THE COSMIC MATRIX
By William F. Hamilton III
The Ancient Canon and Modern Physics
Part 2. The Geometical Interval
The universe begins to look more like a great thought than
a great machine.
-SIR JAMES JEANS
English mathematician and astronomer
(1877 - 1946)
On Geometry and the Physical
World:
Geometry
is the most ancient of mathematical disciplines and
However it was Riemann who understood that this question really does not make
complete sense, and that only a composite structure consisting of geometry and
physics can be tested against experience. Riemann was followed by Einstein who
showed that physical phenomena already required that space be replaced by
space-time and that its geometry was highly non-Euclidean.
“The science of sacred geometry lies in the perfection of
its reflection of the physical world and its representation of how strongly
humanity is governed by geometry. Water molecules, carbon atoms,
proteins, viruses, cells, and tissues are able to facilitate their purpose in
the cycle of life because of their geometrical design. These organisms ability
to stabilize mechanically is due to their connectedness to a frame of
triangles, pentagons and hexagons. In the past, humans have attempted to break
the geometry of the physical world, but it has always resulted in destruction,
rather than re-creation. Rahul Singhvi and others have tried to force living
cells to take on other geometrical shapes because they believed that by
changing the shape of cells, they could switch God’s genetic programming.
Instead the cells became flat away from their geodesic dome shapes and
developed a propensity to divide and activated apoptosis – a death program”.1
Robert James Moon (1911-1989): Arriving at the
Moon earned a Ph.D. degree in Physical
Chemistry under Harkins, and then a doctorate in Physics. He taught in both
departments at the
Moon
developed a geometric theory of the nucleus based on Platonic solids. Further work by Lawrence Hecht has extended
the geometric model to include the electron orbitals.
In the atomic nuclear structure hypothesized by Dr. Robert
J. Moon1 in 1986, protons are considered to be located at the
vertices of a nested structure of four of the five Platonic solids (Figure 1).
From: Advances in
Developing the Moon Nuclear Model by Lawrence Hecht
Eight protons, corresponding to
the Oxygen nucleus, occupy the vertices of a cube which is the first nuclear
“shell.” Six more protons, corresponding to Silicon, lie on the vertices of an
octahedron which contains, and is dual to, the cube. The octahedron-cube is
contained within an icosahedron, whose 12 additional vertices, now totaling 26
protons, correspond to Iron. The icosahedron-octahedron-cube nesting is finally
contained within, and dual to, a dodecahedron. The 20 additional vertices, now
totaling 46 protons, correspond to Palladium, the halfway point in the periodic
table (Figure 2).
Beyond Palladium, a
second dodecahedral shell begins to form as a twin to the first. After 10 of
its 20 vertices are filled at Lanthanum (atomic number 56), a cube and
octahedron nesting fill inside it, accounting for the 14 elements of the
anomalous Lanthanide series.
Next, the
icosahedron forms around the cube-octahedron structure, completing its 12
vertices at Lead (atomic number 82), which is the stable, end-point in the
radioactive decay series. Finally the dodecahedron fills up, and the twinned
structure “hinges” open, creating the instability which leads to the fissioning
of uranium (Figure 3).
The completed
“shells” of the Moon model, correspond to the elements whose stability is
attested by their abundance in the Earth’s crust: Oxygen, Silicon, and Iron.
These elements also occur at minima in the graphs of atomic volume (Figure 4),
and of other physical properties (viz. compressibility, coefficient of
expansion, and reciprocal melting point) as established by Lothar Meyer in the
1870s to 1880s. Palladium, which is an anomaly in the modern
electron-configuration conception of the periodic table— because it has a
closed electron shell, but occurs in the middle of a period— is not anomalous
in the Moon model. Further, as I discovered since my
A
WORKING MODEL OF THE NUCLEUS
In the Moon model of the nucleus, a
nesting of four of the five Platonic solids similar to that conceived by
Johannes Kepler to describe the Solar System, is employed. Also shown is a
photograph of a working model of the nucleus, made for Moon by retired
machinist George Hamann in 1986. The scale model was constructed out of used
aluminum offset printing plates.
TETRAHEDRON INSCRIBED
IN A CUBE
Every
cube implies a tetrahedron. Four diagonally opposite vertices of the cube form
the vertices of the tetrahedron (a). The alpha particle is conceived as a
smaller tetrahedron (b), whose vertices fit at the centers of the faces of the
larger tetrahedron pictured in (a).
Dr.
Robert Moon was not the first scientist to apply Platonic geometry to the
sciences. Johanes Kepler used the
Platonic solids to explain the planets of the solar system.
