Sacred
Geometry , music and a possible correlation with quantum-mechanics
Version 1.2
, July 1997 by Martin Euser.
In this article I will introduce a simple formula of Sacred Geometry
which has
interesting properties. I will not provide an extensive introduction to
the
concept of Sacred Geometry since there are many websites where you can
find
introductory material into this field.
Instead of that I will, however, provide some links to interesting
material
(new developments in this area of research).
The basic idea of Sacred Geometry is summed up by Plato:
'The Demiurgos geometrizes'.
Now, the Demiurgos, often erroneously translated as "God", is simply
the collective
host of beings that is sometimes referred to as 'the Architects' or the
third
(manifest) logos of our cosmos.
In the Biblical Genesis we find the term 'Elohim', which again is
erroneously
translated as "God". Instead, it can be better translated as 'host of
angelic beings' which formed mankind in their image.
This, of course, is a story in itself and more details can be found in
Ralston
Skinner's work 'Source of Measures', Gottfried de Purucker's
'Fundamentals of
the esoteric philosophy', Alvin Boyd Kuhn's writings, Gerald Massey's
writings
and Helena Petrovna Blavatsky's writings such as 'Isis Unveiled' and
"Secret Doctrine'(the last book being rather mystical in nature and
generally being little understood). See my other articles for links to
websites
with writings from some of these authors.
Now, the question is, 'How does the demiurgos geometrize?'
What are the ways and methods which are used in or through the Minds of
the
intelligences that have prepared the design for our earth, man, flowers,
animals, minerals, etc.?
This question has occupied many minds , from
the old Hindus to the Babylonians, Greeks, Jews, to the modern student
of
Sacred Geometry. Some (partial) answers seem to have been found and laid
down
in architectural designs to be found in Hindu temples, Egyptian
pyramids, etc.,
and coded into scriptures as the Bible, the Hindu Puranas, etc.
See e.g. the above mentioned work of Skinner. For the keen observer it
is very
clear that there is an inherent order in this universe although
there
may appear to be chaotic phases due to the interactions of a multitude
of
factors, wills, energies, etc.
The growth and development of a human embryo is a good example of the
harmonious workings of nature. In the beginning stage, the cells are
configured
in a beautiful geometrical pattern, reminding one of the structures that
are
found in the mineral kingdom. Later on in the developmental stage one
can
observe structures reminding one of plants, lower animals, etc.
culminating in
the fully developed foetus. Michael Schneider has written a beautiful
book on
Sacred Geometry which illustrates features such as the above.
He wrote "A Beginner''s Guide To Constructing The Universe"
(HarperPerennial) Some websites relating to the book & Michael:
www article:
Cosmic Dozens: Twelve-Fold Designs of Society
and Art
Workshop:
Sacred
Harmony: The Patterns of Nature and the Ways of Knowing
Book Reviews:
A Beginner's Guide to Constructing the
Universe: The Mathematical Archetypes of Nature, Art, and Science (A
Voyage
From1 To 10)
Book
review by David Fideler, reprinted from PARABOLA (Winter 1995)
Michael's workshops in northern CA
Another classic on Sacred Geometry is Doczi's
book 'The power of limits' which richly illustrates the harmonies
in
nature, art and architecture.
Manifestation means limitation and thus the title of the book was enough
incentive for me to buy it and have a look at it. Doczi develops the
idea of dinergy
and patterns of sharing. Dinergy refers to the working of opposites
united in a
harmonious proportion. An example is the working of the minor and major
parts
of the golden proportion. He invented the word 'dinergy' to refer to the
universal
pattern-creating process.
As a prefatory remark to what follows I want to add that I don't
believe
that one magical mathematical formula exists which explains all and
everything.
The Boundless has an infinite number of attributes and its emanations
vary
accordingly. Yet we are told that analogy seems to be the guiding
principle in
the emanation and building of worlds, forms, etc.
