“Geometry is an art and a science as definite as mathematics.” - Randolph Stone
“The product of mathematics is clarity and understanding. Not theorems, by themselves.” - Bill Thurston


“Geometry is knowledge that appears to be produced by human beings, yet whose meaning is totally independent of them.” - Rudolf Steiner

Mathematics is the art of giving the same name to different things.” - Henri Poincaré



Trinity of polarity - Walter Russell [with some edits to the original]

The three come forth from the one. One is in three and stands between two, forming a triad from which has proceeded everything; so that One is all and all is One.” - Zohar

God never passes nine in any creative expression nor does any phenomena in Nature ever exceed the principle upon which your breathing, or heartbeat is based. Anything in Nature, or in your life's problems, which might have one million parts and one million complexities, if taken apart, will be found to constitute one simple principle of three elements. God multiplies these three by three to make nine, but never more than that. So if you can understand the principle of the basic three, and if you can multiply that three by three, you can become the greatest chemist, metallurgist, electrical engineer, poet, painter, philosopher or inventor in the whole world if no one but you in the world did understand it. This basic THREE, which cannot be added to nor subtracted from, will hereinafter be referred to as THE DIVINE TRINITY.” - Walter Russell

If you knew the magnificence of the 3, 6 and 9, you would have a key to the universe.” – Nikola Tesla

Trefoil knot
This is the simplest example of a nontrivial knot. It can be created by cutting the circle and connecting the two line-ends after one knot-movement.

Its a beautiful, hypnotic example of complex form made from a basic geometry. Nature is simple in its base logic, but infinitely complex in its effects. The whole physical universe is one connected line of (Mind-induced polarized) energy flows formed into a beautiful, hyper-complex knot-shape. Complete separation is an illusion - even birth and death are only changes in the polarity of Life energy.

See also: Celtic knot, Flower of Life, hypertorus.

“In the production of finite existence the first act was the evolution of the Memra, or the Word, which was the first point in the descending series of beings, and from whom in nine other degrees of manifestation emanated those forms which at once compose the universe and express the attributes and presence of its eternal ruler. To these nine forms is given the common name of Sephiroth, signifying Splendors. The whole or some of these Sephiroth constitute the universe, the manifestation of God, (…) The primordial essence is before all things. In his abstract and eternal nature and condition he is incomprehensible, and as an object of the understanding, according to the Zohar, he is nothing, the mystery of mysteries; but he took form as he called forth them all. The ancient of ancients is now seen in his own light; that light is his holy name, the totality of the Sephiroth. The order of their emanation is as follows: From Kether, the Crown, the primal emanation of En Soph, proceed two other Sephiroth–Chocma (wisdom), active and masculine; the other Binah (understanding), passive and feminine, the combination of which results in thought, of which the universe is the effect. The crowned Memro, or Kether, or primordial Logos, is the thinking power in creation, Chocma the act of thinking, and Binah the subject of the thinking.” - Zohar

russel-rodin math

Illustration of the “Russell-Rodin” reduced-doubling-circuit numbers.

Note how the vertically-opposing (male-female) numbers all add up to 9 (the whole): 9, 8+1, 7+2, 6+3, 5+4.

The red-opposing (Russell) numbers always add up to 0 (which is the fulcrum of balance of the 9): 1 -1, 2 -2, 3 -3, 4 -4.

The 4 -4 position is the maximum distance of the polarity tension and simulates rest in the physical universe, but this position is not rest! The 4 -4 positions are also not located on the 0 point, as it may seem from this simple 2D illustration.