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he so-called
Platonic Solids are regular polyhedra.
“Polyhedra” is a Greek word meaning “many faces.” There are five of these, and
they are characterized by the fact that each face is a regular
polygon, that is, a straight-sided figure with equal sides
and equal angles:
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TETRAHEDRON |
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CUBE |
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OCTAHEDRON |
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DODECAHEDRON |
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ICOSAHEDRON |
It is natural
to wonder why there should be exactly five Platonic solids, and whether there
might conceivably be one that simply hasn't been discovered yet. However, it is
not difficult to show that there must be five – and that there cannot be
more than five.
First,
consider that at each vertex (point) at least three faces must come together,
for if only two came together they would collapse against one another and we
would not get a solid. Second, observe that the sum of the interior angles of
the faces meeting at each vertex must be less than 360°, for otherwise they
would not all fit together.
Now, each
interior angle of an equilateral
triangle is 60°, hence we could fit together three, four, or
five of them at a vertex, and these correspond to the tetrahedron, the octahedron, and the
icosahedron. Each interior angle of a square is 90°, so we can fit only three
of them together at each vertex, giving us a cube. (We could fit four squares
together, but then they would lie flat, giving us a tessellation instead of a
solid.) The interior angles of the regular pentagon are 108°, so again we can
fit only three together at a vertex, giving us the dodecahedron.
And that
makes five regular polyhedra. What about the regular hexagon, that is, the six-sided
figure? Well, its interior angles are 120°, so if we fit three of them together
at a vertex the angles sum to precisely 360°, and therefore they lie flat, just
like four squares (or six equilateral triangles) would do. For this reason we
can use hexagons to make a tessellation
of the plane, but we cannot use them to make a Platonic solid. And, obviously,
no polygon with more than six sides can be used either, because the interior
angles just keep getting larger.
The Greeks,
who were inclined to see in mathematics something of the nature of religious
truth, found this business of there being exactly five Platonic solids very
compelling. The philosopher Plato concluded that they must be the fundamental
building blocks – the atoms – of nature, and assigned to them what he
believed to be the essential elements of the universe. He followed the earlier
philosopher Empedocles in assigning fire to the tetrahedron, earth to the cube,
air to the octahedron, and water to the icosahedron. To the dodecahedron Plato
assigned the element cosmos, reasoning that, since it was so different
from the others in virtue of its pentagonal faces, it must be what the stars
and planets are made of.
Although this
might seem naive to us, we should be careful not to smile at it too much: these
were powerful ideas, and led to real knowledge.
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As late as
the 16th century, for instance, Johannes Kepler was applying a similar
intuition to attempt to explain the motion of the planets. Early in his life he
concluded that the distances of the orbits, which he assumed were circular,
were related to the Platonic solids in their proportions. This model is
represented in this woodcut from his treatise Mysterium Cosmographicum.
Only later in his life, after his friend the great astronomer Tycho Brahe
bequeathed to him an enormous collection of astronomical observations, did
Kepler finally reason to the conclusion that this model of planetary motion was
mistaken, and that in fact planets moved around the sun in ellipses,
not circles.
It was this discovery that led Isaac Newton, less than a century later, to formulate
his law of gravity – which governs planetary motion – and which ultimately gave
us our modern conception of the universe. 3
My own Explorations:
Nearly 28 years ago after a number
of insightful experiences I started to search for a unifying principle in
geometry that would lead to explanations and predictions of physical phenomena
without abandoning the necessity for experiment and measurement. I felt that there was a unifying principle in
geometry just as Einstein did many years before me. However, my approach was along the same lines
as Kepler and Moon.
Just as differential calculus is
used for time-dependent phenomena and integral calculus for areas and volumes
of space, I selected the circle as a basis for time geometries and the square
for space geometries and their 3-dimensional counterparts, sphere and cube as
well as other polyhedra.
The drawing I did for calculating
the golden section of a circle in relation to the velocity of light is shown
below.
There is one more piece of the
puzzle that proved to be the unifying factor that resolved my picture of the
universe and catapulted me into the ancient canon of proportion, sacred
geometry, and metaphysics. One of my
drawings shows that equilateral polygons can be circumscribed around one
another ad infinitum separated by a constant spacing interval. A universe needs a few constants to
operate. The importance of these spacing
intervals of equal dimension which were a natural product of geometrical
construction appear as rungs of a ladder so I termed this Jacob’s Ladder. This
interval seemed to unite all regular forms, at least polygons with all natural
numbers which gave me a key to spacing intervals in the real universe. The JL interval is a ratio expressed as
1:3/16 or in decimal form .1875.
