Anyone who has seen a Mandelbrot Set, a visual representation of fractal mathematics, has partaken of an
incredible banquet of spatial nourishment. In the early
1960s Benoit Mandelbrot, developer and popularize of
fractal geometry, used fractals to plot cotton prices
over a 60 year period and found that the patterns of the
daily fluctuation exactly matched the patterns of monthly
changes. If this type of activity sounds familiar to
astrologers, read on -- it gets even better.
Fractal geometry is a part of chaos theory, which defines
a process of becoming, rather than a state of being.
Chaotic processes are not linear nor are they totally
predictable. One tiny change or difference introduced in
an ongoing process can be magnified over a period of
time, creating turbulence and making disruptive and
dramatic changes to the expected outcome. In other words,
chaos theory is the "new science" of living
The 1960s were full of important discoveries in this
"new science", mainly due to the development of
computers, which compressed the amount of time necessary
for advanced calculations. However the entire 20th
century seems to have been dedicated to superseding
Newtonian physics, for example:
- Relativity theory is
puncturing the illusion of absolute space and time.
- Quantum theory is dissolving
the dream of fixed processes of measurement.
- Chaos theory is assaulting the
idea of deterministic predictability.
- Synergy (the combination or
integration of two or more parts equaling more than
simply a sum)
is displacing the philosophy of
reductionism (the analysis of phenomena reducing them to
more than the sum of their parts).
- Holographic universe theory is
disputing the mechanical clockwork model.
The fact that astrology incorporates all of these ideas,
is perhaps why our science/art has survived for so long.
Of all the work done in contemporary physics, the concept
of fractals is the most directly usable in astrology.
Fractals are irregular patterns that repeat in
ever-decreasing magnifications; and most astrological
timing techniques embrace this idea. Fractals were once
termed pathological and monstrous because they upset
established mathematical standards -- sound familiar,
We live in a fractal world. With such a world view, one
can go on exploring ever-increasing levels of existence
that seemingly continue into infinity. Fractal forms show
self-similar designs that continue to repeat patterns
within patterns as you look into greater and greater
magnifications. In other words: AS ABOVE, SO BELOW. Those
who study the basic forms of the cosmos can relate to the
activity of atoms...
Fractal geometry is a method used to define and integrate
distance in different magnifications or scales, and is
used in our everyday physical world for examining such
phenomena as cycles of cotton prices, the atmospheric
patterns on Jupiter, the leaf of a fern, or multiple
magnifications of shorelines and rivers.
Multi-dimensional views of natures fractals range from a
Landsite satellite view of a river to a local road map of
a selected area of that same river. Seemingly linear
fractals, such as the Koch snowflake (a design featuring
an infinite line enclosed within a finite space)
demonstrate repeatable designs hidden in smaller and
smaller scales (called geometric iteration). These
patterns remind us of the music of J. S. Bach, the
mathematical constructs of Kurt Gödel, and the art of M.
Is it any wonder that physical science in its reach and
grasp of ever-refined concepts of space and time is
moving closer and closer to the metaphysical? Scientists
like: Albert Einstein ("...creator of relativity
theory"); David Bohm, author of Wholeness and the
Implicate Order, Ark Paperbacks 1980, "...the notion
of the implicate order in which any element contains
enfolded within itself the totality of the universe ...
including both matter and consciousness." front
page); Gary Zukav, author of The Dancing Wu Li Masters,
Wm. Morrow & Co., Inc. 1979 (a popularization of the
new physics), and The Seat of the Soul, Simon &
Schuster 1989 ("...Just as the Moon orbits the Earth
which orbits the Sun and there are orbits within orbits,
so, too, there are cycles within cycles." Page
204-5); Rupert Sheldrake, author of A New Science of
Life, Houghton Mifflin Co. 1981 ("...a theory
supporting holographic reality and shared consciousness);
Fred Alan Wolf, author of Parallel Universes, Simon &
Schuster 1988 ("...speculations on time travel ...
and the role of consciousness", cover); Fritj of
Capra, author of The Tao Of Physics, An Exploration of
the Parallels Between Modern Physics and Eastern
Mysticism, Shambala 1975 Arthur Young, author of The
Reflexive Universe, Delacorte Press, 1976 ("...which
shows the relationship between new physics and
metaphysics), and others interlink scientific concepts
with the power of consciousness.
