#geometrymatters

Kelvin-Helmholtz clouds

coexistants 

Cognitive gravity and the superposition of perception and memory

Attention acts as a vector, drawn toward regions of relevance shaped by phenomenal bias β and cognitive variability γ: β anchors prior expectations, while γ allows flexibility. After Escher (Liberation).

A polynon is a conceptual geometric entity of which vertices are non-events and its edges holograms.

A polynon contains all the holograms of that which can be projected as a polytope, showing how consciousness can be fundamental.

In the Hexanon, shown above, the Observer O(n) is in superposition with the phenomenal p+, epiphenomenal p-, negative noumena n- and noumenal vertices n+ of the hexanon as a function for self-reflection of consciousness C.

The polynon is built on the lens mechanism of the wavefunction: each layer of the noumenal lens is a wavefunction in itself, providing mechanism for the observer as function of noumenal reflection.

How Consciousness is a fundamental substance and everything else, including the Observer, is it's function of self-reflection within a complex construct and hologram.

Introducing the polynon:

A polynon is a conceptual geometric entity, a polytope of which vertices are non-events and its edges, holograms. A polynon contains all the holograms of that which can be projected as a polytope. From tetranon to dodecanon, and special cases such as dinon and nullanon.

The epistemological foundation of the polynon is a non-event, hence multi (poly) non-events. The ontological foundation is a noumenon, both in its negative and its positive aspect.

Collapsing the noumenal realm into a single dimension results in the conceptual noumenal monad, where the perceptual border is equal to the infinity of noumenal possibilities, as well as the phenomenal something.

Or, as "a cercle with the centre everywhere and circumference nowhere". A monadic embodiment of the thing and the non-thing, the visual expression of the substance of consciousness.

The neutral monistic approach enfolds the ancient knowledge of the self, and the contemporary science of consciousness, from a Vedantic self reflected in Indra's net, to a multi-dimensional implicate and explicate construct of Bohmian orders.

Also encapsulated in the formulation f(f)=f, akin to an Ouroboros mathematical formulation symbolizing a form of autopoiesis or self-production ad infinitum, a self-referential Fichtean process where the self (I) is both the creator and subject of its reality.

The eternal aspect of consciousness, a timeless monadic essence, is described as the perceptual continuum of f=0, signifying pure, timeless potential.

"Eternity isn’t some later time. Eternity isn’t even a long time. Eternity has nothing to do with time. Eternity is that dimension of here and now that all thinking in temporal terms cuts off." - J. Campbell

The wavefunction is defined here as a layer of the noumenal monad, a transition between epistemological and ontological, given by the limits of the Observer's senses and perception.

In this context, |ψ(x)>*n, with n not strictly equal to 2, transcends its conventional mathematical role within Hilbert space and diverges from standard quantum mechanics, suggesting that different values of n represent distinct dimensions or layers of noumenal collapse.

Each wavefunction is a layer of the noumenal lens, of which border is (n+), or consciousness C; it’s contents are negative noumena (n-) and it’s holographic centre “collapsed” onto a phenomenal Observer O(n) as function of noumenal self-reflection of consciousness.

Furthermore, the Observer is envisioned as a holographic projection sitting at the core of the noumenal lens, emerging from the focus and diffraction of noumenal probabilities within the noumenal lens.

The perceptual border of positive noumena (n+) reflects and focuses noumenal probabilities (n-) into the hologram of the Observer O(n).

A rotation of noumenal lens reveals the noumenal essence (n+), hidden under the superposition of the noumenal lenses (n-) and the hologram of the Observer O(n).

The observer’s perceptual focus, directed through Cognitive Gravity G and visualized with noumenal g(n) and cognitive gradients g(r), refracts the essence of Consciousness, transforming noumenal vertices into discernible edges like light through a prism.

As the number of noumenal collapses increases, the complexity and breadth of the observer’s perceptual dimension diminishes, resulting in a progressive refinement and narrowing of perceptual prowess of the observable construct.

A cognitive threshold marks the evolution from a purely phenomenal observer to one with phantasiai and developed cognitive functions. This threshold varies according to the local conditions of the observer’s cognitive gradient and gravity and the overall noumenal function.

The hologram of a polynon enfolds within it the reflection of the noumenal substance, that both originates and reflects consciousness. It encapsulates the entirety of consciousness in each of its vertices and edges, as projections into a perceptual reality.

The concept of “consciousness nodes” describes (n+) zones between noumenal lenses (n−), revealed as vertices of the polynon, equal and identical in nature.