In the above sketch I superimpose
Jacob’s Ladder over a depiction of atomic orbitals, showing here the radius
distance of the K-shell from the center of an atom, then showing the interval
to the L-shell and our grid shows us that there are 8 intervals between the K
and L shells. Each circle’s
circumference is expanded by one JL interval so that 8 of these intervals
separate two adjacent orbitals. Given
that the first circle’s diameter is set to 1, then the distance between the two
shells is 24/16 or 1.5 times a unit diameter.
The Joule is the basic SI unit of energy and is the
energy expended when an object is moved through a distance of one meter as a
result of a force of one
Power is the rate at which energy is expended and is Calculated from;
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The SI unit of power is the Watt and is equal to one
Joule per second ( 1.0 W = 1.0 J s -1 )
If we
divide a circle into degrees, we have 360 degrees and for every degree there
are 60 arc minutes so that we have 21,600 arc minutes in a circle of unit
diameter. The
nautical mile is the most convenient distance unit to use for navigation
because one nautical mile corresponds to one minute of arc as measured along a
great circle. A nautical mile is 6,080 feet, whereas a so-called statute mile
is 5,280 feet. This is a ratio of
1.1515:1 so that if we were to measure the speed of light in an arc circle and
determine how many arc minutes it traversed in one second, we would find that
the speed of light could be converted to 161771.21 arc minutes per
seconds. Bruce L. Cathie in his books Harmonic 33 and the Harmonics of Space found
that dividing the earth’s rotation into 27 segments instead of 24, he found a
number of harmonies between space and time.
Essentially, we have a spacing interval based on 16 and a timing pulse
based on 9. This 3:4 ratio proves to be
a useful key to the Pythagorean physics of the universe.
Cathie postulates
that the maximum velocity of light is 162,000 arc minutes per second on a
24-hour clock, but 144,000 arc minutes per second on a 27-hour clock. When each division of 27 hours is considered
in seconds, it will yield 97,200 seconds instead of the 86,400 seconds of the
day. The 97,200 is a harmonic of the
number 9 which we will call the base of time.
Each minute of
“9-time” is 1620 seconds which is also a harmonic of the number 9. However, is this useful? Can we use these discoveries to recalculate
physical constants to see the relationships that exist between them?
As we have seen
before, when we measure power we take a unit of energy x time or a watt = 1
joule/sec. We have a fundamental
constant in physics that all of quantum theory hinges on and that is Planck’s
constant.
Planck's constant =
6.626068 × 10-34 m2 kg / s
In 1900, Max Planck
was working on the problem of how the radiation an object emits is related to
its temperature. He came up with a formula that agreed very closely with
experimental data, but the formula only made sense if he assumed that the
energy of a vibrating molecule was quantized--that is, it could only
take on certain values. The energy would have to be proportional to the
frequency of vibration, and it seemed to come in little "chunks" of
the frequency multiplied by a certain constant. This constant came to be known
as Planck's constant, or h, and it has the value h= 6.626x10-34 J/s.
This looks like a unit
of power. However, we see it in the form
below of joule-seconds or erg-seconds. The dimension of Planck's constant
is the product of energy multiplied by time, a quantity called action. Planck's
constant is often defined, therefore, as the elementary quantum of action.
Planck's constant, symbolized h, relates the
energy in one quantum (photon) of
electromagnetic radiation to the frequency of that
radiation. In the International System of units (SI), the constant is
equal to approximately 6.626176 x 10-34 joule-seconds. In the
centimeter-gram-second (cgs) or small-unit
metric system, it is equal to approximately 6.626176 x 10-27
erg-seconds.
The energy E contained in a photon, which represents the smallest possible
'packet' of energy in an electromagnetic wave, is directly proportional to the
frequency f according to
the following equation:
E = hf
If E is given in joules and f is given in hertz (the unit
measure of frequency), then: E = (6.626176 x 10-34) f
and conversely:
f = E / (6.626176 x 10-34)
What we want to know is how Planck’s constant
is related to the velocity of light or the spacing intervals of the orbitals or
planetary orbits.
We are also interested in deriving the fine
structure constant and the mass of the electron as one of the elementary
particles.
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References:
1.
http://www.islamonline.net/English/Science/2002/07/article02.shtml
2.
http://www.21stcenturysciencetech.com/articles/drmoon.html
3.
http://www.mathacademy.com/pr/prime/articles/platsol/index.asp