'A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity.'

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In 2008 PBS aired a program on Fractals produced by NOVA - Hunting the Hidden Dimension, which will give the reader a broader appreciation for the significance of the rise of fractal geometry and mathematics:

Part I:Fractal Basics

Part II: The Mandlebrot Set

Part III: On the Defense

Part IV: Fractals in the Body

Part V: Nature's Fractal Nature

Part II: The Mandlebrot Set

Part III: On the Defense

Part IV: Fractals in the Body

Part V: Nature's Fractal Nature

The Phi Ratio or Golden Mean is known as a rudimentary fractal which is expressed in the Fibonacci sequence and in geometry. The following 'Fibonacci spiral' and Pentagram fractal show a progression of 'self-similar' and inter-nesting Phi proportions:

Modern man’s interest in and understanding of Fractal mathematics and geometry has come a long way since 1872 when Karl Weierstrass found a mathematical function that is ‘continuous everywhere but never differentiable’.

This continuity of form was dubbed a ‘fractal’ 103 years later by Benoit Mandelbrot in Les objets fractals (1975). Notable examples of such continuity or ‘self-similarity’ from the early 1900’s were the Koch snowflake and the Levy-C curve. In 1938 Paul Levy published his book Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole. In the 1960’s computers were used to produce self-similar or recursive designs and these designs became expressive of the revolutionary spirit and creativity of that decade. They were, and still are, symbolic of a dramatic shift in perceptions and fit well within the category of Psychedelic Art. Mandelbrot and the IBM computer helped to put fractal geometry on the map so to speak (i.e. in our field of vision/consciousness). It was in many ways the birth of a new geometry for a new age.

‘From the point of view of mathematics … nested shapes tended to be viewed as rare and pathological examples, of no general significance. But the crucial idea that was developed by Benoit Mandelbrot in the late 1960s and early 1970s was that in fact nested shapes can be identified in a great many natural systems and in several branches of mathematics.' [link]

Yet, long before Mandelbrot, computers and the coining of the word ‘fractal’, knowledge of ‘self-similarity’ existed. The Fibonacci sequence, which actually hails from ancient India, is one such example. Apparently Leonardo Fibonacci (1170-1250 C.E.) introduced both the Hindu-Arabic numeral system which is still in use today throughout the world and the Fibonnacci sequence to Europe in his 1202 C.E. publication Liber Abaci (Book of Calculations). Considering the Vedic civilization’s deep understanding of the self-similarity between macro and microcosm and its gnosis of mathematics and measurement, it is not too surprising to find that the Fibonacci sequence is one of the many enduring gifts ancient India has given the world. Knowledge the Golden Ratio is traced back to the Artha-Veda (thought to be written at least 5000 years old) and somewhat more recently to the work of Pingala (450 – 200 BCE), a Sanskrit grammarian who referred to the Fibonacci sequence as matrameru, translated as ‘mountain of cadence.’ Meru means mountain. And matra, mean not simply cadence (measure of movement or rhythmical flow) in poetic or spoken verse, but also a measure of any kind (time, space, number, distance), a particle, an atom, a degree of a circle, or a minute portion. It also means ‘the full measure of anything, the whole or totality’ and ‘mother’.

It is entirely possible that the multi-dimensional, multi-scale meaning of the word matra is expressive of the Vedic knowledge and conception of what we now call fractals or self-similarity, nested structures or recursive design in nature and cosmos. The infinite repetition of the whole (the Material Cosmos) in individual form and throughout time was certainly basic and fundamental Vedic gnosis. On this basis they measured, acknowledged and recorded the inherent harmonies of time and space, and attuned themselves to these inherent harmonies.

Hence in Vedic cosmology, the Earth’s Day is equivalent or ‘self-similar’ to the Earth’s Year which is equivalent to the Precession of the Equinoxes which is, in turn, equivalent to even larger cycles of time. The tradition of dividing the Precession of the Equinoxes into 12 ages, the Year into 12 months, the Day into 24 (2 x 12) hours is another testimony of the ‘fractal’ nature or ‘fractal’ geometry of Time. The same 360 degree structure and the same stages and flow of development apply equally to the micro-cycle of time as to the macro-cycle. Out of this holistic thought or consciousness, come many fascinating equivalences and correspondences such as the Precession of the Equinoxes being measured in terms of the Vedic Yuga which is equivalent to the radius of the Sun in miles (432,000), and the 86,400 seconds of the Earth’s day being ‘self-similar’ to the 864,000 mile diameter of the Sun.

