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 systems.

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 no
    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, astrologers?

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. C. Escher.

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 consciousness.

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 ascendant.

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 space.

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 astrology.

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 and fractals:

  James Gleick        Chaos -- Making a New Science, Viking Penguin, NY 1987

  John Briggs  &   David F. Peat          Turbulent Mirror, Harper & Row, NY 1989

  Benoit B. Mandelbrot,          The Fractal Geometry of Nature, W. H. Freeman & Co., 1977