Conceptual impressions surrounding this post have yet to be substantiated, corroborated, confirmed or woven into a larger argument, context or network. Objective: To generate symbolic links between scientific discovery, design awareness and consciousness.
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The mathematical challenge of squaring the circle originated with the ancient Greek mathematicians and geometers. The problem was to construct for a given circle a square with the same area as the circle, all to be done with only a compass and straightedge. It was repeatedly demonstrated that the challenge was impossible to do.
The circle represented woman, spirit and infinite wholeness, the square man, earthly circumstances and all things material. Ancient Greek philosophers distinguished all things material within the context of fours, e.g. the seasons of the year, the elements of earth, air, fire and water. Squaring the circle was proven to be algebraically impossible until 1882 when the Lindermann-Weierstrass theorem proved that pi π is transcendental and not an algebraic irrational number.
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“Philosophically and spiritually, to square the circle means to see equally in four directions—up, down, in, and out—and to be whole, complete, and free. Circles often represent the spiritual because they are infinite—they have no end. The square is often a symbol of the material because of the number of physical things that come in fours, such as four seasons (4), four directions (4), and the four physical elements (4)—earth, air, fire, and water, (4x3=12) according to ancient Greek philosopher Empedocles—not to mention its solid appearance.
The union of man and woman in alchemy is a merging of spiritual and physical natures. The triangle is then a symbol of the resulting union of body, mind, and soul. In the 17th century, squaring the circle had not yet been proved impossible. However, it was a puzzle no one had been known to solve. Alchemy was viewed very similarly: It was something few if any had ever fully completed. The study of alchemy was as much about the journey as the goal, as no one might ever actually forge a philosopher’s stone.”
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In reference to Design Consciousness, the concept of “squaring the circle” is based upon the pursuit of balance and harmony during the process of manifestation and transfiguration (design). The square symbolizing the Earth (Yin) being made tangible and formative and the circle symbolizing the Heavens (Yang) being intangible and formless. Two opposing geometries meant to describe two opposing principles in balance, whole and complete. Together this relationship extends beyond their simple geometry to represent a profound philosophical and spiritual journey, i.e. The Path.
Union of opposites and the path to wholeness
To begin, all that can be known in 3D spacetime is first conveyed symbolically in the form of a point (.) From another perspective in relative perpendicularity this same point will appear as a line ( _______ ). A point in motion creates a line or vector.
All lines can also be prescribed as being a vector, i.e. a measurable quantity having direction as well as magnitude, especially if/when determining the position of one point made in reference to another point along that same line ( < ---------- > ).
This additional quality imbues the line with meaning and purpose, which effectively assigns it a value. In this same context, a vector may be interpreted as a scalar, depending upon the frame of reference of the observer.
A circle represents the infinite, divine, and spiritual: Circles have no beginning or end, making them a natural representation of the eternal, the spiritual realm, and the divine or subconscious aspects of the self.
Metaphysically, the process by which a point expands to become a circle can be thought of as a shift from pure potentiality (point) into a manifested form (circle)—a metaphysical leap from the concept of unity or singularity towards multiplicity and structure. This transformation carries deep symbolic and philosophical implications, often explored in metaphysics, geometry, and mysticism.
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Metaphysical Implications, Differences and Interpretations
• Origination: In Platonic thought the point overflows into circle; in Hermetic thought the point manifests the circle in cycles; in Vedanta the point realizes it is the circle.
• Hierarchy vs. unity: Platonic/Neoplatonic emphasize levels; Hermetic emphasize correspondence and cycles; Vedantic emphasizes radical unity.
• Motion and vector analogy: Hermetic is perhaps strongest in suggesting motion (energy in motion) (circle → spiral) and thus vector-like expansion. Platonic uses more static metaphors of “participation”. Vedanta uses expanding awareness rather than physical vectors.
• Symbolic use of circle: For Platonists the circle symbolizes perfection and ideal form; for Hermeticists it symbolizes cycles of manifestation; for Vedantins it symbolizes infinite consciousness.
• Centre/circumference relations:
Platonic: centre (One) transcends, circumference is the many.
