By: Arnold Keyserling & R.C.L.
Dimensions, Mandelbrot, Chaos, 4 Attractors, Music and Color
The Enneagram and The Wheel are the bridge between awareness and consciousness, sense and meaning. They constitute the original Cabala. Out of the nine parts of speech, and twelve components of the mind, all possible meaning arises. The structure of the inner world of mind and language mirrors that of the outer world of matter and vibration. The outer world, including music and color, follows the same laws of number which govern the inner world. The laws of number bridge the inner and outer worlds. By understanding number we understand ourself. We gain a powerful Wisdom tool with which we can make sense of our world.
Pythagoras discovered long ago that number leads to structure, and that the structure of the mind is also the structure of the world. In essence all is number. If you truly understand number you will have the key to all Wisdom. Number is composed of the nine numerals, basic to all information, uniting geometry and arithmetic, space and time. This may seem difficult, but its really as simple as: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It becomes complicated when you begin to consider the reciprocal relations of the nine numerals. These relations are clarified in the following chart which shows the circle of 5 dimensions, based on the relation of infinity to one. The circle of dimensions summarizes the basic laws of number and mathematics. The eight points shown on the dimension circle (0 - 7) correspond to the eight directions on the outside of the Wheel discussed further in Chapter Eight.
THE FIVE DIMENSIONS
Including the zero dimension there are five dimensions: 0 - 4. The zero, first second and third dimensions have long been accepted as true, but the reality of the fourth dimension was questioned. Since Einstein, however, the existence of the fourth dimension is now an accepted fact. The first, second and third dimensions are now understood as imaginary. We live only in the fourth dimension, but in order to understand our dimension, our reality, we must also understand each of the other imaginary dimensions. Moreover we must realize how infinity permeates each dimension, including the fourth. So we begin our exploration of Number as a Wisdom tool by understanding the dimensions and their relation to the infinite.
(0) - The Zero Dimension is the POINT, the infinitely small place holder. •
(1) - The First Dimension is the LINE, which consists of an infinite number of points. --------------------------------------------
(2) - The Second Dimension is the PLANE, such as a square. It contains an infinite number of straight lines.
(3) - The Third Dimension is the SOLID, such as a cube. It contains an infinite number of planes or squares.
(4) - The Fourth Dimension, SPACE-TIME CONTINUUM, is reality. In the fourth dimension the infinite number of solids in the Universe are in relationship with each other through time and energy. The Fourth Dimension is portrayed geometrically by fractals and by the Hypercube. The Hypercube is the symbol used in mathematics to try and represent the fourth dimension in two dimensions (a drawing on a piece of paper - a plane). From the center of the Hypercube through its 8 diagonals the Hypercube is related to everything in the Universe. The infinity in the Fourth Dimension lies in the infinity of relations. This can be expressed in terms of "fractal scaling", from the infinite small to the infinite big, perpendicular to the other dimensions and including the intervals or fractal dimensions between them. The meaning of fractal scaling is explained later in this Chapter, for now it is sufficient to understand this as scales of magnitude, as for instance from the size of the atom to the size of a galaxy. The Hypercube is cut by 4 diagonals constituting the central point. In consciousness this center point represents the identity or the Self. According to the Pythagorean theorem, the number of the diagonals - four - times the square root of three, equals nine (4 * 3 = 9). The four diagonals on the Hypercube are 5-1, 6-2, 7-3 and 0-4.
5 - 1: Matter
6 - 2: Consciousness
7 - 3: Energy
0 - 4: Self organization
The Hypercube and its diagonals are shown in the diagram.
METHODS/EXPERIMENTS: Try to think about the concept of infinity, from infinitely small, to large, to never ending. Try to visualize and imagine each of the five dimensions, starting with the infinitely small point, on up. Try to visualize simple geometric figures, see them in your minds eye. After you have practiced with and mastered the simple forms, circle, triangle, square and pentagram, move them around at will and view them from all different perspectives.
Then try to visualize a hypercube and see it from different angles. Try to visualize, imagine and feel the interrelatedness of all matter which Science has discovered as a cold fact. Sense and feel how your body, like all matter in the Universe, is in energy interchange with all other bodies in the Universe. Start off with the physical objects immediately around you, and then expand outward in ever larger vistas, to finally include the whole Universe.
Then when you are ready try the Meditation of the Hypercube exercise created by Wilhelmine Keyserling. She is a teacher of Yoga and philosophy in Vienna, Austria who is the author of many books on these subjects. She also happens to be Arnold Keyserling's wife.
MEDITATION OF THE HYPERCUBE
Sit comfortably on the floor in an erect position. Take your time and visually build a Hypercube all around you. Imagine the corners with three, four, five and six in front of you, and with one, two, seven and zero behind you. You are sitting right in the middle and your heart is in the center of the cube, the point where all four diagonals intersect. Focus you attention on the center, the inner Self. Then visualize the 0 point - Awareness behind and to the right going up. Then go there mentally while inhaling very slowly. When exhaling draw a diagonal through your center to 4 - Willing in front of you to the left going down. Repeat this a few times if necessary. Inhale going out to the zero point, exhale going down through the center to the 4 point.
Then return to your center and simultaneously expand to the two points (0-4) at once - inhaling your breath while you do this, then exhale and come back to the center. Do it a second time. The third time, if you wish, you can pierce through the Will-point into the Earth and through the Awareness point towards the infinite. Then follow the same process with 7 - 3, 6 - 2, 5 - 1. End by remaining in your center, in your Whole Self which includes all numerals and the Zero Awareness.
THE DIMENSIONS AND TIME
(0) - Zero dimension, a single point, exists not in space, but in time only. It is the moment in the present between past and future, the subject, zero. It constitutes potentiality, the four space dimensions constitutes actuality. ï
(1) - Future. The moments create the future, forming a trajectory. Trajectory
(2) - Present. The trajectory is seen like a disc, or revolution.
(3) - Past. The disc turns one half time around its axis and fills out the sphere of the past.
(4) - Wave. The movement continues to form a wave, constituting fractally the space-time continuum.
THE DIMENSIONS AND MATHEMATICS
(0) - The Zero Dimension is the home of Natural numbers. The subject point, the moment, is Zero, pure Awareness. Its numbers are the natural numerals: Zero
The nine natural numbers are the basis of quality and invariance. All numbers can be reduced to the nine: for example by addition: 365 = 14 = 5. Becoming aware therefore means deducing or abstracting to the nine fundamental criteria. In the Jewish Cabala this is the nine names of the divine, in Chinese it is the nine forms of the Tao.
(1) - The first dimension is the home of the Whole or Integer numbers. The points, natural numbers, have no extension. Integer numbers unite positive and negative up to ten and create the number line.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10
The Whole numbers are created through addition, basic to sensing, and through subtraction, basic to the spirit, the path in the unknown future.
(2) - In the second dimension we have the Rational numbers, based on three points, are visualized on the plane with a vertical and horizontal axis. The number plane was known to the Pythagoreans and called the Chi:
Rational numbers are produced by division in the positive field - FRACTIONS - and multiplication in the negative field - PRODUCTS. Zero is the center of the CHI, which for Pythagoras and Plato was the tool of the Demiurge, the Creator. The fields contain only the fractions and products inside the ten numbers. Division is the basis of thinking, multiplication the synergy of the soul.
(3) - The third dimension brings out the Real numbers. Real numbers start from zero and connect fractions of the same numerical value, leading to the proportions and functions.
The proportions are the basis of continuity and harmony. They connect fractions of the same value to zero. The functions are the basis of discontinuity. They connect products by which bodies are in relation, as for instance in the atom, where the distances of the electron shells follow the numbers of the central diagonal 1 - 4 - 9 - 16, and the possible number of electrons in each shell, the capacity, follow the diagonal 2 - 8 - 18 - 32.
The rational numbers of the second dimension, and the whole and natural numbers of the first and zero dimension, all have a fixed place on the number line. The real numbers in the third dimension are, however, fundamentally different. Although they are located somewhere on the number line, they have no fixed place there.
