THE COSMIC MATRIX
By William F. Hamilton III
The Ancient Canon and Modern Physics
Section 1. The Golden Section
The universe begins to look more
like a great thought than a great machine.
-SIR JAMES JEANS
English mathematician
and astronomer
(1877 - 1946)
Abstract: Ancient texts and traditions refer to a canon of
numbers, geometry, and measurements that were enshrined in ancient works of
stone and represented the cosmos. Many
allude to the ubiquitous presence of the golden section throughout nature as
evidence for divine geometry. As far back as Greek Mystery schools
2500 years ago, we learn that the ancients were taught that there are five
perfect 3-dimensional forms -The tetrahedron, hexahedron, octahedron,
dodecahedron, and icosahedron. Collectively these are known as The Platonic
Solids -- and are the foundation of everything in the physical world. Modern
scholars ridiculed this idea until the 1980's, when Professor Robert Moon at
the
Introduction: We begin our examination of nature using the golden section, also
known as PHI, the golden mean, and the golden proportion which we find ever
present in the fractalality of nature.
The self-similar repetition of a proportion using the golden mean or the
Fibonacci sequence of numbers is found in physical constants, plants, animals,
man, and ancient architecture. As we
investigate the ubiquitous presence of the golden section in nature, we are
impressed with the idea that it is an essential and integral part of the Cosmic
Matrix.
The Golden Section is a ratio based on a phi
The
Golden Section is also known as the Golden Mean, Golden Ratio and Divine
Proportion. It is a ratio or proportion defined by the number Phi (
=
1.618033988749895... )
It can
be derived with a number of geometric
constructions, each of which divides a line segment at the unique
point where:
the
ratio of the whole line (A) to the large segment (B)
is
the same as
the
ratio of the large segment (B) to the small segment (C).
In
other words, A is to B as B is to C.
This
occurs only where A is 1.618... times B and B is 1.618... times
C.1
Leonardo Fibonacci discovered the series which converges on phi
In the
12th century, Leonardo Fibonacci discovered a simple numerical series that is
the foundation for an incredible mathematical relationship behind phi.
Starting
with 0 and 1, each new number in the series is simply the sum of the two before
it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
. . .
The
ratio of each successive pair of numbers in the series approximates phi (1.618.
. .) , as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60.
The
table below shows how the ratios of the successive numbers in the Fibonacci
series quickly converge on Phi. After the 40th number in the series, the
ratio is accurate to 15 decimal places.
1.618033988749895
. . .
Compute any number in the Fibonacci Series
easily!
You can
use phi to compute the nth number in the Fibonacci series (fn):
fn
= Phi n / 5½
(This
provides an estimate which always rounds to the correct Fibonacci number.)
The DNA spiral is a Golden Section
The DNA
molecule, the program for all life, is based on the golden section. It measures
34 angstroms long by 21 angstroms wide for each full cycle of its double helix
spiral.
|
|
|
34 and
21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476
closely approximates phi, 1.6180339.
Living organisms grow and growth
can be shown to evolve in fractal cycles.
The golden section produces growth in form with self-similarity of form
from one stage to the next stage. A gnomon is a portion of
a figure which has been added to another figure so that the whole is of the
same shape as the smaller figure (Huntley, 1970,169). The concept of the gnomon illustrates
the fact that the ratio represented by the spiral constant is the only growth
ratio by which a new unit can grow in proportion to the old unit and still
retain the same shape. This is exactly the process of growth in a spiral shell.
As noted by D'Arcy Thompson (1961, 179):
In the growth of a shell we can
conceive no simpler law than this, namely that it shall widen and lengthen in the
same unvarying proportions: and this simplest of laws is that which Nature
tends to follow. The shell, like the creature within it, grows in size but does
not change its shape; and the existence of this constant relativity of growth,
or constant similarity of form, is of the essence. and may be made the basis of
a definition, of the equiangular spiral.