Out of stillness comes seeming motion:

  • 0 represents the stillness of the fulcrum
  • 9 represents motion and encompasses the whole inward-outward swing of the wave. This seeming motion consists of an inward-outward wave:
    • 3 represents the inward, generative motion (3 overlapping center points of stillness)
    • 6 represents the outward, degenerative motion (6 outer points of polarity, because of the 3 x 2 extensions from the still centers)

Seeming motion (9) arrises from the polarization of stillness (0) into 7 distinct octave notes:

  • the maximum pressure note between the male and female side corresponding to the ±4
  • 1, 2, 4 corresponding to +1 +2 +3 represent the male polarity charging side
  • 8, 7, 5 corresponding to -3 -2 -1 represent the female polarity discharging side
“Universal Mathematics” - Walter Russell

The cubical 8 and 9:

  • The one inner-point of the cube and its 8 corner points also add up to 9.
  • Every cube can be sub-divided in 8 sub-cubes.
    • Each of these 8 sub-cubes can be defined using 3 vectors, with each vector having 2 directions: x +-, y +-, z +-. So each sub-cube is addressable using these 3 —> 2^3 = 8. Meaning the whole cube is controlled from the positive polarity (3) and the negative polarity (6), which makes 9 —> 3^2 = 9.

ancient art


cave drawings

common cave drawings in the world - *
32 most common cave drawings in Europe - *

jain religion

Raj Lok symbol

“Jain cosmology considers the universe, as an uncreated entity, existing since infinity, having neither beginning nor end. Jain texts describe the shape of the universe as similar to a man standing with legs apart and arm resting on his waist. This Universe, according to Jainism, is broad at the top, narrow at the middle and once again becomes broad at the bottom.”

note: how this slight asymmetry might indicate the male-female polarity. - wikipedia

note: the Swastika vortex symbol and the Ahisma hand symbol for “non-violence”
Jain time cycle

“According to Jain cosmology, the universe does not have a temporal beginning or end. Its “Universal History” divides the cycle of time into two halves (avasarpiṇī and utsarpiṇī) with six aras (spokes) in each half, and the cycles keep repeating perpetually. Twenty-four Tirthankaras appear in every ara, the first Tirthankara founding Jainism each time. In the present time cycle, Rishabhanatha is credited as being the first tīrthaṅkara, born at the end of the third ara ” - wikipedia
Raj Lok art

note: the balanced position of the two hands on the equator and the central Axis mundi
Jain cosmography illustration part (17th century) - source

note: the resemblance to the generative (vertical, Solar) and degenerative (horizontal, Lunar) poles in the Walter Russell cosmology

vedic religion

Sri Yantras are a type of sacred geometry symbolism from ancient India.

“Most yantras include triangles. Downward pointing triangles represent feminine aspect of God or Shakti, upward pointing triangles represent masculine aspect such as Shiva.” - wikipedia

note: the cube and circle shapes (the polarity of spiritual-physical, silence-rest, formless-form and stillness-motion), the 8 polarized triangles representing the 2 sets of 4 motion states plus one inner triangle representing the undivided (non-dual) whole of the 9, the 4 openings (vertical and horizontal connection to the outside).

See also: the Vedic Swastika, Jain religious art, Buddhist Mandala.

written language



western alphabet development


10 digits

the efficient 10-digit numerical system has its origin in Ancient India
10-digit numerical writing comparison

divination systems

(todo: exploration of possible fundamental connections to the previous geometry)

i ching


0 +1 +2 +3 ±4 -3 -2 -1 0

  • 8 primary trigrams in two polarities (male and female) leads to 64 hexagrams (double trigram) combinations.
    • The male-female wave motion is balanced for both the generative and degenerative wave-cycle.
    • See also: “I Ching: Book of Changes”.
    • “64 codons, 61 represent amino acids, and the remaining three represent stop signals, which trigger the end of protein synthesis. Because there are only 20 different amino acids but 64 possible codons, most amino acids are indicated by more than one codon. (Note, however, that each codon represents only one amino acid or stop codon.) This phenomenon is known as redundancy or degeneracy” - *
      • 8 fundamental amino acids?? (8 human-essential amino-acids)

see also

geometry.txt · Last modified: 2021/08/02 14:10 (external edit)