To the ancient Greeks who first developed mathematics to a high art in the West, all numbers had to have a fixed location somewhere on the number line. The existence of the Real numbers, with no fixed location, was known only to a few high initiates in the Pythagorean brotherhood who swore to keep it secret. It can be easily understood today by way of the Pythagorean theorem illustrated here:
The Pythagorean Theorem exemplifies the rational numbers. But what happens if A and B both equal 1? In this case C must equal the square root of 2. But 2 is an irrational Real Number. It is a number which goes on and on with no repetition into infinity. 1.41421... It is a never ending number and has no fixed place on the number line. Unlike an infinite rational number which goes on and on, but repeats, such as a third (.3333333...), where we can know the exact location on the number line, with a Real Number, we can only know its approximate location. There are other examples of Real Numbers, such as Pi (the ratio of a circumference of a circle to its diameter), the square root of any prime number, e, . etc. These Real Numbers never end and never repeat.
(4) - The fourth dimension is the home of the Complex numbers and Fractal geometry. Unlike the other dimensions, the fourth is the real world in which we live, the meso-cosmic world. It is the space time continuum of Man and Nature where there is constant change based on feedback. As Mandelbrot recently discovered the fourth dimension includes not only the first three dimensions, but also the gaps or intervals between them, the fractal dimensions.
MANDELBROT'S CONTRIBUTION TO THE WISDOM OF THE DIMENSIONS
Complex numbers and fractal geometry are the most important to Man, yet they were the last to be discovered by reason, and the most difficult to grasp. The full significance of the mathematics of the fourth dimension could not be realized until the downfall of the Newtonian-Euclidian mind-set and its replacement in the 1970's and 1980's with the Science of Chaos. The Chaos insights were led by Benoit Mandelbrot, an IBM scientist and Professor of Mathematics at Yale, with many other scientists close behind. Computers helped Mandelbrot realize the full significance of the formula which now bears his name.
The Mandelbrot set is a dynamic calculation based on the iteration (calculation based on constant feedback) of complex numbers with zero as the starting point. The order behind the chaotic production of numbers created by the formula z -> z2 + c can only be seen by the computer calculation and graphic portrayal of these numbers. Otherwise the formula appears to generate a totally random and meaningless set of numbers. It is only when millions of calculations are mechanically performed and plotted on a two dimensional plane (the computer screen), that the hidden geometric order of the Mandelbrot set is revealed. The order is of a strange and beautiful kind, containing self similar recursiveness over an infinite scale. See the graphic below of the Mandelbrot set, and the others which follow in this chapter. These graphics provide an accurate right brain glimpse into this world.
Mandelbrot's formula summarizes many of the insights he gained into the fractal geometry of nature, the real world of the fourth dimension. This contrasts markedly with the idealized world of Euclidian forms of the first, second and third dimensions. These forms had preoccupied almost all mathematicians before Mandelbrot. Euclidian geometry was concerned with abstract perfection almost non-existent in nature. It could not describe the shape of a cloud, a mountain, a coastline or a tree. As Mandelbrot said in his book The Fractal Geometry of Nature: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
Before Mandelbrot mathematicians believed that most of the patterns of nature were far too complex, irregular, fragmented and amorphous to be described mathematically. But Mandelbrot conceived and developed a new fractal geometry of nature based on the fourth dimension and Complex numbers. The fractal geometry can mathematically describe the most amorphous and chaotic forms of the real world. As Mandelbrot said: "Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently."
Mandelbrot discovered that the fourth dimension of fractal forms includes an infinite set of fractional dimensions which lie between the zero and first dimension, the first and second dimension and the second and third dimension. He proved that the fourth dimension includes the fractional dimensions which lie between the first three. He calls the in between or interval dimensions the "fractal dimensions". He has shown mathematically and graphically how nature uses the fractal dimensions and what he calls "self constrained chance" to create the complex and irregular forms of the real world.
Thanks to Mandelbrot and the recent insights of the science of chaos we now have a mathematical understanding of some of the heretofore secret workings of Nature. We understand for the first time why two trees growing next to each other in the forest at the same time, from the same stock, with the same genes, will still end up unique. They will be similar to be sure, but not identical. Just so every snow flake falling from the same cloud at the same time under identical conditions is still unique, different from all of the rest. This is only possible because of the infinity which lies in the dimensions and the interplay of chance - the unpredictable chaos.
METHODS/EXPERIMENTS: An understanding of how the fourth dimension includes the infinity of intervals between the other dimensions can be gained by visualizing a few of the better known fractal dimensions (sometimes called Hausdorff dimensions by mathematicians). One of the most famous fractal dimensions lies between the zero dimension and the first dimension, the point and the line. It is created by "middle third erasing" where you start with a line and remove the middle third; two lines remain from which you again remove the middle third; then remove the middle third of the remaining segments; and so on into infinity. What remains after all of the middle third removals is called by Mandelbrot "Cantor's Dust". It consists of an infinite number of points, but no length. An example of the process (not exactly to proportion) is shown here.
The Cantor's Dust which remains is not quite a line, but is more than a point. The dimension is calculated to have a numerical value of .63 and was discovered by mathematician George Cantor in the beginning of the Twentieth Century. It was considered an anomaly and was avoided by most mathematicians as a "useless monstrosity". In fact this fractal dimension is a part of the real world of the fourth dimension and corresponds to many phenomena of Man and Nature. For instance, Mandelbrot cracked a serious problem for IBM by discovering that the seemingly random errors which always appeared in data transmission lines in fact occurred in time according to the fractal dimension illustrated by Cantor's Dust. Knowing the hidden and mathematically precise order behind the apparently random errors allowed IBM to easily overcome this natural phenomena of data transmission by simple redundancies in the transmission.
Another well known fractal dimension lies between a line and a plane, the first and second dimension. It is called the Sierpiniski Gasket after mathematician Waclaw Sierpiniski and has a fractal dimension of 1.58. Create it by starting with an equilateral triangle and remove the open central upside down equilateral triangle with half the side length of the starting triangle. This leaves three half size triangles. Then repeat the process on the remaining half size triangles, and so forth ad infinitum. The remaining form has infinite lines but is less than a plane.
There are many other illustrations given of fractal dimensions in most of the Chaos references mentioned in this book, particularly in Mandelbrot's books. An excellent reference in this area is Michael McGuire's book An Eye For Fractals containing both computer graphics and photographs of nature. McGuire's photographs show the fractals all around us in Nature, the trees, clouds, mountains, rivers, stones and kelp. Another helpful albeit very technical reference is Manfred Schroeder's Fractals, Chaos, Power Laws which shows how even noises follow the fractal laws.
Try to visualize the many different fractal forms given above and in these books to get a feel for the infinity which lies in between the first three dimensions and is thus part of the fourth. The computer program and manual which goes with Gleick's book called Chaos the Software is also very helpful for it contains a section which allows you to create fractal forgeries of nature such as a cloud, a mountain or even an entire planet. Think about one of the questions which led Mandelbrot to his discoveries: how long is the coastline of England? The closer you look at the coast line, the closer you measure, the longer it gets, and then some!
Fractal forms are also found in the body. The best known example are the arteries and veins in mammalian vascular systems. The bronchi of the human lung are self similar over 15 successive bifurcations. This area of biological research is just beginning. McGuire refers to recent discoveries in brain research which suggests that a fractal structure based on hexagons may be how the receptive fields of the visual cortex are organized.
Fractal geometry and the insights of the science of Chaos are based on Complex Numbers - the numbers of the fourth dimension which are capable of modeling the dynamics of chaotic systems. Unlike all other numbers, the Complex Numbers do not exist on the number line at all, even with an approximate location like the Real Numbers. The Complex Numbers only exist on an x-y time plane involving the so called Imaginary Numbers. They have only indirect reference to the number line.