Hermetic: centre and circumference interplay, movement outward/inward, spiral.
Vedantic: the centre is everywhere, the circumference nowhere – emphasizing the infinite immanence of the centre.
In reference to design consciousness the transition, translation and transformation of energy in motion (EIM) from a point into a circle embraces all three interpretations brought to mind by these philosophies.
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If you adopt a metaphorical or geometrical reading, vectors (i.e., directed lines from a centre to the periphery) can be included in all three traditions, though to varying degrees and with different emphases:
• In Platonic/Neoplatonic scheme: one might imagine countless vectors from the One to each emanated being or Form; these vectors represent the mode of participation or derivation.
What might be considered a Hermetic interpretation of a circle.
• In Hermetic tradition: more overtly you have radial vectors (emanation lines), circular/spiral motion (vectors along a circumference plus vectors advancing the centre). So vectors symbolize movement of the centre of motion, or advance of life.
• In Vedantic scheme: you could metaphorically think of vectors of consciousness radiating outward (individual to universal) or inward (recognition of the universal in the individual). However, the primary emphasis is not on physical vector motion but on awareness expanding.
What might be considered both Platonic and/or Vedantic.
In each case, the “point to circle” metaphor can be viewed as "center plus radial vectors gives circumference/multiplicity". How one interprets or emphasizes those “radial vectors” differs:
• Are they arrows of emanation (Platonic)?
• Are they arrows of cyclical motion and transformation (Hermetic)?
• Are they arrows of conscious expansion or realisation (Vedantic)?
Summary & Conclusion
• The image of a point expanding into a circle is a rich metaphysical metaphor used in multiple traditions to depict the transition from unity to multiplicity, from potential to manifestation, from singular awareness to universal being.
• In the Platonic / Neoplatonic tradition it captures emanation from the One into the many, with a hierarchical structure.
• In the Hermetic tradition it captures manifestation in cycles, with motion (including radial and spiral aspects) emphasized, and the interplay of center and circumference.
• In the Vedantic tradition it captures the infinite nature of consciousness: the individual (point) realizes it is the universal (circle), with the center everywhere and circumference nowhere.
• Vectors — understood as directed lines of movement, connection, or expansion — can play a role in all three, but are most explicitly used in Hermeticism. In Platonic thought they are more implicit, and in Vedanta they serve as metaphors for awareness rather than literal motion.
Four elements (the square) shifting from potential (point) into form (circle)
about a source of observation
Earth, Air, Water, Fire
Squares representing the finite, earthly, and material: Squares are defined by four sides and corners, signifying the physical world, the tangible, the rational mind, and the grounded, conscious aspects of the self.
Eight elements (an octagon) in full manifestation about a
source of observation
North, South, East, West
The concept of squaring the circle is not a mathematical or geometrical problem, but rather the expression of a "metaphysical principle" where duality creates the context where the concept of unity is observed within the parametric constraints of human consciousness.
Two overlapping squares surrounding the observer within the margins of a circle make reference to the challenges about to be experienced within the earthly sphere of three dimensional spacetime. They likewise attend to a context that inspires creativity while simultaneously accommodating growth and expansion through the experience of the observer.
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In metaphysics, the octagon symbolizes the transitional space between the earthy square and the spiritual circle, representing a state of renewal, rebirth, and the unification of the material and the divine. It signifies new beginnings, regeneration after a cycle, and the interconnectedness of heaven and earth. The number eight (8) is also associated with cosmic balance and eternal life in various traditions, further enhancing the octagon's meaning of transformation and spiritual growth.
Design Metaphysics: The Octagon
The eight-sided polygon known as an octagon, signifies harmony and resonance. The intersection and relationship of two overlapping squares creates an eight-pointed star symbolized by the sun, i.e. the star that represents the center of our solar system.