B-DNA has spirals in phi proportions
|
DNA in the cell appears as a double-stranded helix referred to as
B-DNA. This
form of DNA has a two groove in its spirals, with a ratio of phi in the
proportion of the major groove to the minor groove, or roughly 21 angstroms
to 13 angstroms. |
|
Insight on B-DNA proportions contributed by Melih Yazici.
The DNA cross-section is based on Phi
A
cross-sectional view from the top of the DNA double helix forms a decagon:
A
decagon is in essence two pentagons, with one rotated by 36 degrees from the
other, so each spiral of the double helix must trace out the shape of a
pentagon.
|
|
|
The
ratio of the diagonal of a pentagon to its side is Phi to 1. So, no
matter which way you look at it, even in its smallest element, DNA, and life,
is constructed using phi and the golden section!
The
Phi-cycle of the Planets:
Even the orbital radii of the planets are related to PHI as is the
Astronomical Unit, the radius distance from the Sun to the Earth. We will look at that in a moment, but the
purpose of this research is to find the metric
that will unify the physics of the cosmos.
Mercury
is the starting point.
Perihelion = 2- ½ (Ö2+1) = 0.792893
Aphelion = ½ (Ö2+1) = 1.207107
Start with a circle for Mercury's mean orbital
distance, and then construct a square with sides of the same length. The
midpoint between the radius of the circle and the diagonal of the square is
Mercury's aphelion (A), the outside point of its orbit. The same distance
towards the Sun is Mercury's perihelion (P), the inside point of its orbit.
John N.
Harris, M.A. has done an amazing re-analysis of the solar system based on
logarhythmic and phi-based proportions.
His log spiral form with phi-proportions is illustrated beautifully in
the following graphic.2
This new work fits perfectly with the work I started 28 years ago
that I recorded in three notebooks filled with calculations in order to find a
universal metric that tied everything together.
We will soon see the same type of geometrical framework applied to the
atomic world find that it is only a matter of scale and that the universe is
self-similar in all of its parts and appears like a divine plan, a cosmic
program, or geometrical construction of the highest order.
The Speed of Light
Viewing the planetary distances, another relationship emerges that
suggests a deeper significance to the velocity of light with respect to
planetary orbits. Taking the value of
PHI as 1.6180339 and the orbital distance to Mercury in English miles and
converting the speed of light to miles per minute, we get the following
calculation:
PHI x 2 = 3.236678
Mercury’s mean
distance from sun = 36,270,000 miles
Md/c in minutes = 3.24
minutes
Setting Mercury to 1
so that Mercury becomes the new standard of astronomical units, i.e. sun to
Mercury = 1 AU, we then calculate the relative distances to outer planets.
Venus = 1.86859426 Au’s
1.86859426 x 3.24 = 6.055 and Vd/c = 6.0637 minutes
Earth = 2.583325718 x
3.24 = 8.3699 and Ed/c = 8.38308 minutes
Mars = 3.936156024 x
3.24 = 12.753 and Md/c = 12.7731
Considering margins of
error in these planetary ratios, there is a fairly close match between figures
to indicate that the speed of light and the golden section are related to the
harmonics of the universe. 3
AU Ratios Expressed in
Miles:
|
Mercury |
1.0 |
|
|
Venus |
1.86859426 |
|
|
Earth |
2.583325718 |
|
|
Mars |
3.936156024 |
|
|
|
[Planetoids; asteroids] |
|
|
Jupiter |
13.44000495 |
|
|
Saturn |
24.68341859 |
|
|
Uranus |
49.64745547 |
|
|
|
77.78485764 |
|
The Golden Ratio and
the fine structure constant:
The following calculations were supplied to me by theoretical
physicist Tony Bermanseder from his study of the dimensions in the Great
Pyramid of Cheops. There are some who
feel that the dimensions and ratios of the pyramid represent a record of an
advanced science in stone.