Astrology is the study of
In our search for meaning, astrologers study the fractal
quality of space when we explore the inner dimensions of
the zodiac. We divide each sign into decanates 3 x 10
degrees and dwads 12 x 2˝ degrees. We divide the aspects
into scalar families, or harmonics, such as the eighth
harmonic family of degrees: conjunctions (0),
semi-squares (45) squares (90), sesqui-quadrates (135),
and oppositions (180). We create harmonic charts (a
derived chart within a chart) by dividing the chart (360
degrees) by a significant harmonic and magnifying it to a
circle. This reminds me of the biblical passage
"wheels within wheels".
Time has its fractal qualities as well. In the world of
computers, they speak of "real time" and
"virtual (hypothetical) time". Adding to our
studies of space, astrologers examine the fractal
qualities of time. We look at a horoscope as an arbitrary
point, the birth of a process within ongoing processes,
designating "real time" transits as a basic
temporal standard of movement -- a day for a day in the
ephemeris -- with which to monitor the chosen process.
By increasing the magnification of a real time period,
astrologers move into another fractal dimension. We use
what we call correspondences or derived times. A virtual
(derived) year can be represented by a real time day in
an ephemeris (day for a year progressions), or one degree
of the rotation of the Earth on its axis (primary
directions), or a full lunar cycle (minor progressions),
We explore another fractal dimension when we use a real
day to symbolize a virtual month (tertiary progressions);
and yet another fractal perspective is investigated by
applying the Sun's (or another planet's) arc to all the
natal charts planets, keeping the natal charts exact
pattern (Solar arc directions).
Instead of following one dimensional line in space and
time, by telescoping time inter-dimensionally, we follow
multi-dimensional threads within a whole life process.
Each thread is a wave of consciousness, and carries its
own meaning. Arthur Young, designer & developer of
the Bell Helicopter, author of "The Geometry of
Meaning", and founder of the Institute for the Study
of Consciousness once said that if astrology was ever to
be "proven" to science, it would be through the
understanding of progressions.
Many astrologers realize that in dividing time in these
and many other ways, we are creating journeys into the
many dimensions of consciousness itself, where additional
meaning and understanding may be uncovered.
When I look at an Astro*Carto*Graphy map, I see a
representation of the cosmos superimposed over a
flattened out map of our Earth, so that I can see all
time zones at once. I am able to isolate a particular
moment in time, and observe where the planets' positions
in the heavens correspond to the geographic locations on
the Earth. The map's planet and angle alignments give me
anchors of intensity. However, I may choose any Earth
longitude and latitude to focus in on the next smaller
level of the same pattern.
When I look at a horoscope wheel of those chosen Earth
coordinates, I see the same sky pattern from that
geographic perspective, which includes celestial anchors
for that location, such as the midheaven and the
These are two scenes which show us the same frozen moment
of time, from different spatial perspectives. During one
moment of time, the time zones on the Earth become
"virtual time modes" with the emphasis on
When I look at a numeric ephemeris, I see the fragmented
threads of bodies moving through time and space (the
zodiac), bundled by sequential dates. An analogy would be
an early motion picture with its spasmodic, jerky motion.
When I look at a graphic ephemeris for some period of
time, I see planets smoothly flowing through real time
(horizontal axis) and real space (vertical axis). If I
overlay the static lines of natal planetary positions to
the graph, I can identify aspects in motion which may
have potential impacts on that natal entity. I can expand
or contract the time element by changing the duration of
the graphic ephemeris, thereby changing the "virtual
time mode" with the emphasis on time.
Each of these astrological tools gives us a different
perspective -- all are valid examples of fractal tools of
My hope in presenting this essay is only to explore the
edges of these ideas, and that more questions will arise
in the minds of astrologers about the concepts upon which
astrology has had its strong, long-lived basis.
Birth dates of scientists
important in the development of fractal geometry.
Gaston Maurice Julia Born: 3 Feb. 1893 in Sidi Bel
Abbčs, Algeria Died: 19 Mar. 1978 in Paris, France
Benoit Mandelbrot Born: 20 Nov. 1924 in Warsaw, Poland
(Who's Who in America)
Waclaw Sierpinski Born: 14 Mar. 1882 in Warsaw, Poland
Died: 21 Oct. 1969 in Warsaw, Poland
Niels Fabian Helge von Koch Born: 25 Jan. 1870 in
Stockholm, Sweden Died: 11 Mar. 1924 in Stockholm, Sweden
Books about chaos theory
Making a New Science, Viking Penguin, NY 1987
John Briggs & David F. Peat
Turbulent Mirror, Harper & Row, NY
Benoit B. Mandelbrot,
The Fractal Geometry of Nature, W. H. Freeman & Co.,