Number is composed of the nine numerals, basic to all information, uniting geometry and arithmetic, space and time. Their reciprocal relation is clarified through the circle of 5 dimensions, based on the relation of infinity to one. The circle of dimensions summarizes the laws of number and mathematics. The eight points shown on the dimension circle (0 – 7) correspond to the eight directions on the Vector Equilibrium of space time .
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 and 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 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.
The Zero Dimension is the point, the infinitely small place holder. It exists not in space, but in time only. It is the moment between past and future, the subject, zero. It constitutes potentiality, the four space dimensions constitute actuality. The Zero Dimension is the home of Natural Numbers. The subject point, the moment, is Zero, pure Awareness. Its numbers are the natural numerals.
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.
The First Dimension is the LINE. It contains an infinite number of points. Temporally, it represents the Future. The moments of Zero Dimension create the future, forming a trajectory.
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.
The Second Dimension is the PLANE, such as a square. It contains an infinite number of straight lines. In Time, the Second Dimension represents the Present.
The trajectory of moments in the First Dimension (Future) appears like a disk or revolution.
In the second dimension we have the Rational numbers, based on three points, 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 and multiplication in the negative field. 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.
The Third Dimension is the SOLID, such as a cube. It contains an infinite number of planes or squares. Temporally, the Third Dimension represent the Past.
The disc of the Second Dimension (Present) turns one half time around its axis and fills out the sphere of the past.
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:
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 the square root of two 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.
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. In the Time domain, the Fourth Dimension continues the movement of the Third Dimension (Past) to form a wave, constituting fractally the space-time continuum.
THE HYPERCUBE 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 , 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. The number of the diagonals is 4 X 3 = 9, according to the Pythagorean theorem. The four diagonals are 1-5, 2-6, 3-7 and 4-0.
The Four Diagonals: 1-5 Matter 2-6 Consciousness 3-7 Energy 4-0 Self-Organization
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. 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.
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 do this by visualizing simple geometric figures, see them in your minds eye.
After you have practiced with and mastered the simple forms, circle, triangle, square and pentagram, try moving them around at will and view them from all different perspectives. Then go on to the more complex forms shown next, the 6, 7, 8 and 9 sided forms.
Then try to visualize a hypercube, and see it from different angles.
With this visual basis, 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 on the Hypercube exercise given below .
- 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. (Cubes have eight corner points; four below and four above.)
- You are sitting right in the middle and your heart is in the center of the cube, the point where all four diagonals intersect (shown in white circle above).
- 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.