To understand Complex Numbers you must first understand Imaginary Numbers with which they combine. Imaginary Numbers can be understood with the simple formula: X2 + 1 = 0. The only solution to this formula is that X equals the square root of negative one: X = -1. X in this formula is an Imaginary Number, because according to convention governing all other numbers, a negative number times a negative number produces a positive number. The square root of a negative number is therefore an impossibility, yet nevertheless it exists (ie. x2 + 1 = 0), and mathematicians routinely use and refer to such numbers as Imaginary.(1)
With Imaginary numbers a negative times a negatives creates a negative, not a positive. Is that so illogical? Without Imaginary numbers the complex dynamics and turbulence of the real space/time world could not be described mathematically. Imaginary numbers combine with real numbers to create Complex numbers. Complex numbers are the basis of much of higher math. They allow mathematicians to see many essential connections and relationships in mathematics which would not otherwise be possible. Complex numbers allow an algebraic understanding of the hidden unity in the ideal world of numbers. They also provide a geometric description of the fractal beauty of the real world, the zig-zag world of nature and other very complicated systems. This is not possible with the other, non-complex numbers, that exist alone without Imaginary numbers. For more on imaginary numbers see the Appendix section "The Mathematics of the Mandelbrot Formula and the Workings of Numbers and Vectors in the Complex Plane".
Complex Numbers are a combination of Imaginary numbers which have no place on the number line, and any other type of number which does have a place on the number line - like the real, rational and natural numbers. Mathematics symbolizes the Complex Numbers with a letter z and defines a Complex Number as follows:
z = a + bi
a = real number, and bi = imaginary number
Both the real and imaginary parts of the complex number can be either positive or negative and either whole numbers or decimals. The complex numbers can be easily added and subtracted, and almost as easily multiplied and divided. For examples of how the Mandelbrot formula works showing the mathematics of complex numbers see the Appendix section on the math. The interested reader will there find a more complete, but still simple introduction to this mathematics.
The alternative symbols x and y are also sometimes used instead of a and b to symbolize a complex number. The use of x and y provides reference to a familiar grid of a plane and so facilitates a geometric understanding of complex numbers. The x axis stands for the regular number line, shown horizontally. The y axis stands for the imaginary number line of negative square roots. The y axis is shown vertically to create a plane of complex numbers.
z = x + yi
x = real number, and yi = imaginary number
Every non-complex number has its place on the one dimensional number line. But every complex number has a place on a vast two dimensional plane of numbers called the complex plane. Thus to locate a particular complex number you have to refer to both a horizontal axis of real numbers and a vertical axis of imaginary numbers. This contrasts with all other numbers which can be located as a point on a one dimensional line.
The line of real numbers shown on the x axis combine at a right angle with a line of imaginary numbers on the y axis to form the complex plane. This is further explained in the Appendix section "The Mathematics of the Mandelbrot Formula and the Workings of Numbers and Vectors in the Complex Plane".
As shown above z = 2 can be anywhere on the circle. The point for the complex number 2 -3i is also shown.
The Complex Numbers when iterated - subject to constant feedback - produce Fractal Scaling as is shown by the Mandelbrot set:
z -> z2 + c
c = any complex number.
Written out this formula is equivalent to:
z -> (x + yi) + (x + yi)
-> means iteration, the feedback process where the end result of the last calculation becomes the beginning constant of the next: z2 + c becomes the z in the next repetition. Like life it is a dynamic equation, existing in time, not a static equation.
When iteration of a squaring process is applied to non-complex numbers the results are always known and predictable. For instance when any non-complex number greater than one is repeatedly squared, it quickly approaches infinity: 1.1 * 1.1 = 1.21 * 1.21 = 1.4641 * 1.4641 = 2.14358 and after ten iterations the number created is 2.43... *
10 to the 42nd power which written out is 2,430,000,000,000,000,000,000,000,000,000, 000,000,000,000. A number so large as to dwarf even the national debt. Mathematicians say of this size number that it is approaching infinity.
The same is true for any non-complex number which is less than one, but in reverse; it quickly goes to the infinitely small, the zero. For example with .9: .9.9=.81; .81.81=.6561; .6561.6561=.43046 and after only ten iterations it becomes 1.39...10 to the negative 47th power, which written out is .000000000000000000000000000000 0000000000000000139..., a very small number indeed.
With real, rational or natural numbers the squaring iteration must always go to infinity unless the starting number is one. No matter how many times you square one, it will still equal one. But just the slightest bit more or less than one and the iteration of squaring will attract it to the infinitely large or small. The same behavior holds true for complex numbers: numbers just outside of the circle z = 1 on the complex plane will jump off into the infinitely large, complex numbers just inside z = 1 will quickly square into zero.
But the magic comes by adding the constant c (a complex number) to the squaring process and starting from z at zero: z -> z2 + c. Then stable iterations - a set attracted to neither the infinitely small or infinitely large - become possible. The potentially stable Complex numbers lie both outside and inside of the circle of z = 1; specifically on the complex plane they lie between -2.4 and .8 on the real number line, the horizontal x grid, and between -1.2 and +1.2 on the imaginary line, the vertical y grid. These complex numbers in effect stay within the meso-cosmic realm, the world of Man, even if the z -> z2 + c iteration process goes on forever. These numbers are contained within the black of the Mandelbrot fractal.
MANDELBROT SET SHOWN ON THE COMPLEX PLANE
In the Mandelbrot formula z -> z2 + c, where you always start the iterative process with z equals zero, and c equaling any complex number, an endless series of seemingly random or chaotic numbers are produced. Like the weather, the stock market and other chaotic systems, negligible changes in quantities, coupled with feedback, can produce unexpected chaotic effects. The behavior of the complex numbers thus mirrors the behavior of the fourth dimension, the real world where chaos is obvious or lurks behind the most ordered of systems.
With some values of c the iterative process immediately begins to exponentially increase or fall into infinity. These numbers are completely outside of the Mandelbrot set of "meso-cosmic" dynamics. With other values of c the iterative process is stable for a number of repetitions, and only later in the dynamic process are they attracted to infinity. These are the unstable strange attractor numbers just on the outside edge of the Mandelbrot set. They are shown on computer graphics with colors or shades of grey according to the number of stable iterations. The values of c which remain stable, repeating as a finite number forever, never attracted to infinity, and thus within the mesocosmic set -- the Mandelbrot set -- are plotted as black.
Illustrations of how the calculation of z -> z2 + c works with simple values for c are contained within the Appendix section "The Mathematics of the Mandelbrot Formula and the Workings of Numbers and Vectors in the Complex Plane". There you will see how some iterations of complex numbers like 1 -1i run off into infinity from the start, just like all of the real numbers. Other complex numbers are always stable like -1 +0i. Other complex numbers stay stable for many iterations, and then only further into the process do they unpredictably begin to start to increase or decrease exponentially (eg. .37 +4i stays stable for 12 iterations). These are the numbers on the edge of inclusion of the stable numbers shown in black. Chaos enters into the iteration because out of the potentially infinite number of complex numbers in the window of -2.4 to .8 along the horizontal real number axis, and -1.2 to 1.2 along the vertical imaginary number axis, there are an infinite subset on the edge which are subject to the unpredictable strange attractor. All that we know about these edge numbers is that if the z produced by any iteration lies outside of a circle with a radius of 2 on the complex plane, then the subsequent z values will go to infinity, and there is no need to continue the process. This is further explained in the Appendix section on the math.
By using a computer you can escape the normal limitations of human time. You can try a very large number of different complex numbers and iterate them to see what kind they may be. Under the Mandelbrot formula you start with z equals zero and then try different values for c. When a particular value of c is attracted to infinity - produces a value for z greater than 2 - then you stop that iteration, go back to z equals zero again, and try another c, and so on, over and over again, millions and millions of times as only a computer can do.
Mandelbrot was the first to discover that by using zero as the base z for each iteration, and trying a large number of the possible complex numbers with a computer on a trial and error basis, that he could define the set of stable complex numbers graphically by plotting their location on the complex plane. This is exactly what the Mandelbrot figure is. Along with this discovery came the surprise realization of the beauty and fractal recursive nature of these numbers when displayed graphically. The black parts of the Mandelbrot fractal plot the stable iterations on the complex plane. When a complex number is attracted to infinity, small or large, it is either not plotted on the graph or is shown as a color or shade of grey according to the number of iterations it takes before the complex number begins its exponential spiral into infinity.