What is interesting to me is that in the Vedic civilization the infinite replication of the macrocosm in microcosmic forms was understood as a whole, divinely orchestrated, harmonic perfection. The minute portion of the whole was understood as an individual expression of the whole and inseparable from the whole … in a sense, unbreakable. Yet the word modern man has adopted word for these minute portions or repetitions of the whole – ‘fractal’ – is derived from fractus, meaning broken, fractured, shattered, vanquished or defeated.

To name a repetition of the whole after a word that means broken, shattered and defeated seems to me to be quite an oxymoron … a reflection of a isolated consciousness that feels irrevocably broken off from the Universal Mother and hence shattered. It is sad really, to contemplate the loss of knowledge of and harmony with the Divine Matrix that we have endured over how ever many millennia – the fall from the Divine Matra (the Universal Mother or Matriarch and all her measured parts and expressions) that leaves us with such a shattered identity. But, according to Sri Aurobindo and the Rig Veda, even our apparent set backs are useful stages in the evolutionary journey towards recovering our true identity, recovering our self-similarity and harmony with the Earth, the Cosmos, and all of creation.

A new-found appreciation of self-similar or recursive forms or designs of the Divine Matrix emerged in the art, architecture and science of European Renaissance, in part due to the 1509 publication of Luca Pacioli's Divina Proportione whose subject was the Golden Mean or Golden Ratio. Leonardo da Vinci (1452 – 1519) abundantly used the Golden Mean in his creations. Johannes Kepler (1571-1630) is another man whose appreciation of the Golden Mean and self-similar or nested structures has lived on through his work to illuminate our modern understanding of our Earth and Cosmos. His idea that the structure of the solar system was a macrocosm of nesting platonic solids was apparently incorrect, but out of his appreciation for a soulful Earth and Cosmos and the repetitive geometries of nature came his discovery of the Kepler solids and three laws of planetary motion. [Pictured on the right is Kepler's Platonic solid model of the solar system from Mysterium Cosmographicum (1596).]

In our modern world fractal geometry and mathematics (symmetrical and asymmetrical) are applied towards computer technology, communication, data storage and the transmission of data, graphic arts, animation, wireless technology (via smaller and smaller antennae), nano-tecnology, energy technology and diagnostic medicine. Fractals are also the building blocks of modern Physic’s Chaos Theory which strives to explain the unpredictable nature of whole systems.

This continuity of form was dubbed a ‘fractal’ 103 years later by Benoit Mandelbrot in Les objets fractals (1975). Notable examples of such continuity or ‘self-similarity’ from the early 1900’s were the Koch snowflake and the Levy-C curve. In 1938 Paul Levy published his book Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole. In the 1960’s computers were used to produce self-similar or recursive designs and these designs became expressive of the revolutionary spirit and creativity of that decade. They were, and still are, symbolic of a dramatic shift in perceptions and fit well within the category of Psychedelic Art. Mandelbrot and the IBM computer helped to put fractal geometry on the map so to speak (i.e. in our field of vision/consciousness). It was in many ways the birth of a new geometry for a new age.

‘From the point of view of mathematics … nested shapes tended to be viewed as rare and pathological examples, of no general significance. But the crucial idea that was developed by Benoit Mandelbrot in the late 1960s and early 1970s was that in fact nested shapes can be identified in a great many natural systems and in several branches of mathematics.' [link]