The sun is an energy source that is continuously regenerating itself every moment. The sun exemplifies change. The eightfold pattern repeatedly vibrates in sympathy within the source of its own frequency. The octad is structured about the parameters of an even greater network.*
The sun represents the beginnings of a new source being "formed" within a field of greater complexity. The sun is perceived as a keynote and single point of origin. The sun is a source of harmony, a symbol of energy in motion, an involuntary force finding its own equilibrium within the infinity of the cosmos. The sun takes on the form of a nucleus and at the center of a system of energy that harbors a complex mathematical geometry. Multiple configurations and patterns of energy in motion are in perpetual transition and transformation as they swirl about this central core of attraction.
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Key Metaphysical Interpretations
• Transition and Unity: The octagon bridges the square (representing the material world, stability, and earth) and the circle (representing the spiritual world, eternity, and heaven). It is the point where one energy begins its transformation into another, uniting these two fundamental concepts.
• Sacred geometry: The most common metaphysical interpretation is that the octagon is a mediating form between the square (Earth, matter) and the circle (heaven, spirit). This makes it a symbol of the spiritual path toward enlightenment or the unification of material and spiritual existence.
• Regeneration and Cycle Completion: The shape is a symbol of regeneration and structured renewal, marking the completion of a universal cycle and the commencement of a new one.
• Spiritual Growth and Defense: In Feng Shui, an octagonal mirror is recommended for protection against negative spiritual activity. The form's multiple angles are also associated with comprehensive awareness, defensiveness, and a readiness to assess and yield momentum.
• The Bagua Map: The Western Bagua Map, a tool for self-development, is also octagonal, with each of the eight sections relating to a different aspect of life that supports the individual.
Balance and Cosmic order
Wholeness: The eight sides of the octagon can represent a quest for wholeness by balancing different aspects of existence, such as intellect, emotion, body, and spirit.
Google Research
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The mark of Design Consciousness is based upon the overlapping of two squares (regular quadrilaterals) of equal size representing the Yin and Yang energies of energy in motion. Together they symbolize the concept of duality, i.e. that which permeates all that can be known. Together they create the points and framework of an octagon. The thrust is of this energy is towards the northeast.
In the I Ching, the directions are often associated with elements, energies, and specific qualities. The northeast has a strong connection to the Earth element and the concept of stillness or quiet wisdom. It is linked to the idea of stability, grounding, and inner strength.
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Specific meanings associated with the northeast direction in the I Ching:
• Stability & Grounding: The northeast represents a place of solid foundation, where one can find inner clarity and balance. It is about being grounded, firm, and having a strong base from which to act.
• Quiet Reflection: This direction is also tied to introspection and calm, where one can find peace and clarity. It's a space for reflection, connecting to one's deeper wisdom, and evaluating situations without rushing to decisions.
• Transformation & Preparation: The northeast is seen as a transitional space, where you prepare for new cycles and developments. It has a connection to learning from past experiences and gaining insight to apply in the future.
In many classical Chinese traditions, the northeast is linked with the 8th trigram, Ken (☷), which symbolizes mountains and stillness. The mountain represents immobility, which can help in fostering patience, stability, and mental fortitude.
In practical terms, the northeast is often considered an auspicious direction for cultivating wisdom, understanding, and a strong sense of self before taking action.
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3 squares = 12 points (4 directions + 4 elements + 4 seasons)
Earthly experience
The dodecagon, as a twelve-sided polygon, offers the observer a holistic perspective of the three-dimensional world, creating a map for consciousness that reveals holographic pathways and non-measurable parameters. Such a synthesis reflects both ancient symbolic geometry and modern theories of holographic information (Bohm, 1980; Talbot, 1991). Light energy and information overlap, creating luminous layers that blur distinctions between measurable and immeasurable qualities—an interplay reminiscent of the observer effect and the entanglement of perception and matter (Barad, 2007; Wheeler, 1983).
That which can be measured influences that which cannot, and vice versa. This mutual influence echoes the quantum indeterminacy that underpins consciousness theories linking cognition and energy fields (Penrose, 1994; Hameroff & Penrose, 2014). Within the linear 3D time spiral, each imagined state of consciousness gives birth to the next, forming a holographic sequencing of present, past, and future — an idea also found in process philosophy and cybernetic design (Braidotti, 2019; Hayles, 1999).
