“The Integratron Root Square”
6” x 6”, acrylic on canvas, 2022
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“Geometry is knowledge that appears to be produced by human beings, yet whose meaning is totally independent of them."
— Rudolf Steiner
In the process of geometrically exploring the relationships between the first three root ratios (2, 3, 5) and finding a way to integrate them visually in a meaningful way into a kind of symbol, or put them in a relationship that could be simplified or contained within a square, I discovered this overall way of integrating the three roots, what I am calling the "Integratron Root Square."
What I'm interested in are how these fundamental root relationships with the 2, 3 & 5, and their shapes (the square, triangle, and pentagon), are interconnected and inter-generating universal constants. It's hard to imagine them as separate functions or laws, especially when the certain extended geometrical properties of lines of different ratios or grid patterns or circles overlap almost magically to create just the exact different ratios, points, and angles needed, and integrated in perfect alignment.
“The dynamics generated within the three primary shapes are illustrative of the self-organizing forces of nature.” - Jonathan Quintin
These three roots, the square root (2), cube root (3) and phi (5) encompass both even & odd numbers, with 5 being an integration of both, altogether form a trinity of generative principles or roots (Jonathan Quintin). Phi or 5 has the quality of something extra it seems, it has the uneven/indeterminate generative/emergent element or creative and self-balancing dynamic seen by living systems, a self-referencing feedback property, or a "dynamic fracticality", in relation to the Fibonacci sequence, seen in both animate and inanimate pattern formations in nature.
“You will know, as far as it is allowed to a mortal, that nature is from all points of view similar to itself.” - Pythagorus
The Pythagorean Trinity, Fibonacci Sequence, and Creation Geometry
To me this trinity of generative root ratios seems to be connected to what is called the Pythagorean Trinity (Ken Wheeler), and the relationships between the first five numbers in the Fibonacci sequence, (1, 1, 2, 3, 5). In the metaphysics of the Pythagorean Trinity, the One is considered ultimately transcendent and absolute, beyond any representation, but is represented as an indefinite dyad (1,1), as an inseparable transcendent principle and manifest attribute of the One, that makes up the first two numbers of the Fibonacci sequence and the first part of the Pythagorean Trinity. The first 1 represents the intrinsic transcendent in principle, and the second 1 its inseparable extrinsic attribute. Thinking about this, the two initial overlapping circles of the vesica piscis come to mind as a kind of visual, with both representing a "one."
The golden ratio emerges from the two systems of geometry generated out of the vesica pisces, the ad triangulum and the ad quadratum (root 3 and root 2), and these two geometries were generated from two perfect spheres or circles or infinities or "ones" overlapping at their midpoints. So two circles (pi/infinities) overlap to create root 2 & root 3, which then those geometries overlap to create root 5 (phi/golden ratio). This relationship between the square and cube, (2, 3), is the second part of the Pythagorean Trinity, and represents the emergence of the principle of duality and its relationship with the One, which links back and creates a trinity, representing the manifest universe and all things in 3 dimensions.
The third part of the Pythagorean Trinity are the relationships between 3 and 5, the cube and golden ratio/phi, the trinity and life (being/ontology), and is the final integration or "consubstantiation" of the first two parts of the trinity (Spirit-Matter/One-Many/Transcendent-Immanent).
The feedback of the Fibonacci sequence seen in this way is the feedback of life or phi with the continuum of the present moment, the One, or the "irreducible One" as R. A. Schwaller de Lubicz phrased it; the self-balancing and self-organizing and spiraling of polarities and dualities back into the constant creation of the present moment, the infinite one-continuum of the eternal ever-present, with its many uncertain and emergent outcomes, bifurcations, and growths.
In this way, geometry is an interactive and deductive metaphysical tool we can use for understanding and discovering the inner "mechanics" of the universe, its generative archetypal constants, and the ultimate source of the transcendent and its connection to all things.
This is the metaphysics of the origin of process, the origin of order, and the origin of origin; the fact that all logic, order, and rationality in the universe originates from numbers and shapes with irrational roots and principles is quite the curious mathematical mystery.
“The compass is the most important scientific instrument that humanity ever invented.” — Keith Critchlow
"The study of geometry is a meditation on the harmony of the cosmic order." — Pythagoras
“Details make perfection, and perfection is not a detail.” ― Leonardo Da Vinci
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Initial study for Integratron, compass & pencil
Extended circles & lines
Fractally embedded root ratios
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"I have a distinct feeling that number is a key to the mystery, since it is just as much discovered as it is invented. It is quantity as well as meaning." —Carl Jung
"Number thereby throws a bridge across the gap between the physically knowable and the imaginary. In this manner it operates as a still largely unexplored mid-point between myth (the psychic) and reality (the physical), at the same time both quantitative and qualitative, representational and irrepresentational." —Marie-Louise von Franz
"It may well be the most primitive element of order in the human mind... thus we define number psychologically as an archetype of order which has become conscious." — Carl Jung
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Alternative study with root ratios and
pentagonal geometry (Albrecht Dürer method)
The three roots overlaid and aligned with Leonardo da Vinci's Vitruvian Man geometry
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"Unity is plural and, at minimum, is two." — Buckminster Fuller“One becomes two, two becomes three, and out of the third comes the one as the fourth.” — Maria Prophetissa"The one engenders engenders the two, the two engenders the three and the three engenders all things." — Tao te Ching"All the effects of nature are only the mathematical consequences of a small number of immutable laws." — Pierre Simon de Laplace"Where there is matter, there is geometry." — Johannes Kepler"The harmony of the world is made manifest in Form and Number." — Sir D'Arcy Wentworth Thompson"Geometry purifies the eye of the soul, since it is by it alone that we contemplate the truth." — Plato“There are fractal codes containing the laws of creation and the more we understand these laws the more we can apply the harmony that is apparent in nature as a more powerful force in our own lives. There is truth and purity in natural things and our contact with them nourishes the soul and illumines the mind.” - Jonathan Quintin"Music is the arithmetic of sounds, as optics is the geometry of light." — Claude Debussy“With light poise and counterpoise, Nature oscillates within her prescribed limits, yet thus arise all the varieties and conditions of the phenomena which are presented to us in space and time.” — Goethe
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