Selected quotes:
p. 4
In science today we are witnessing a general shift away from the assumption
that the fundamental nature of matter can be considered from the point of
view of substance (particles, quanta) to the concept that the fundamental
nature of the material world is knowable only through its underlying patterns
of wave forms.
Both our organs of perception and the phenomenal world we perceive seem
to be best understood as systems of pure pattern, or as geometric structures
of form and proportion. Therefore, when many ancient cultures chose to examine
reality through the metaphors of geometry and music (music being the study
of the proportional laws of sound frequency), they were already very close
to the position of our most contemporary science.
Professor Amstutz of the Mineralogical Institute at the University of Heidelberg
recently said: 'Matter's latticed waves are spaced at intervals corresponding
to the frets on a harp or guitar with analogous sequences of overtones arising
from each fundamental. The science of musical harmony is in these terms
practically identical with the science of symmetry in crystals.'
The point of view of modern force-field theory and wave mechanics corresponds
to the ancient geometric-harmonic vision of universal order as being an
interwoven configuration of wave patterns. Bertrand Russell, who began to
see the profound value of the musical and geometric base to what we now
call Pythagorean mathematics and number theory, also supported this view
in The Analysis of Matter: 'What we perceive as various qualities
of matter,' he said, 'are actually differences in periodicity.'
p. 5
Within the human consciousness is the unique ability to perceive the
transparency between absolute, permanent relationships, contained in the
insubstantial forms of geometric order, and the transitory, changing forms
of our actual world. The content of our experience results from an immaterial,
abstract, geometric architecture which is composed of harmonic waves of
energy, nodes of rationality, melodic forms springing forth from the eternal
realm of geometric proportion.
p. 16
Ancient geometry rests on no a priori axioms or assumptions. Unlike Euclidean
and the more recent geometries, the starting point of ancient geometric
thought is not a network of intellectual definitions or abstractions, but
instead a meditation upon a metaphysical Unity, followed by an attempt to
symbolize visually and to contemplate the pure, formal order which springs
forth from this incomprehensible Oneness. It is the approach to the starting
point of the geometric activity which radically separates what we may call
the sacred from the mundane or secular geometries. Ancient geometry begins
with One, while modern mathematics and geometry begin with Zero.
p. 21
Unity is a philosophic concept and a mystic experience expressible mathematically.
The Western mentality, however, withdrew its discipline of acknowledging
a supra-rational, unknowable mystery as its first principle. But in rejecting
this reverence to a single unknowable unity, our mathematics and science
developed into a system requiring complex, interconnected hypotheses, imaginary
entities such as those mentioned above [negative numbers, infinite decimal
numbers, algebraic and transcendental irrational numbers, imaginary and
complex numbers] and unknown x quantities which must be manipulated, quantified
or equalized as in the algebraic form of thought. So the unknown appears
not just once but at every turn, and can be dealt with only by seeking quantitative
solutions.
Our present thought is based on the following numerical and logical sequence:
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
With zero in the centre, there is a quantitative expansion 1, 2, 3 ... and
our sense of balance requires having -1, -2, -3 ... on the other side, giving
a series of non-existent abstractions (negative quantities) which demand
an absurd logic. The system has a break-point, zero, disconnecting the continuum
and dissociating the positive numbers from the negative balancing series.
In the ancient Egyptian numerical progression, beginning with one rather
than zero, all the elements are natural and real:
1/5, 1/4, 1/3, 1/2, 1, 2, 3, 4, 5
All the elements flow out from the central unity in accordance with the
law of inversion or reciprocity. The Egyptians based their mathematics on
this simple, natural series of numbers, performing sophisticated operations
with it for which we now need complex algebra and trigonometry. We have
already seen the natural demonstration of this series in the physical laws
of sound. The plucked string, when divided in half, produced double the
frequency of vibrations. Thus this series expresses the essential law of
Harmony.