Yet, long before Mandelbrot, computers and the coining of the word ‘fractal’, knowledge of ‘self-similarity’ existed. The Fibonacci sequence, which actually hails from ancient India, is one such example. Apparently Leonardo Fibonacci (1170-1250 C.E.) introduced both the Hindu-Arabic numeral system which is still in use today throughout the world and the Fibonnacci sequence to Europe in his 1202 C.E. publication Liber Abaci (Book of Calculations). Considering the Vedic civilization’s deep understanding of the self-similarity between macro and microcosm and its gnosis of mathematics and measurement, it is not too surprising to find that the Fibonacci sequence is one of the many enduring gifts ancient India has given the world. Knowledge the Golden Ratio is traced back to the Artha-Veda (thought to be written at least 5000 years old) and somewhat more recently to the work of Pingala (450 – 200 BCE), a Sanskrit grammarian who referred to the Fibonacci sequence as matrameru, translated as ‘mountain of cadence.’ Meru means mountain. And matra, mean not simply cadence (measure of movement or rhythmical flow) in poetic or spoken verse, but also a measure of any kind (time, space, number, distance), a particle, an atom, a degree of a circle, or a minute portion. It also means ‘the full measure of anything, the whole or totality’ and ‘mother’.

It is entirely possible that the multi-dimensional, multi-scale meaning of the word matra is expressive of the Vedic knowledge and conception of what we now call fractals or self-similarity, nested structures or recursive design in nature and cosmos. The infinite repetition of the whole (the Material Cosmos) in individual form and throughout time was certainly basic and fundamental Vedic gnosis. On this basis they measured, acknowledged and recorded the inherent harmonies of time and space, and attuned themselves to these inherent harmonies.

Hence in Vedic cosmology, the Earth’s Day is equivalent or ‘self-similar’ to the Earth’s Year which is equivalent to the Precession of the Equinoxes which is, in turn, equivalent to even larger cycles of time. The tradition of dividing the Precession of the Equinoxes into 12 ages, the Year into 12 months, the Day into 24 (2 x 12) hours is another testimony of the ‘fractal’ nature or ‘fractal’ geometry of Time. The same 360 degree structure and the same stages and flow of development apply equally to the micro-cycle of time as to the macro-cycle. Out of this holistic thought or consciousness, come many fascinating equivalences and correspondences such as the Precession of the Equinoxes being measured in terms of the Vedic Yuga which is equivalent to the radius of the Sun in miles (432,000), and the 86,400 seconds of the Earth’s day being ‘self-similar’ to the 864,000 mile diameter of the Sun.

What is interesting to me is that in the Vedic civilization the infinite replication of the macrocosm in microcosmic forms was understood as a whole, divinely orchestrated, harmonic perfection. The minute portion of the whole was understood as an individual expression of the whole and inseparable from the whole … in a sense, unbreakable. Yet the word modern man has adopted word for these minute portions or repetitions of the whole – ‘fractal’ – is derived from fractus, meaning broken, fractured, shattered, vanquished or defeated.

To name a repetition of the whole after a word that means broken, shattered and defeated seems to me to be quite an oxymoron … a reflection of a isolated consciousness that feels irrevocably broken off from the Universal Mother and hence shattered. It is sad really, to contemplate the loss of knowledge of and harmony with the Divine Matrix that we have endured over how ever many millennia – the fall from the Divine Matra (the Universal Mother or Matriarch and all her measured parts and expressions) that leaves us with such a shattered identity. But, according to Sri Aurobindo and the Rig Veda, even our apparent set backs are useful stages in the evolutionary journey towards recovering our true identity, recovering our self-similarity and harmony with the Earth, the Cosmos, and all of creation.

A new-found appreciation of self-similar or recursive forms or designs of the Divine Matrix emerged in the art, architecture and science of European Renaissance, in part due to the 1509 publication of Luca Pacioli's Divina Proportione whose subject was the Golden Mean or Golden Ratio. Leonardo da Vinci (1452 – 1519) abundantly used the Golden Mean in his creations. Johannes Kepler (1571-1630) is another man whose appreciation of the Golden Mean and self-similar or nested structures has lived on through his work to illuminate our modern understanding of our Earth and Cosmos. His idea that the structure of the solar system was a macrocosm of nesting platonic solids was apparently incorrect, but out of his appreciation for a soulful Earth and Cosmos and the repetitive geometries of nature came his discovery of the Kepler solids and three laws of planetary motion. [Pictured on the right is Kepler's Platonic solid model of the solar system from Mysterium Cosmographicum (1596).]