The Observer O(n) is in superposition with the phenomenal p+, epiphenomenal or phantasiai p-, negative noumena n- and noumenal vertices (and centre) n+ of the hexanon as a function for self-reflection of consciousness C.

The polynon is abstract and elusive, residing beyond human imagination yet within its grasp. No heavier than a fleeting idea, but far too heavy to be understood. A palindrome for language and thought, mirroring itself in perpetual paradox. Beginning the same way it ends.

FRACTAL TREES – BASIC L-SYSTEM- EXAMPLE 9.4

Fractal processes have practical applications in creating 3D vegetation for models using plant generators like Xfrog and Laubwerk. These tools utilize mathematical and botanical principles to produce realistic plants. A key algorithm family used in these generators is Lindenmayer systems (L-systems), which generate complex structures through iterative production rules. Understanding L-systems can enhance the use of these tools or inspire custom design applications. The simplest L-system involves drawing lines from a point at variable angles and reducing line length proportionally with each generation, creating intricate plant-like structures.

L-systems, such as those created with Grasshopper and Anemone, start with a set point and draw lines at defined angles and lengths, doubling the number of lines with each generation. Varying angles and lengths produces diverse structures. The book "The Algorithmic Beauty of Plants" is a valuable resource for understanding these systems. By creating simple tree generators with random seeds, users can design non-repetitive, impressionistic vegetation models for various applications.

The geometry of the Borromean Rings

Borromean rings are a captivating geometric structure composed of three interlinked rings. What makes them unique is their interdependency; if any one ring is removed, the entire structure collapses. This fascinating property, known as "Brunnian" linkage, means that no two rings are directly linked, yet all three are inseparable as a group. This intricate dance of unity and fragility offers a profound insight into the nature of interconnected systems, both in mathematics and beyond.

Borromean Rings and Mathematical Knots

Borromean rings also find a significant place in the study of mathematical knots, a field dedicated to understanding how loops and tangles can be organized and categorized. The intricate relationship among the rings provides a rich visual and conceptual tool for mathematicians. Knot theorists use these rings to explore properties of space, topology, and the ways in which complex systems can be both resilient and fragile. The visual representation of Borromean rings in knot theory not only aids in mathematical comprehension but also enhances our appreciation of their symmetrical beauty and profound interconnectedness.

Symbolism and Divinity in Borromean Rings

Throughout history, Borromean rings have been imbued with symbolic significance, often associated with divinity and the concept of the trinity. In Christianity, they serve as a powerful visual metaphor for the Holy Trinity – the Father, the Son, and the Holy Spirit – illustrating how three distinct entities can form a single, inseparable divine essence. This symbol is not confined to Christianity alone; many other cultures and religions see the interconnected rings as representations of unity, interdependence, and the intricate balance of the cosmos.

Borromean Rings as a Metaphor for Illusory Reality

The Noumenal Monad

Within the Polynon framework, nothingness is an undifferentiated cognitive space that precedes geometry and structure. It exists as a pre-geometric source, housing the unformed essence of all phenomena from which spacetime, and all its manifestations, arise. Imbued with primal noumenal potential, it is denoted as the state where all possible realities are latent, waiting to be expressed into an observable world.

A circle whose center is everywhere and circumference is nowhere.

The monad is a geometric construct that acts as a universal grammar of existence, mapping how the boundless noumenal potential transitions into perceptual and phenomenal dimensions while retaining coherence across all levels of manifestation.

The centre being “everywhere” signifies the Monad’s omnipresence as the locus of all potential states, embedded in every point of reality. The circumference being “nowhere” reflects its boundless architecture, transcending the constraints of space, time, and materiality, integrating infinite possibilities within a singular, cohesive structure.

The compactification process begins with the noumenal everything compressing into a singular phenomenal something, reflecting a specific instance or manifestation. This phenomenal something is inherently equal to the noumenal everything because it retains the entirety of noumenal potential, either in its un-collapsed state of infinity or as a collapsed singular “ring” dimension, compacted into a single cognitive focal point.

The only attribute of a point is that it marks position. Take away this attribute and in the unposited point we have a symbol of pure Being, the abstract noumenon, that which underlies every mode of phenomenal manifestation, every form of existence. It is at once All and Nothing, at once Absolute Consciousness and Unconsciousness.

B.W. Betts, Geometric Psychology or the Science of Representation

Or, as Fichte envisioned it, the line symbolizes the progression of consciousness—linear and sequential—while the circle represents its completeness and self-enclosure, encompassing all its dimensions. Thus, both the noumenal everything and the phenomenal something are expressions of the same essence, differing only in their state of manifestation and representation.