Infinite variation along analogical lines (nature repeats itself in
endless
variations) may be the answer to the question of the operations in/of
the
Divine Mind and more specifically pertaining to the operations of the
Builders
, but more research is necessary. The emergence of fractal theory seems
to be
an interesting development in this respect. Fractals embody the
principle of
repetition so abundantly present in nature.
The theosopher Gordon Plummer has shown that the dodecahedron is capable
of
generating the icosahedron, and these two Platonic solids together, the
other
Platonic solids (tetrahedron, cube, octahedron) through the working of
the
well-known number Phi (the golden section number).
The dodecahedron and icosahedron can generate each other in an infinite
series
through a repeated geometrical process where the Phi proportion plays a
dominant role.
His "Mathematics of the Cosmic Mind' is interesting food for thought.
Some
independent researchers have followed up on his ideas and also try to
combine
ideas from Sacred Geometry with chaos theory and fractals.
An example of Sacred Geometry , applied to the Platonic solids and the
Kabbalistic Tree of Life , can be found at: Sacred
Geometry discovery
Now I come to my own , thus far limited research in the area of Sacred
Geometry.
While studying the Secret Doctrine of Blavatsky, I discovered her
mentioning
something about seven being the root number of formation. Being engaged
in a
little study of Greek geometrical concepts, like arithmetic mean,
geometrical
mean and harmonic mean, I decided to do a little research on the root of
seven
( 7 ) and this proved to be the start of some interesting discoveries.
Figure 1 shows the relation of these means to each other: 
I will sketch some highlights of my findings, which form only the
proverbial
tip of the iceberg.
The next section presupposes a basic interest in and acquaintance with
high
school mathematics. I'll skip the derivations of the results (which are
very
easy to obtain, however).
First of all, I started with the idea that Pythagoras' famous musical
proportions might be found by using the idea of arithmatical,
geometrical and
harmonic means generated from a pair of numbers (two poles or a
pair of
lines) , thus:
A - n .V(7) and A + n .V(7) <P> where A is the arithmetical mean
of this
pair and V is the symbol for taking the square root (in
this case
of the number seven).
I found out that the following two poles had a relation to the
fourth
(which is a proportion of 4:3) :
16 -
4.V(7) and 16 + 4.V(7) ( the '.' is a symbol for multiplication)
This pair of numbers yields the following means:
A = 16 ; G = 4 * 3 = 12 ( '*' is symbol for multiplication) and H = 9
G means the geometrical mean for the two poles and H indicates the
harmonic
mean.
The formulas for G and H are in this case:
G = V( 16² - 16 * 7) and H = G²/ A
Now, 16:12 is 4:3 and 12:9 is 4:3 , so the 4:3 proportion is implied.
This result encouraged me to research further pairs of numbers and I
found some
interesting poles with properties as follows:
9 - 3. V(5) and 9 + 3.V(5) have A= 9, G= 6 and H= 4 which imply
a relation of 9:6 = 3:2 and 6:4 which again is 3:2. This is the
proportion for
the quint or fifth in music!
I'll spare you some details of research and present the abstracted
general
formula capable of generating musical proportions:
The poles N² - N * V(2N-1) and N² + N * V(2N-1) generate the following
means:
A = N² , G² = N² * (N -1)² , H = (N-1)²
G = N * (N -1)
Note the beautiful interplay of terms here: substitute N + 1 for N and H
becomes N which is the value of A for N proper.
For N=2 we get : A = 4, G = 2, H= 1 which implies 4:2 = 2:1
In other words, the idea of an octave!
N =3 yields the proportion of the fifth as I've shown already, N = 4
yields the
proportion of the fourth in music. N = 5 yields the 5:4 proportion which
Pythagoras seemed to dislike, but was incorporated in the musical scheme
of
another Greek School of philosophy as being a natural proportion.
However the case may be, the famous 9:8 proportion (the distance between
two
notes on the major scale) is generated when N=9.
So far for music. As an aside: the root-scheme includes all the roots of
uneven
numbers including the famous ones in Sacred Geometry. The only root
missing is
2 which is symbolical for a square. I suspect that there is another
principle
to be synthesized into my little theory - some formula containing this
root-number and maybe some squaring function. Systems interacting, so to
speak.