Every point in the plane of complex numbers is either outside the Mandelbrot set, infinite, or inside of it, finite. The Mandelbrot fractal thus portrays two-dimensionally the infinity between the whole numbers zero and one, the potential and the actual. This is the meso-cosmic world of Man and the basis of all computer operations. The border which defines our area between the finite and the infinite - where our potential can come into actuality - is impossible to determine exactly. It is subject to the strange attractor. You never know when you may fall into or out of it, or how. The closer you look, the more you magnify by choosing a new c close to the last one chosen, the more the fractal complexities repeat with recognizable patterns - but rarely identical - to define an infinitely irregular border.
Only by plotting these numbers in time using an iterative process and two-dimensional representation is the hidden order and great beauty of the fourth dimensional complex numbers revealed. The infinitely recursive nature of the Mandelbrot fractal is the truly astonishing feature of the Mandelbrot and other fractal sets. Infinitely recursive means that the basic shapes of the overall form repeat themselves, but with variations, no matter how close you look at the detail. There is self similarity or self affinity from one scale to the next. As you magnify and look deeper and deeper into the microcosm of the figure, you find the same basic forms are repeated, but are still different and unique. At each scale the fractal is viewed there is a consonance of similarity with the original form, a repeating self similarity.
The primary shape of the Mandelbrot fractal are the two black blobs or warts, called "atoms" by Mandelbrot. The large kind of heart shaped black blob on the right is called a "cardioid", and the smaller black figure on its left is "disk" like. Both the cardioid and the disk each have an infinity of smaller black disk like shapes surrounding them, and each of these smaller black disks in turn has an infinity of similar and still smaller black disks around it, and so forth ad infinitum.
To the left of the large atom, extending from a line to the left of the large sphere you will find another smaller cardioid; magnifying you will see more and more cardioids radiating out all over the large atom, and out again from each of the smaller atoms, and so forth, again to infinity. The black atoms, which plot the complex numbers within the stable set, are infinitely recursive, or self similar. So too are the colored shapes next to the black atoms. The geometric shapes repeat with slight variations in various sizes approaching the infinitely small as the details of the edge of the set are magnified.
Study of the images makes this all clear. Pictures of the Mandelbrot set and others fractals can now be found in many books and videos. We suggest you seek them out and immerse yourself in this beautiful geometric worlds. To provide some immediate, direct visual input we include here a few of our favorite images of the Mandelbrot fractal. These are just a few out of the billions of different Mandelbrot shapes, but they show some of its beauty, and illustrate fractal recursiveness. The colors on these illustrations help us to appreciate the beauty and magnificence of the Mandelbrot. The sequences on the next two pages show a zoom into the depth of infinite detail of the Mandelbrot.
The Mandelbrot set is holistic and continuous. All of the black atoms of the Mandelbrot fractal are touching and connected. Most of the connections are too small to be visible. They are connected by extremely thin lines or filaments that require millions of scales of magnification to become visible. Mandelbrot contains an infinite number of these black cardioid atoms connected by filaments.
Other sets of complex numbers use the same iterative formula z -> z2 + c to produce sets of static complex numbers and fractals, but they do so without falling back to zero. In other words, the iteration does not start with z = 0 after the complex vector falls into infinity. Instead, the formula keeps the same value for c and uses a new value for the beginning z. These other sets and fractals not using zero are called Julia sets. They are named after the French mathematician Gaston Julia. He was the first person to begin studying iteration with complex numbers in the 1920's.
Unlike the Mandelbrot set which samples all values of c to test whether they are attracted to infinity or not, the Julia sets are based on a fixed value for c and the value of the beginning z less than 2 is varied over time. There are an infinite number of different Julia sets possible. But unlike the Mandelbrot atoms which are all grounded in zero and connected with each other in the Complex Plane, the different Julia sets are disconnected with each other. Further, some of the Julia sets are internally disconnected, falling apart like Cantor's dust. For that reason the internally disconnected Julias are sometimes called Cantor sets. Here is an example of an internally connected Julia set.
Again, there are many good books on fractals available with beautiful illustrations of Cantor and Julia sets. The following is another of the better known Julia sets, called the "Dragon". It is accompanied with a close up of the basic fractal pattern. Again, the branching swirls shown in the close up are infinitely recursive. The deeper and closer you look, the more swirls you will find. This same pattern repeats forever over infinite scales of magnitude. It goes on and on forever. You can see this recursive self similarity throughout the designs of all fractal patterns.
Even without the advantage of computer plotting Julia and a few other mathematicians in the 1920's knew that iteration of complex numbers produced fractals with recursive features. They did not, however, comprehend the full significance of the process, nor did they think to stabilize the dynamics in zero. Mandelbrot was the first to realize that this was the geometry of nature, the reality of the fourth dimension, and not just some meaningless bizarre fluke of mathematics. His discovery was based on the eighth criteria - ZERO - AWARENESS - tying all of the finite complex numbers together by grounding the z in zero and floating the c.
Mandelbrot discovered that his holistic fractal governs and defines all of the Julia sets. Julia sets whose value of c lie within the set of the Mandelbrot fractal, within the black atoms, are internally connected, holistic. But Julia sets whose value of c lies outside of the Mandelbrot fractal on the plane of complex numbers are fragmented into infinitely many pieces. The further from the black edge, the quicker the Julia sets break up and fall into dust. The Julia sets with a value of c near the inside and outside of the black border, the edge of the Mandelbrot set, are the most complex and beautiful of all. The next diagram shows where several different connected Julia sets are located just inside the black zero of the Mandelbrot. Yes, as shown, Julia sets include a straight line, as well as a circle.
METHODS/EXPERIMENTS: Many computer programs exist today to create a variety of different fractals, including the Mandelbrot set. This is a popular topic on the Internet. For instance, the School of Wisdom websites contain more information and pictures on fractals, including animated zooms of the Mandelbrot set. Many fractal generating programs can be obtained for free, or a modest charge. Just run an Internet search of "fractals" for the latest webs. One popular program you will probably find is "Fractint" by a team of programmers available on the Net. A book with great pictures Fractals: The Patterns of Chaos by John Briggs references several more programs for different computers. The catalogue Media Magic contains a complete selection of Chaos books, software, calendars, videos, etc. from Nicasio, California, 800-882-8284.
By experimenting with fractals, and magnification over scales, you gain a firsthand experience of fractal scaling, self similarity, recursiveness and infinity. The great beauty and infinite complexity of the Mandelbrot set and the Julia sets are intriguing to all who see them, even if they are not aware of their philosophic significance. It is also important to see many of the computer fractals and the fractals in the world of Nature. This will provide a deeper understanding and intuitive sense of what is meant by fractal recursiveness. This is a key to understanding the hidden order which appears out of the Chaos of our everyday lives.
Think about events which have happened in your life where seemingly random events later took on meaning, or situations where order appeared out of chaos, or the reverse, where a hidden chaos appeared out of what seemed to be perfect order.
Read some of the many books out on the Chaos Theories, and study the color pictures of fractals included in most of these books. Good books to start with are: The Turbulent Mirror by John Briggs and F. David Peat, and CHAOS: Making A New Science by James Glick. When you are ready try the beautiful, important, but difficult books by Benoit B. Mandelbrot himself: the book written for the "general reader" The Fractal Geometry of Nature, or his even more technical work Fractals, Form, Chance and Dimension. Mandelbrot has also contributed essays to An Eye For Fractals by Michael McGuire, and The Beauty of Fractals by H.-O. Peitgen and P.H. Richter, which has the best color prints now available of the Mandelbrot and Julia fractals.
Video tapes of fractals are also helpful. There are many available, for instance Fractals: An Animated Discussion with computer graphics and interviews of Mandelbrot and Edward Lorenz. The Fractal Universe video has excellent animations from a variety of scientists and video artists.