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Earthly Parameters
Being drawn to this center is an integral part of the design process. This focal point is symbolic of a holographic source and precursor to the entrance into another universe. After certain attributes are fully experienced, explored and understood as a single strand of consciousness woven within a string of universes, certain mysteries collapse upon themselves, which in turn reveal other attributes of a unique geometry, holographic representations and dimensional interpretation.
At the hub of the dodecagon lies an unknown center — a source, akin to a black hole of awareness, symbolizing a metaphysical origin point. Recognition of this “center” parallels both mystical traditions and contemporary design thinking that seeks emergent meaning through iterative feedback loops (Verbeek, 2011; Latour, 2005). The pull toward this enigmatic center mirrors the creative process — a holographic design journey between order and potentiality.
" The dodecagon symbolizes time, its relationship to space and the location of energy in the form of holographic light and information. Together these referential points of quantum energy describe the qualities of the solar field. Holographic in character and designed by attraction and intention, each component triad (4) is an expression of a micro system that internally symbolizes the achievements of a larger macro system functioning in apparent balance. When the dodecagon is viewed within the constraints of this universe both time and space take on a new meaning and purpose. Time becomes the thread that sews all holograms together by making them apparent within the parameters of a virtual field of quantum energy (QFVPP). Time creates the condition. Time is the tool that brings meaning and purpose (design) into context when experiencing the three dimensional universe."
The symbolic completeness of twelve—found in DNA, the zodiac, time cycles, and cultural cosmologies—represents wholeness and systemic balance (Eliade, 1959). Within this geometry, the dodecagon becomes both an archetype and a design framework connecting microcosm and macrocosm, each reflecting the other (Floridi, 2014). When viewed through the lens of metaphysical design, time itself becomes a tool—a medium that weaves holographic experiences into a unified quantum field (Bohm, 1980; Wheeler, 1983).
12 Transcendent Design Principles
Truth, Beauty, Goodness
Light, Power, Love
Emotion, Intuition, Intelligence
Believing, Feeling, Knowing
Humanity Transcendent
The author generated this text in part with GPT-3, OpenAI’s large-scale language-generation model. Upon generating draft language, the author reviewed, edited, and revised the language to their own liking and takes ultimate responsibility for the content of this publication.
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References (APA 7th Edition)
Barad, K. (2007). Meeting the universe halfway: Quantum physics and the entanglement of matter and meaning. Duke University Press.
Bohm, D. (1980). Wholeness and the implicate order. Routledge.
Braidotti, R. (2019). Posthuman knowledge. Polity Press.
Eliade, M. (1959). The sacred and the profane: The nature of religion. Harcourt.
Floridi, L. (2014). The fourth revolution: How the infosphere is reshaping human reality. Oxford University Press.
Hameroff, S., & Penrose, R. (2014). Consciousness in the universe: A review of the ‘Orch OR’ theory. Physics of Life Reviews, 11(1), 39–78.
Hayles, N. K. (1999). How we became posthuman: Virtual bodies in cybernetics, literature, and informatics. University of Chicago Press.
Latour, B. (2005). Reassembling the social: An introduction to actor-network-theory. Oxford University Press.
Penrose, R. (1994). Shadows of the mind: A search for the missing science of consciousness. Oxford University Press.
Talbot, M. (1991). The holographic universe. Harper Perennial.
Verbeek, P.-P. (2011). Moralizing technology: Understanding and designing the morality of things. University of Chicago Press.
(Moralizing Technology and the ethics of things / Peter-Paul Verbeek / TEDxTwenteU)
Wheeler, J. A. (1983). Law without law. In J. A. Wheeler & W. H. Zurek (Eds.), Quantum theory and measurement (pp. 182–213). Princeton University Press.
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"To believe is to accept another's truth.
To know is your own creation."
Anonymous
Edited: 10.30.2025
Find your truth. Know your mind. Follow your heart. Love eternal will not be denied. Discernment is an integral part of self-mastery. You may share this post on a non-commercial basis, the author and URL to be included. Please note … posts are continually being edited. All rights reserved. Copyright © 2025 C.G. Garant.