Tony is the author of QR (Quantum Relativity) Theory and has
worked out the mathematics that can unify the Quantum Theory with the Theory of
Relativity. Please have patience with the
mathematical calculations as in further sections I am going to bring more
geometry and calculations to demonstrate our universe is indeed a cosmos
and reveals an underlying design.
We proceed to closer examine
the seeming arbitrariness of J. Iuliano's work to derive those seemingly
ad hoc approximations to the face-pyramid angle and a possible connection
between the transcendental p and the fine structure constant Alpha.
We apply our own process
of QR's 'reverse engineering' to show the fine structure connection previously
discussed in the Alpha-Variation paper.
I refer all readers to
that paper for the background data.
Much of Quantum
Relativity and its superbrane parameters is based on the fine structuring of
fundamental constants, such as Planck's Constant h=lps/2Re.c^3 and
Re=10^10.lps/360 as the classical Electron Radius, related to the Compton
Radius in the proportionality given by Alpha as Alpha=Re/Rcompton, Rcompton
relating the de Broglie matter wave momentum via p=h/l.
The basic QR parameter is
the magnetic permeability constant mo=120p/c defining Maxwell's Constant
moeo=1/c^2 in dimensionless (say superconductive form for the Action-Law
of Action=Charge^2 and for unitary resistance in the Quantum Hall effect,
Josephson Junctions and the Conductance Quantum 2e^2/h).
The fourfoldedness of the
Unified Field of QR (UFoQR) also defines mo=[8p/360][Ne*/Re], where
N is a counter for the magnetocharge e*=lps/hc=2Re.c^2.
This allows determination
of N=2700/c^3 in unified action units and independent of experimental
measurement.
Because
N=2700.e/{[ec].c^2}=2700.{e/c^2}/[ec], with [ec] the unified magnetic
monopole-mass in superbrane space and the Planck-Length LPdefining that
superbrane space via the Planck-Length-Oscillation: OLP={e/c^2}=LPSqrt(Alpha);
N MUST be the decisive and precise wormhole-perimeter lps=2700/(27x10^24) as
the Weyl-Geodesic assuming its linearised form at this threshold between the
metricated field equations of General relativity and its nonmetricated
encompassement and quantum substructure described by QR.
This linearization of
superstring class I as the Planck-Boson so is physically-metrically manifested
in the heterotic superstring class HE(8x8) as the Ng Van Dam macroquantisation
of the Planck-Scale then further amplified by the Electron Radius and its
associations with the atomic and subatomic realms of the Compton-de Broglie
scenarios.
So this linearization
from the preBig Bang (preinstanton and preinflaton) epochs of the circularized
'higher' superstring dimensions unfolded the scale-invariance of angular
displacement.
This unfolding then
crystallizes the 360 factor as a degree-measure, which is NOT arbitrary, but
associated with the QR definition for the approximation for the transcendental
number p, defining circularity.
This we shall now discuss
in some detail.
Draw a circle radius R for x-axis diameter given by
coordinates (-R,0) & (R,0) and y-axis diameter given by coordinates (0,-R)
& (0,R).
Draw the line segment (R,0)-(0,R) with length
Sqrt[2].R and midpoint (R/2,R/2) for the half segment length RR1*=Sqrt[2].R/2.
Now project this midpoint radially to the circle's
perimeter as point R1 to form the angle (45+45/2)=67.5 degrees=ORR1.
This defines the half angle f1=45/2 degrees as the
angle R1*RR1 in cosf1=RR1*/RR1 and sin(2f1) =RR1*/OR=RR1* for the unit
square OR=1=unit radius.
So sin(2f1)=RR1.cosf1.
Generally, a polygonal line-segment keeps doubling
from a square with 4 sides to an octagon with 8 to a hexadecagon with
16 and so on.
The generalized length for the line-segment is
R.Rn=sin[2fn]/cos[fn].
The perimeter for the circle then sums the line
segments in n and approaches 2p as n gets larger and larger.