The recent breakthroughs in the new interdisciplinary science of Chaos, and its discovery of the four "Attractors" (formerly called forces) which make sense of the Chaos, help us to understand the basic criteria of Wisdom and make sense of our world. The hidden order and similarity over scales revealed graphically in the otherwise random collection of numbers in the Mandelbrot and Julia sets is based on one of these four Attractors, the Strange Attractor. The other three attractors, which likewise bring hidden order out of chaos, follow the first, second and third dimensions. They are called the Point Attractor, the Circuit (or sometimes Cycle) Attractor and the Torus Attractor. As humans living in the fourth dimension we are at our best when we avoid their influences and follow only the spontaneity and freedom of the Strange Attractor. Only in this way can we live autonomously in the moment, in tune with what the Chinese call the Tao, the Way, the flow of forces in the fourth dimension.
The four attractors act on all levels of reality to form cosmos out of chaos. By understanding these attractors, and how they work, we can more easily make sense of what is happening in the real world. The world is not really ordered as previously believed, it is fundamentally disordered - chaotic, but it contains forces or attractors of cosmos that create patterns of order over time.
The four attractors correspond to the four basic ordering principals of reality: Energy, Consciousness, Matter and Self Organization. The four fundamentals can in turn be understood as summations of the eight criteria:
Energy as Feeling and Spirit;
Consciousness as Thinking and Soul;
Matter as Sensing and Body; and,
Self Organization as Willing and Awareness.
This knowledge is summarized in the following chart.
To really understand the Attractors we have to have a spiritual understanding of space and time. As to space, we must understand how space is the original force - in Sanskrit called Brahman, in Chinese Wu Chi, in Peruvian and Japanese Ki - which creates the world through the point. Real spiritual insight into this can only come from direct experience of CHI. The rest of this book, particularly in Chapter Four, contains information on CHI, sometimes called kinesthetic body in psychology. A few methods and exercises from psychology and the martial arts are also provided to allow a direct experience of CHI. Try these experiments to gain an experiential understanding of space as original force. Then your understanding of the attractors will improve.
A spiritual understanding of time entails realization that time is rhythmical. It is the connections or order you make in the fields of the four attractors: Point attractor - Energy - Feeling/Spirit; Cycle attractor -Consciousness - Thinking/Soul; Torus attractor - Matter - Sensing/Body; Strange attractor - Self Organization - Willing/Awareness.
In the computer the electric current automatically creates the iteration. With Man it is not so easy. We ourselves have to return to Zero - Awareness - to start a new iteration. Awareness is sacred space (called "Wakan" by the Native American Lakota tribe; "Mana" by the Polynesians). You connect with as soon as you attain the center of your true Self, called by the Japanese - Hara. Cosmologically Hara corresponds to the center of the Earth. That is why traditional initiations awaken the force of gravity. Gravity, according to Newton's law, is the result of reciprocal mass attraction. The nearer you get to the center, the weaker the force will be. At the very center it will be nil. But if the light touches from outside the Self, then the center - Self - will radiate.
The Other sees the Self, but the Time-Ego does not, because the Self is hidden from the Ego behind dream and sleep. Only in yoga and meditation with PrimaSounds and the like can this be overcome and the immortal Being be energized.
Thus the difficulty which many scientists have in understanding Chaos is not mental, but existential. Their consciousness (Cycle attractor) refuses the Strange attractor, which means total individual responsibility. They cannot ground themselves in Zero and experience the true meaning of space and time. As they cannot find their inner core - Awareness/God - they can only see Chaos from the outside. They cannot make the jump from knowledge to Wisdom, to inside the black. Without this anchor they lack the experiential insight - and the confidence and Wisdom this brings - needed to live on the edge where great discoveries are made. They only see isolated Julia and Cantor sets. They miss the pattern which only comes from awareness of the whole, the Mandelbrot fractal.
The Point Like attractor corresponds to Energy, the function of Feeling and the realm of Spirit. With this attractor in play a person is invariably drawn to one particular activity, or repelled from another, like the positive or negative poles of electromagnetic energy. There is also sometimes a point just in between attraction and repulsion, the saddle point, when the energies are in balance, just before one force becomes stronger than the other.
With the Point attractor there is typically a fixation on one desire, or revulsion, and all else is put aside until it is satisfied or destroyed. With the positive attraction force all roads seem to lead to the same destination. With the negative repelling, all lead from the same place. A positive magnet drawn to negative, a pendulum slowing down with friction and air resistance, or more graphically, a young male dog around a bitch in heat, all demonstrate the workings of the point attractor. It is a black-white, good-bad, single minded attractor, except in the rare instances of the saddle point.
The Circuit or Cycle attractor corresponds to Consciousness, the function of Thinking and realm of Soul. With this attractor a person is drawn first to one thing and then to another, like a circling magnet, first attracting then repelling then attracting again. Here there is a cycling back and forth from a set of two or more activities. There is some regularity and simplicity to the cyclic events. An example is a desire to sleep at the end of a day, which when gratified naturally leads to a desire for activity at the beginning of a new day, followed much later by a desire to sleep again, etc. In Nature it can be seen in many ways; for instance, the predator prey systems where the respective predator prey populations cycle up and down in relation to the other.
The Cycle attractor is more complex than the simple attraction or repulsion type point attractor. Like thinking it sees both sides and tends to include a third; for example, the synthesis coming out of the thesis and anti-thesis.
The Torus attractor corresponds to three-dimensional Matter, the function of Sensing and the realm of Body. This is a more complex cycling which moves forward and so is different while it repeats itself. There is a high degree of irregularity and complexity in the pattern of the Torus attractor, particularly when compared to the Circuit or Point attractors. But unlike the Strange attractor, a pattern can still be found and predictions made. Mathematically the Torus is three dimensional and is shaped like a large donut or bagel. It is made up of a spiraling circle on many planes which may, or may not, eventually hook up with itself after completing one or more full revolutions.
An example would be the more complex set of attracting events which occur to a person on many levels over a course of a year, and repeat again, year in and year out, like the desire to swim each summer, hike each fall, and eat and drink too much on holidays. In Nature it is shown for instance by the complex interaction of a number of interdependent species: the population of one predator species relates to that of the prey of its prey. For example, the size of the insect population effects the size of the frog population, which effects the size of one of their predators, the trout, which in turn effects their predators, the pike. Unfortunately, most humans are also subject to the complex but predictable influences of the Torus attractor, or the even more simplistic influences of the cycle or point attractor.
We need to escape from the deterministic influence of the point, circuit and torus attractors into the freedom and spontaneity inherent in the Strange Attractor. The Mandelbrot fractal is one of an infinite number of forms which the Strange Attractors assumes. The Feigenbaum fractal shown above is another geometric form found hidden in time throughout nature which is based on the Strange Attractor. Being fractal, all of these geometric forms are recursively self similar over infinte scales of magnitude.
There is no apparent order at all to the actions of the Strange attractor. On the surface it appears to be pure chaos, but nevertheless there is order of a subtle kind which only appears over time when looked at in the right perspective. It is an order of self similarity, sometimes even idenity, where a geometric shape allows for comprehension. It is an order where infinity is constrained, but never contained. The Strange Attractors correspond to Self Organization, Willing, Awareness and the fourth dimension of fractal forms where chance is inherent. Feigenbaum and Mandelbrot Although these Strange Attractors may look very different from each other, they frequently have a hidden relationship with each other. This is shown for instance, in the diagram to the left, comparing the Feigenbaum and Mandelbrot fractals. The relationships can take near infinite forms and are not always as obvious as the proportional one shown here.
Chaos, with a subtle unpredictable order by the Strange Attractor, is also called "turbulence". One of its important characteristics is great sensitivity to initial conditions. This is exemplified by the famous example of Edward Lorenz: the wing movements of a butterfly in Peru may later through an extremely complex series of unpredictably linked events magnify air movements and ultimately cause a hurricane in Texas. The so called "Butterfly Effect" has been proven to apply in many dynamic systems, including the weather, where even the smallest of changes can trigger a chain reaction of unexpected exponential consequences.
