.5 degrees.
These parts of 22.5 (seen by you in degrees) appears also in the
Geometric Point ( which is a Square),and when taken 16 times creates
THE PERIMETER of The Geometric Point of the English units of measures
decoded by me, and published in the 1st edition of The American
Civilization's Geometry during 2001 and given as a gift to the British
Academy.
Please be aware that the Square of perimeter 16 (side=4)hiddens in it
the Circle of radius=2 whose circumference equals also the area
(12.58505748) by the Geometric PI value of 3.146264371..which is the
base of my discovery of the Geometric Equlibrium.
See at this MathForum my presentation of the Geometric Equilibrium.I
suspect that further research of the Nautilius Shell Curve can give
acces to a "natural evolution" of the Spirale that may have
correspondence in some ancient time messages.
I suspect that the research of the increase of the curve's length
or"something else around it" can give excellent ressults. Myself in my
research after I create the approaching geometrical modells, I many
times work by trial and error and the geometrical proportion indicates
alone her direction or the false direction.
As I see these angles of 22.5 of the Nautilius Shell Curve I clear
have a hint that is more hidden in the Evolution of this Curve.Try see
it in both direction: Increasing and also Decreasing of 22.5.
Something is clear hidden in the Spirale since it attracted the
attention of the ancients, too...
Notes:
(1)Thank you for you informing me of the meeting of this month. Since
I am very busy in making the next edition of The American
Civilization's Geometry, I have very little time even for myself. On
top of all this I have to keep my Business of "Refrigeration -
Heating" going since this is my source of money for life and
research. However, I already registered with the International
Congress of the Mathematicians from 2006 in Spain. By then I should
have ready published this edition to which I am now working.
Sincerely yours,
Ion Vulcanescu - Philosopher
Eur Ing Panagiotis Stefanides
I, thank You for Your interesting account,about the
Nautilus Shell Curve and The Geometric Point.