(a small part of this has been solved by me now, but that's material for
a
second article on Sacred Geometry).
I did some more research on this generative set of poles and discovered
some
more interesting properties. When researching the presence of cyclicity
in the
generated means by use of this type of scheme (I call it the
'root-number
scheme ') I discovered that it seemed to have a relation to the famous
periodic
shell-number in quantum-mechanics:
2 * N²
First electronic shell for N =1 has 2 'positions' for electrons (1S-
shell)
Second electronic shell for N=2 has 8 'positions' for electrons
(2S-shell
allows two orientations of spin) and 1P shell (allows 3* 2 or six
orientations)
And so on.
What I discovered was that by varying on the above mentioned scheme by
introducing
a factor m a new feature arose:
N² - m.N * V(2N-1) and N² + m. N * V(2N-1)
where m is an integral number .
Now, G and H depend on m besides N as to their value. I discovered that
by
setting m = N (which is the maximum value of m that is possible
without
making G negative) and substituting 2.N² for N yields integral values of
G and
H :
(2*N²)² +- N.2.N² * V(4N²-1)
which renders:
A= (2N²)² ; G = (2N²)*N ; H = N²
which again shows an interesting relation between A and H: H has now the
same
numerical value as A has in the previous scheme. Somehow there seems to
be an
interplay between the schemes in such a way that the arithmetic mean and
the
harmonious mean take each other's value.
This makes sense to me, however, as I think that everything in nature is
connected to each other and the above mentioned interplay would fit
perfectly
into nature's wondrous works. BTW, if you care to research the values
for A, G
and H in this case you will find musical harmonies again!
I have not as yet deeply investigated the relation between the above
scheme and
quantum-mechanics (QM) except for noting the interesting results for H
being
equal to N² for certain values of N , namely where A=(2N²)², and
discovering
the limited allowed values for m: m >= -N and m <= N which reminds
me
somewhat of the number of allowed values for the projection of orbital
angular
momentum of a particle, say an electron, on the z-axis of a Cartesian
coordinate-system (although we need a third variable to establish this
connection - the angular momentum quantum number l). This is
speculative, of
course, but not as weird as it may seem to be. Motion might be described
by a
new kind of formalism which connects natural phenomena in a clearer way
than
has been established up to now, i.e. underlying forces, patterns, etc.
may be
supposed to exist in order to describe the connections between the rich
variety
of phenomena that we observe (and who says that we are aware of all
these phenomena?)
Now, N² is a term which appears in the equation for the energy states of
a
hydrogen atom :
EN = constant/ N²
Spectral emission occur when an electron falls from a higher energetic
level to
a lower one. The frequency of the spectral line (for hydrogen) is simply
given
by:
frequency = C ((1/ N)² - (1/ M)²) where N and M are integral numbers
> 0 and
M > N.
Actually, I suspect that relations such as I discovered may be connected
to
orbital motion, be it of electrons or of planets and possibly with
rotations
around the own axis of these bodies (which are called 'spin' in the case
of
subatomic particles such as electrons).
Time and inspiration permitting I may do some more research in this
area.
I need to mention another relation between G² and A:
(G² )² / A ³ = ¼ which is interesting since it reminds me of the famous
law of
Keppler: the quotient of squared time of revolution and long semi-axis
of orbit
of the planets around the sun to the third power is constant. He derived
the
elliptic motion of the planets from this empirical fact and some other
ones. Don't
bother about the number ¼: the original form of the root-scheme can be
multiplied by a constant without any essential change as to the results.
Admittedly
speculative, but a whole field of research opens to the open-minded
researcher.
After all, titanic intellects such as those of Pythagoras spoke of the music
of
the spheres and this is just what the root-scheme formulas seem to
be
about: harmonic proportions in nature.
The same proportion for G and A can be derived for subsystems of the
root-scheme.
I leave this as an exercise to the mathematically inclined readers.