In our world, the fourth dimension of complex dynamics, there is extreme sensitivity to initial conditions. This means that even the smallest effort can unexpectedly multiply and have a great impact. It is beyond our capacity to predict what will happen, what little action might, or might not, lead to profound change. The insights of Chaos, where it has been proven that negligible changes in chaotic systems can produce significant unexpected results, stands as a new inspiration to all individuals. If you are in the world starting something new, you might make a big difference. No one can know for sure - their straw, no matter how small, might just be the one to break the camel's back. Your beginning efforts of personal transformation may be important to the entire world - like the butterfly's wings. If the timing and connections are just right, your new little work may well lead to a hurricane of change. History is replete with examples of this, both for good and bad.
When we are subject to the first three attractors we are manipulated; only in the Strange attractors can we be free. So we all have to become Strange attractors as shown for instance by the Lorenz fractal below. Here a point cycles back and forth in the figure eight shape of infinity, but it never repeats its tracks. Even if the process were to go on to infinity, the lines would never cross over or repeat each other. There is always a new path to reach the other side. The Lorenz fractal is a geometric depiction of a spiral loop into infinity, the never ending cycle back and forth, yet forward and different. This is the back and forth cycle between Yang and Yin, left and right brain, the real and the possible.
We must learn to exist at the meeting point between the real and the possible, or as Don Juan explained to Castenada "the crack between the worlds". We are then within the force of the Strange attractor and can see the hidden order behind the seeming chaos. It is a time of no time, a flowing peak experience where all seems to go right by itself, effortlessly. It is a time when dreams and wishes are fulfilled that you didn't even know you had.
The Feigenbaum, Lorenz, Mandelbrot and other fractals portray in two dimensions the infinity between zero and one, the potential and the actual. Mandelbrot's formula goes even further to provide a mathematical map to navigate in the crack between the worlds, to cope with Chaos and bring our potential into actuality. Mandelbrot's inspiration 0 : z -> z2+ c brings order from chaos in the fourth-dimensional world of Complex numbers in time, in iteration. We too live in the complex world of the fourth dimension, in a space-time continuum of turbulence and chance. Thus Mandelbrot's discovery should also apply in a fractal recursive manner to provide us with the formula for coherent living in a chaotic world. The philosophy suggested is a dynamic and pragmatic process of feedback, experimentation, detachment and grounding in Awareness for constant renewal.
If an action - Will - goes astray, does not work, falls off into infinity, then stop that activity. Once the mistake is obvious and certain, choose to let go of the failure. Then choose again to take a chance - to return to Zero, pure Awareness - God - where inspiration for a new action will come again. When the idea comes, go for it, don't wait for certainty or you may never act at all and life will surely pass you by. Seldom does inspiration come with the certainty of a burning bush. Choose to take a chance, try it without attachment, just for the hell of it.
Then travel the new vector - the new complex number - the new activity - to see whether or not it is part of a larger order, or part of chaos - whether or not it is within the black of the Mandelbrot fractal. Only time - constant iterations - will tell. You can only discover the force of the Strange attractor by doing -- trial and error with feedback - learning from your mistakes and always beginning anew from Awareness.
If the new action is successful - leads to greater order and coherence - continue to follow it. It may be perfectly stable, well within the black, and like a successful business after it is well started, the activity can eventually go on with others, or by itself, without your attending to it. So again the activity may be finished for you. There are probably many other things remaining for you to do. Keep experimenting, changing, try many things at once. Otherwise you may stagnate in success - drown in black ink - and never see the big picture, or again experience the fractal beauty and excitement of living on the edge. Thus an established success, like failure, should lead to freedom, to a new activity into the unknown future. It is the destiny of actualizing Man to pioneer the frontiers of cosmos in the midst of chaos.
Sometimes a new activity starts off ordered and successful, but later falls apart and iterates into chaos. Don't give up at the first sign of difficulty, because the beginnings are always hard. But when you are sure that it will not work, that it is definitely on its way to nowhere, then leave it. Re-center yourself in Awareness to find a new direction, and try again with something different. Don't be frustrated, eventually even Edison found the light.
Even when a person is not consciously participating in this process, is uncentered and unaware, the actions of the Strange attractor can still manifest in the desire to do the unexpected - the wild hair, fluke decision. When under its influence the pull seems to be towards disorder and serendipity. The hidden order lurking behind the Strange attractor may only appear much later, or through synchronicity with other events. An example might be a desire to make a career change which makes no sense at the time, but later in life is recognized as an essential step to a larger order; or perhaps you are seized by a sudden eccentric desire to go to a place never seen before and there meet your future wife who is also there by chance. Examples of this hidden order in chaos abound in life and nature, and even in manmade things such as fluctuations in the price of cotton as the research of Mandelbrot, Lorenz and others has shown.
In addition to fractals, the other symbol of the fourth dimension - the Chaotic real world where we all live and take chances - is the Hypercube which we first discussed. It uses the Euclidian forms to bridge the third dimension with the fourth. The full meaning of the hypercube can now be shown. The four diagonals of the Hypercube represent the four attractors (Point, Circuit, Torus, Strange) and four basic Elements (Energy, Consciousness, Matter, Self Organization). The eight directions created by the four diagonals represent the seven basic functions and realms (sensing, thinking, feeling, willing, body, soul, spirit), plus the zero dimension of Awareness. Another alignment of the directions created by the Native Americans will be discussed in Chapter Eight. The Native Americans and others think in terms of ten directions, adding up and down, and for that and other reasons the alignment with the function and realms is different. The chart below shows the Eight Directions of the hypercude.
METHODS/EXPERIMENTS: Think of all of the chaos in your world today, in your past. Is your life too ordered and rigid, or too chaotic? Which do you think you need more of now? Which do you think the world needs more of now? Look back on your life and recognize how the four different attractors were in effect at various times to bring order out of the chaos. Then examine your current life situations and analyze how each type of attractor is working to bring order to your life. Is any attractor prevalent, are any missing? How can you recognize when a strange attractor is at work? Does it have a special signature, or pattern, or sense, which can allow you to recognize it when it first appears? Do any of the other attractors?
A word of warning here about the superstition trap. Attractors are real and can be experienced directly. This is scientific fact. Chaos is ordered by the attractors. When we act with the attractors to bring order, even with the Strange attractor, we are, like Nature herself, using constrained chance, structured chance. There is a fundamental and important difference between the reliance upon constrained chance and Strange attractors and reliance upon blind luck and superstition. The lazy will take the latter course and convince themselves they are living on a higher plane. These same people will complain to God when their luck turns. Do not confuse Chance and Choice with chance and more chance. There is a fine line here; gambling and superstition are a real danger for some and must be avoided.
We are not advocating gambling in any form, nor chaos for the hell of it, nor silly superstitions. Strange attractors are very real phenomena; they can be observed over time and are precisely, mathematically calculable. We are pointing to an open door, but counseling a philosophy of direct perception and verification for yourself. Make your own luck. Work to be prepared to take advantage of the chances when they come. Look for the chances, know where and how to look. Know how to make a choice. This is a philosophy of self reliance and inner coherence based on reason, but it goes beyond the limits of reason to embrace the whole world, to know Chaos.
The full dynamics and meaning of the fourth dimensional numbers, the complex and fractal numbers, was only recently discovered with the Science of Chaos. The Chaos theories in turn did not come about until man had the ability to create computerized graphic representations. These graphics revealed the geometric scaling inherent in Complex Numbers. But even before computers and the Chaos discoveries, Nature had shown its fractal character to Man. We discovered it through sound and music. It has to do with the phenomena of overtones and undertones and the octave.
A string (1 dimension) in vibrating will form a series of sub-vibrations, called overtones
1 2 3 4 5 6 7 8 9 10
c c g c e g x c d e
The overtones are the places on the string where the nodes and sub-waves or harmonics naturally form. In other words, the sounding of any one tone will naturally create certain fractal tones as shown above with the overtones.
The fractal tones follow the fractions of the rational numbers. For example, when the tone C is played all of the overtones of C are also created. The string vibrating at the frequency called C includes fractal waves such as a third that of C - called G or a fifth, and a fifth that of C - called E or a major third.