Archimedean Circumscription/Inscription
approximation for transcendental p then is:
p=2^[n+1].sin{45 degrees/2^[n-1]}/cos{45
degrees/2^n}
For n=1; the approximation is: p(1)=3.061467458...
For n=2; the approximation is: p(2)=3.121445152...
For n=3; the approximation is: p(3)=3.13654849...
For n=14; the approximation is: p(14)=3.141592653...correct
to 8 decimal places.
Now express the trigonometric functions as expanded
series:
sinx=x-x^3/3!+x^5/5!-.....in odd terms for x
and
cosx=1-x^2/2!+x^4/4!-....in even terms for x to form
the limit
lim(n->oo){2^[n+1].[45.2^[1-n]-45^3/3!.(2^[n-1])^3+45^5/5!.(2^[n-1])^5-..]/[1-45^2.2^[2n]/2!+45^4.2^[4n]/4!-....]
=lim(n->oo){2^[n+1].45.2^[1-n]}=2^2.45=180.
lim(n->oo){1/n}=0, therefore all the
trigonometric terms of the form1/2^n vanish.
So we have indeed justified J. Iuliano's bold assumption
to simply form the identity 360=2p or p=180 to freely interchange radian
measure with the angular degrees.
Of course this works only for the demetricated
superbrane scenarios and requires thus a dimensionless setting for the
spacetimes as the ontological foundation for mensuration physics.
But the 360 factor in the Electron Radius definition
now becomes the necessary circularization for the wormhole perimeter lps.
Precisely 10^10 wormhole perimeters fit into the
linearised and metricated macroquantised wormhole perimeter represented
by 2p.Re AS 360.Re circularized.
This also unitizes our four folded UFoQR in 8p/360
as the A/4 factor in Stephen Hawking's Black Hole event horizon entropy counter
for the information mapped surface area as hologram, measured in Planck-Areas.
In short the entire quantum geometric
structures described by QR fall into place.
J.Iuliano's pyramid angle now is justifiably 'reverse engineered'.
Recalling our Fine Structure Constant derivations; we
have the Universal Wave function approximation for AlphaB=1/137.0470721 and the
little gamma approximation as Alphag=1/137.0471224.
Using the Iuliano formulations, we then compute the
pyramid angle for those two cases as:
arc cosB{-137.0470721/10^4}=90.78524646 degrees or
(16p+1.584501463) radians=51.84998392 radians and representing 51 degrees
50' 59.94" as the pyramid angle.
arc cosg{-137.0471224/10^4}=90.78524675 degrees or
(16p+1.584501468) radians=51.84998393 radians and
representing 51 degrees 50' 59.94".
Both values represent J. Iuliano's calculation to
that precision in his pyramid face-angle of (51.849982814). 4
Tony B.
The
great genius Leonardo Da Vinci observed the golden section in nature and
incorporated it in his remarkable paintings.
Addendum:
I
have consulted my notebook for early calculations on the speed of light and PHI
and have found the following which will be elaborated on in subsequent
sections.
A circle, divided by a decagon, has a base which is a
phi-ratio of the circle. If we consider
time to be one turn of the circle, then we can relate time to PHI. 360/10 = 36 degrees or the phi angle. Log 5 (Golden Spiral logarithmic expansion
factor) equals 100,000. The base of time
in this schema is 1/6 of a circle X 10.
The speed of light is given as 186,282 mps and when converted to arc
minutes per second (divided by 1.1516264), it equals
1.6175558 (which is a division by Log 5).
360/phi equals 222.49224 degrees of our time circle. This is a difference of .9865509 which is
accounted for by the time difference in a year circle which is .9863003. In other words, there is a deep relationship
between our measurement of time and the golden section. Is that why the golden section is found so
pervasively in the process of growth?
To be continued in Section
2.
References:
1. http://goldennumber.net/goldsect.htm
2. http://www.spirasolaris.ca/index.html
3. http://www.earthmatrix.com/orbital/astronomical_unit.html
4. http://www.worldnpa.org/main/