Schemes as the above are known (in a general sense) in the algebraic
theory of
groups. I've not yet researched all the mathematical properties (as to
group
properties, subgroups, cyclicity, existence of unity element, etc.) of
this
type of system. The system will probably have to be extended with other
factors, elements of interaction, etc.
To the student of Sacred Geometry: there seems to be a beautiful element
of
squaring involved here. Finding a root is related to squaring, since the
squaring of a root yields the original number from which the root has
been
extracted. Squaring and taking the square root of numbers are inverse
functions, which very much seem to have a place in sacred-geometry. The
play of
nature (according to the pattern in the mind of the Demiurgos), I
suppose.
This squaring and finding a root must be brought into relation with
circles and
spheres, or circular/ spherical motion to make it produce natural
phenomena
such as harmonic motion, spirals, etc. This should be the subject of
deep
investigation, I think, the more so since scientists have got stuck in
their
paradigms and may need to get an impulse from alternative researchers to
establish new paradigms pertaining to nature.
I almost forgot to mention that the ancient Greeks seemed to attach much
value
to surface and perimeter of geometrical figures. I have related these
two with
the harmonic mean as follows (for square and rectangle):
Let a be the side of a square, b and c the sides of a rectangle.
Demand the invariance of proportion between surfaces and surfaces and
perimeter
to perimeter of square and rectangle.
Then it follows that: 4a / (2b + 2c) = a²/ (bc)
Elaborating this gives: 2bc/(b+c) = a which is equivalent to:
a = bc/((b+c)/2) which is the formula for the geometrical mean
for the
poles:
½(b+c) +/- ½(b-c)
So, ½(b+c) is the arithmetical mean here, being the ¼ part of the
perimeter of
a rectangle, so to speak an average of the four sides..<P>
H = a =
2bc/(b+c) which is the side of a square <P>
and G²
= bc which is the surface of a rectangle, which means that G = V(bc)
which
starts one thinking about the relation of square to rectangle again.
Could it be that nature prefers certain numbers a, b and c so that
integral
numbers and proportions of integral numbers are preferred in its work?
Certainly
I see no way in which a transcendent number by itself could appear in
manifested form, the perfect circle being a mathematical idea and
not a
physical fact. What else could be meant by "squaring the
circle" than the operation of a principle of limitation in
nature?
Remember that pi is a transcendent number which means that it cannot be
written
as a proportion of two integral numbers. Nature (the divine mind to
others)
must have found a way to manifest itself (by limitation), thus using
approximations of the famous numbers pi and phi (and other numbers).
Ralston
Skinner has decoded a system for the use of approximations of pi in his
~Source
of Measures~ and I think he is right in many of his assertions.
Madam Blavatsky has confirmed some of his central assertions and she
also noted
that:
'the whole of the work of Euclides was meant to demonstrate the
properties of
the dodecahedron' (a quote from memory). A quote which is certainly
memorable
when viewed in the light of some discoveries in Sacred Geometry (see
Gordon
Plummer's book referred to previously).
So, I suspect that the principle of quantization is inherent in nature,
even
more than scientists have already discovered in the realm of
quantum-mechanics.
Much research has still to be done in this regard and I would invite
others to
engage in this type of research!
An interesting link in this regard is:
Cycles in the universe
To end this article I want to express my hope that geometry will one day
again
play a dominant role such as it once had in mathematics. Clerk Maxwell
(from
the famous Maxwell equations in electromagnetism) used Thomson's
geometrical
proofs of his equations, but nowadays this geometrical knowledge and
these
skills seem to be lost to the scientific community to a large degree.
Add to
this an integration of Phenomenology (as Goethe very much wanted) and
Sacred
Geometry with science, and, a new science, more true to nature, may be
born!
Martin Euser
To main site
Bibliography
Ralston Skinner: The source of measures; Wizards bookshelf, San
Diego,
Cal., USA
ISBN:0-913510-47-5
Gordon Plummer: The mathematics of the cosmic mind; Theosophical
Publishing
House; Wheaton, Ill., USA
ISBN: 0-8356-0030-0