Third |____|___|____| (called "fifth" in music)
Fifth |_1/5 _2/5_3/5 _4/5__| (called "major third")
The animated graphic should help non-musicians get a better feel for this phenomena by showing the movements on a string:
The undertones are harmonics that appear if the string is lengthened:
c c f c a f x c b a 1 2 3 4 5 6 7 8 9 10
The tones are vibrations and intervals. The intervals are based on scaling. Between the first nine tones the following intervals occur:
0 - 1 Original Tone
Octave 1 - 2
Fifth 2 - 3
Fourth 3 - 4
Major Third 4 - 5
Minor Third 5 - 6
6 - 7 (within hearing limits of major third)
7 - 8 (within hearing limits of major second)
Major Second 8 - 9
The vibrations of sound creating the octave and consonant notes exemplify aurally the fourth-dimensional fractal laws of scaling and self similarity.
The Major Second, Major and Minor Third reach the identity of the octave inside of one span. The cycle of fifths reaches identity after 12 steps covering 84 half tones; the cycle of fourths with the same tone values covers 60 half tones.
But in order to reach the octave exactly, this interval having no tolerance, the musical scale of 12 tones has to be tempered adding the triton 2: 1, and the minor second 12 2: 1.
The twelve tone circle has its center in the 1/1 diagonal, and its beginning in the zero point of CHI. The 12 notes and intervals are used on any modern piano keyboard. This is shown on the Music Wheel which follows:
The second law of music is the physical scale, based on the major second. The tolerance of resonance is 9/8 + 10/9. They are experienced as the same interval. Based on this hearing tolerance the musical scale is made up of seven whole tones.
Hearing is experiencing our world: the lowest tone, 16 hertz (low C), has a length of 22 meters; middle C with 512 hertz has a length of .69 of a meter; and the highest perceptible tone of around 20,000 hertz has a length of about one half of a millimeter.
TONE SCALE (hertz)
. . .
Tone 16 Beta range Waking
12 Alpha range Reflection
4 Theta range Dream
2 Delta range Deep Sleep
METHODS/EXPERIMENTS: Try making and hearing the lowest and highest tones you can perceive. Try listening as hard and deep as you can; how many sounds can you hear; what is the source of each. With a metronome or timepiece try drumming at the beta, alpha, theta and delta rates and see if continual drumming at one rhythm induces the state of consciousness associated with the rate. See Chapter Four for information on these brain wave states of consciousness. Play with a piano keyboard and hear the seven and twelve tone scales.
One interval does not fit in the 12 tone system, the natural seventh. Its frequency and harmonics are dissonant with all of the other basic fractions. For this reason it is said to have no place in Music and is either excluded completely, or when rarely used it is called the "leading tone" and sets up dissonant tension.
When the natural seventh is taken as the basic interval, to the exclusion of the others which normally make up our musical scale, a completely new scale is created whereby the octave is divided into five intervals. The new tone frequencies and intervals created thereby we call PrimaSounds. This new pentatonic scale opens hearing to the inner Universe and in most ancient civilizations this music was held sacred. This type of music is fractal, based on zero, the fourth dimension and the Strange attractor. It has no melodies, rhythms or other forms or order found in other music. It sounds almost completely chaotic, unpredictable, yet there is a fractal order with the link to your own being which makes the sounds soothing and leaves you serene. Being attuned to the primal energies of the soul it has the power to throw you into the zero dimension, to open you to the healing influences of the Strange attractor.
PrimaSounds are able to unite all dimensions and states of consciousness through opening the being to Awareness, the Zero dimension. The five notes in the PrimaSounds scale are named after the vowels: A E I O U. (Note that the vowel scale designation is arbitrary. This designation was made by Keyserling in 1971 before the discoveries of Tomatis, and does not correspond with Tomatis vowel/chakra alignments.) This new/ancient music is the tonal key to becoming human, to the realization of our full potential in all dimensions. More information on PrimaSounds and its discovery can be found in Chakra Music: the story of PrimaSounds (Keyserling-Losey) (School of Wisdom, Volume 4). PrimaSounds work in the element of Energy. Our center of attention normally exists in another element, in Consciousness. In Consciousness the seven basic components are the four functions and three realms. These seven have a natural internal relationship whereby Sensing corresponds with Body, Thinking with Soul, and Feeling with Mind. Willing stands on its own in the center, connected with Zero, Awareness. For example, the senses are sharpest when perceiving the physical world, as compared to sensing other people (soul) or an idea (spirit). Conversely the Body is most directly apprehended by sensing. Thinking is most acute when in dialogue with other people, as opposed to thinking with a thing or abstraction. Conversely the Soul is understood best by thinking. Feeling, and its twin sister imagination, are at home in the Spirit, where free reign is given to it. Feeling in the body or soul is more limited and often negative. Conversely the Spirit is most easily accessed by the Feelings. Thinking about the Spirit frequently leads to erudite nonsense. Spirit must first be felt before it can be seen or put into words.
When these seven components are moved from the scale of Consciousness to the scale of Energy, there is a fractal variation in the internal alignment. In the world of Energy where PrimaSounds vibrate in accord with the seven energy centers, or chakras, there is a new alignment of the original seven. Now Sensing corresponds with Soul, not Body, and Thinking corresponds with Spirit, not Soul. Feeling now acts on Body, instead of Spirit. Only Willing remains the same, in the center related to Awareness, the Zero outside of and underlying the seven.
As a consequence of this fractal variation, PrimaSounds should be listened to by Feeling the effects of the vibrations on the body, Sensing the energy of soul stimulation, and Thinking of the Spirit, a holistic idea of a concept, of infinity, or no-thinking. This realignment is shown on the following chart.
METHODS/EXPERIMENTS: Try experiencing this now by listening to one of the School of Wisdom PrimaSounds CDs. Detailed listening instructions are included with the music, and with the book Chakra Music (vol.4). For more guidance on PrimaSounds see the PrimaSounds webs and Chapter 7 of Chance and Choice.
The most reliable way we know to directly experience the four chaos attractors - as opposed to just understanding them - is through sound. The attractors are perceived as forming order out of chaos in the following manner:
- Body-Sensing. Torus. RHYTHM.
Based on 9 Rhythms.
- Soul-Thinking. Circuit. HARMONY.
Based on the tenfold and sevenfold overtone and undertone scale.
- Spirit-Feeling. Point. MELODY.
Based on the temperated cycle of fifths.
- Awareness-Willing. Strange. FRACTAL.
Based on the nodes of the natural seventh.
(1) - RHYTHM. Rhythm follows the law of the octave so that there are 9 fundamental rhythms, including their squares, ie:
2 - 4 - 8 - 16 - 32
3 - 6 - 12 - 24 - 48
5 - 10 - 20 - 40 - 80
Listening to good rhythms, as in Africa, your body yearns to participate and all tiredness goes away. You move into the sway of the Torus attractor and naturally invigorate the body, matter.
(2)- HARMONY. The intervals of harmony are heard "in tune" or "out of tune"; it hurts the ears if somebody plays false. Its basis is the overtone and undertone scales.
Undertones 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10
Overtones a b c x f a c f c c c g c e g x c d e
The ninth tone is the whole tone. Thus from the interval point of view the harmonic scale is visualized in the Enneagram shown below in the inside of the Wheel. The two angles of the triangle in the Enneagram constitute halftones - in Gurdjieff terms the change of direction. The physiological scale thus has seven tones, starting and ending with c. Major c d e (f) (g) a b c Minor
Undertones are space like chords, overtones are time like sequences. According to great Austrian composer Joseph Matthias Hauer, one of Keyserling's primary teachers, each person has their own fundamental harmonic - bass, baritone, tenor or soprano. For this reason in 12 tone music there are four "voices". By tuning into your inner harmonic, which is in one of these four voices, you can reach the Cycle attractor. Your inner harmonic tunes you into the Cycle attractor. If you awake to its influences, you can sublimate your natural instincts. In this way you can escape from manipulation by the instincts and associative thinking. You can instead find your inner centeredness and autonomous thinking. The instincts are sublimated by being made conscious and harnessed to the will. The stream of associative ego thinking, which is normally driven by the instincts, can then come to an end. When it does, real thinking can take over. Thinking controlled from a harmonic center - the Self in the center rather than the ego on the periphery. The complementary nature of polar opposites can then be realized. Once we are freed from enslavement to unconscious instincts -- the ego and the stream of egocentric mental chatter that goes with it - the Cycle attractor can be used to clarify and integrate Consciousness with insightful, holistic thinking. All this can be aided by the harmonics of music with which you can awaken to the Cycle attractor.