In our modern world fractal geometry and mathematics (symmetrical and asymmetrical) are applied towards computer technology, communication, data storage and the transmission of data, graphic arts, animation, wireless technology (via smaller and smaller antennae), nano-tecnology, energy technology and diagnostic medicine. Fractals are also the building blocks of modern Physic’s Chaos Theory which strives to explain the unpredictable nature of whole systems.

‘Fractal geometry and chaos theory are providing us with a new perspective to view the world. For centuries we've used the line as a basic building block to understand the objects around us. Chaos science uses a different geometry called fractal geometry. Fractal geometry is a new language used to describe, model and analyze complex forms found in nature.’I have recently read that the essence of Chaos Theory is really about finding the underlying order underneath the apparent chaos. It seems then that like the name ‘fractal’, “Chaos Theory’ is something of a misnomer. The human mind is faced with the task of fathoming the pervasive and integral order and multi-layered dynamism of the whole and of self-similar sub-systems of the whole. We labor (even if unconsciously) to find the perspective from which all dynamism and form are known, seen and experienced as a harmonious expression of, and re-creation of the One Universal Being. This was the perspective achieved by the Vedic seers and needs to be restored in our new age.

At some point our modern understanding of fractals and Chaos Theory must come full circle and reunite with the Vedic gnosis of macro and microcosm and gnosis of a divinely or supramentally orchestrated Order. For this to happen there must be some recognition that a point-less or soul-less system is not capable of order. There also must be some recognition that ‘fractal’ geometry applies to the structure of Time as well as to the structure of material creation (the Matrix). The ancient calendar systems of the Vedic and Mayan civilizations and their various epochs, ages and cycles of time, including the 12 month Vedic Year – which is one and the same as our modern Tropical Zodiac, will begin to make much more sense to us as our modern understanding of self-similar or nested micro and macrocosmic structures in both time and space begins to expand. The world-wide and world-unifying cultural renaissance or rebirth that will result from this expansion in understanding and consciousness is something that sages and prophets have been predicting for ages.

The immortal’s thoughts displaced our bounded view, …

All things now bore a deeper heavenlier sense;

A clear harmony marked their truth’s outline,

Reset the balance and measures of the world.

Each shape showed its occult design, unveiled

God’s meaning in it for which it was made

And the vivid splendour of his artist thought.

– Sri Aurobindo, Savitri Book VII, Canto V

All things now bore a deeper heavenlier sense;

A clear harmony marked their truth’s outline,

Reset the balance and measures of the world.

Each shape showed its occult design, unveiled

God’s meaning in it for which it was made

And the vivid splendour of his artist thought.

– Sri Aurobindo, Savitri Book VII, Canto V

Additional Notes: Readers interested in the Geometry of Time should read The Gnostic Circle by Patrizia Norelli-Bachelet in which the 9 year cycle of time and larger cycles of 9 such as the 25,920 year Precession of the Equinoxes are seen as an expansion of the 12 month year. The larger cycles of time are seen as ‘self-similar’ to the 12 month year in that one journeys through the same 12 stages/month of development (i.e. signs of the zodiac) in larger scales of time and space.

Related Links:

* Number and the Supramental Matrix by Lori Tompkins

* The Simultaneity of Time, Spirit and Matter by Lori Tompkins

* Macrocosm, Microcosm & the Laws of Time and Space (Quote from PNB's

*The Gnostic Circle*)

* A Higher Vision of Time Is Needed... (Quotes from PNB's

*Time & Imperishability*and some commentary)

*

*The ALL-pervading Energy of Time - the New Time (Quote from PNB's*

*The New Way, a study in the rise and establishment of a gnostic society*).

* The Geometry and Superstructure of Time

* 'Time as a Power (not an Enemy) of Self' by Lori Tompkins

* Learning the One Song, Learning the Universe by Lori Tompkins

* In Harmony's Way by Lori Tompkins

* God Forever Geometrizes by Lori Tompkins

**S U B S C R I B E ( v i a R S S ) and / or**

**S H A R E IT using A N Y of the I C O N S below**

extreamly fascinating! i have seen a woman in portland selling fractal art and always wanted to know a bit more. Intense.

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