(3) - MELODY. In traditional European music theory, major and minor scale, the overtones and undertones, are the basic foundation of music. For instance, a particular symphony is identified as being in D minor. Joseph Hauer showed that this was wrong, that the real basis of music is in Melody - not melody as mere thinking combinations, but Melody as inspired "tone gestalts". This Melody is heard all at once as inspiration, an event with which every good composer is familiar. Thus with Melody you can reach the Point Attractor, truly feeling the spirit. You thereby open up to inspiration, drawn to the zero point of Awareness - the formless infinite, from which new form emerges. With melody and the Point attractor your Energy can be strengthened and tuned.
Taking the twelve tone scale, and without regard to the tone values, Hauer discovered that there are 479,000,000 possible melodies in time. The melodies can be harmonized because of the "diatonic comma", 81/80, in the tempered scale. This scale was first invented in China 4,000 years ago, and in Europe for keyboard instruments by Werckmeister in the Seventeenth century. Bach explained that he created the well tempered piano as a basis for future spiritual democracy. He wanted to show that there are only melodical voices and no accompaniment. Thus the whole melodic structure is shown on the outside of the Wheel as the cycle of fifths. Each of the twelve tones has seven parts - the seven octaves. The melodies which can be produced by these tones in time are the doorway to the Point attractor.
(4) - FRACTAL MUSIC: 7th HARMONIC/PRIMA SOUNDS. The natural seventh interval, excluded in the diatonic and twelve tone scales, is the secret basis of Esoteric Music. This was described but not explained by Keyserling's teacher, George Gurdjieff. The quintessence of true fractal Music lies in its attunement to the seventh harmonic. The seventh overtone, when tuned to alpha, and played next to a human, produces longitudinal sound energies which interface with transversal energies of the human energy field. The result is the creation of standing wave patterns - a sound vortex around a point of transversal energy vibration.
Resonance with the seventh harmonic sounds can tune you into the Strange attractor. The point and hypercube - the zero dimension and the fourth dimension - originate the geometry of the Strange attractor. Together as previously shown they fill out the intervals between the dimensions, the fractal dimensions between 0 and 1, 1 and 2, 2 and 3, and 3 and 4. The tones which are based exclusively on the natural seventh - the pentatonic scale of Prima Sounds - can be combined spontaneously from out of Awareness to create pure Fractal Music. Fractal Music, previously known as Esoteric Music, when so created can then relate the being living in the fourth dimension with the infinity in the zero dimension and the fractal dimensions. Listening to such music invokes the Strange attractor. This experience can liberate you from past habits and the other attractors. Then you can self organize autonomously, in tune with the entire Universe and the spirit of the times.
With Prima Sounds and the Strange attractor that comes with it, the entire Mind can be cleansed. All centers - sensing, thinking, feeling, willing, body, soul, spirit - can be changed from dependence on the existing cosmos, to ordering the chaos. The chakras are then opened, re-tuned and integrated as is further explained in Chapter 7 and in Chakra Music (Vol. 4 School of Wisdom). Then you can participate in the emerging Cosmic Humanity where there are no elites, no sacred way, but only different styles of being, living and working: the polyphony of the rainbow civilization.
Fractal scaling is not only revealed in Sound and Music, but also in Light and Color. The transversal waves of light have three systems. They can be visualized in the rainbow, the globe of the Earth, and the primary colors.
The primary colors in pigments are red, blue and yellow.
Out of these all other colors can be mixed.
The rainbow consists of seven colors between 3,800 and 7,600 Angstrom and the reciprocal vibrations.
The synthesis is visualized in the globe.
Between north and south of the color globe is the grey axis. The center is grey in matter. In light opposing colors make white. The clear colors are on the surface. The unclear ones are underneath towards to the center. Every possible color can thus be defined by its location on the color globe.
The Eye follows the parameters of black and white (focus), and of red and green, violet and yellow (peripheral vision). Some people have a red-green blindness, or a blue-yellow blindness. Purple is absent in the spectrum, but the purple body is the basis of color vision.
Thus the structure and systemics of the senses constitute the phenomenological truth, the sole experienced reality. Goethe said correctly: "The highest Wisdom would be to grasp that facts are the theory. Don't look beyond the phenomena, they are the teaching."
METHODS/EXPERIMENTS: Try some experiments with color and quartz crystals. View the splitting of the colors from white light as a crystal is held in the sun. Try gazing at crystals and other gems. You may not see the future, but some subtle inner changes frequently occur. Beyond the ordered world of crystals and gems, look afresh at the natural world - vegetation and weather - and see the fractals, the interplay between cosmos and chaos.
As another experiment create the blackest most perfectly dark room you can one night. Then look hard into the darkness for an extended period of time. Do you see the sparkling red and other colored points of light. It looks something like dust particles, or like what you see when you rub your eyes. It is easiest seen in total darkness, but after you learn to recognize it, you can see it in semidarkness or even in the light. Watch what it does, the hallucinatory shape-shifting qualities. What are you seeing?
Do not take any of your senses for granted, including seeing, and assume that they can not be improved and refined. In reality very few people have fully developed their senses. With some effort and appropriate exercises most of us could perceive the world far better than we do now. Try for instance the yogic "tratak" meditation exercise where you practice silent meditation with the eyes open. Your gaze is fixed on the single point of an external object, such as a flower. You stare silently at the flower, focusing all of your attention on truly seeing it, putting all else aside. Slowly you become one with the flower and see it in a new way.
Also recall the way of "seeing" described by Carlos Castenada in his books on Don Juan. When you shift into the Zero/Fourth dimension it is possible to see energies, the otherwise invisible forces existing in the space time continuum. For instance there are the "luminous egg and fibers" described by Don Juan which surround all humans. Don Juan said that the luminous fibers of some people are strong and bright, others are dark and weak. In Western esotericism we refer to this as seeing auras. In science we refer to this, if at all, as bioenergies or brain waves. "Kirlean" photography developed by East European scientists in the 1970's appears to be able to photograph this energy. In India this energy is called the Chakras, in Japan, Ki. In China it is called Chi and is the basis of all oriental medicine. Many practitioners of Oriental medicine claim that they can see Chi and so diagnose illness. Acupuncture manipulates the flow of Chi by sticking tiny needles in key points in the body where the flow of Chi tends to get blocked. A few specialist in China can not only perceive the Chi of another, they can touch the other's Chi with their own Chi . By infusing their Chi in this way they can cure serious illnesses and disease without any physical contact with the patient at all. Western doctors who have recently been allowed to study oriental medicine in China confirm these incredible reports. The use of Chi is also the basis of the martial arts. We will explore this further in Chapter Five. Can you sense your own Chi, or the Chi of another? Use PrimaSounds and try to see it, to touch it, to move with it.
You may also want to try experimenting with some of the new technologies in the area of vision and consciousness alteration such as Mind Machines, Color Goggles and Ganzfield Effect Glasses. There are many books on the subject and catalogues offering some of the equipment, ranging in price from $50 to $5,000. The best known book is Michael Hutchison's Mega Brain, also try Would The Buddha Wear A Walkman? by Hooper and Teresi.
Do not forget the ancient visual technologies used by the Buddha and others, such as Mandalas. Remnants of Mandalas are found in almost all ancient cultures. Mandalas are geometric patterns, frequently painted with many colors. They are designed to be stared at - contemplated - so as to effect subtle inner changes. They use color and form to directly impact the psyche. Some of the most spectacular Mandalas were created in Tibet where this type of consciousness tool was refined to a high art. There are many books on the subject. The best have many color pictures so that you can try it out yourself.