Recently, I embarked on an intriguing philosophical exploration, assisted by AI, to ponder how the term "fractal," introduced by Benoit Mandelbrot in 1975, could be translated from ancient Indo-European roots, particularly within the Indo-Iranian branch, into Sanskrit. This endeavor stemmed from my belief in the fractal essence of Vedantic teachings, prompting me to seek linguistic parallels.

Vedanta represents one of the six classical schools of Hindu philosophy, signifying "the conclusion of the Vedas" and is associated with the Upanishads, the concluding sections of the Vedic texts. It centers on the essence of the self (atman) and its connection to the ultimate reality (Brahman). At its core, Vedanta asserts that the true self, atman, is one with the all-encompassing Brahman. Our ignorance of this unity brings suffering, while understanding this oneness facilitates spiritual liberation (moksha) from the cycles of birth and death. The philosophy emphasizes logical reasoning and discourse as pathways to achieving liberating knowledge.

The Vedantic concept of Visvatomukham, translating to "with the face on all sides," resonates deeply with the fractal geometrical perspective of the universe. Visvatomukham suggests that divine reality is expressed through all manifested forms, paralleling how fractals exhibit self-similar patterns across different scales, where each microcosm mirrors the macrocosm.

The Upanishads declare:

“From fullness comes fullness. When fullness is taken from fullness,

Fullness remains.”

This statement emphasizes that the infinite continuously emanates from the eternal source, which remains inexhaustible. Similarly, in mathematics, fractals exhibit infinite iterations, yielding vast complexity from simplicity, with the core pattern preserved in each iteration.

In Vedanta, Brahman is perceived as the ultimate reality that transcends and manifests the diverse universe. Fractals symbolize this dynamic interplay between unity and diversity, illustrating the profound interrelationship between the whole and its parts.

The hypothesis that the universe resembles a fractal dodecahedron posits that the cosmos is finite and curved, contrary to the conventional view of it being infinite and flat. This idea emerged from cosmologists aiming to clarify anomalies observed in cosmic microwave background radiation, the remnant radiation from the Big Bang. They contended that a dodecahedral model could account for the observed uniformity in temperature fluctuations, which would be unexpected in an infinite, flat universe.

Noteworthy scientists who contributed to this theory include:

• Jean-Pierre Luminet, a French astrophysicist and author, who published research on the dodecahedral model in 2003;
• Glenn Starkman, a professor of physics and astronomy at Case Western Reserve University, who co-authored a paper on the dodecahedral model in 2003;
• Jeffrey Weeks, an American mathematician and MacArthur fellow, who developed software to visualize potential cosmic shapes.

Fractals are mathematical constructs defined by a key characteristic: self-similarity. This property means that the fractal consists of parts that echo the structure of the whole at varying scales. Mathematically, a fractal exhibits scale symmetry, ensuring that enlarging or reducing components of the object yields similar details to the overall structure.

Fractals can be generated through algorithmic iterations but are also prevalent in nature as outcomes of complex dynamic processes. Here are ten natural examples of fractal patterns:

• Tree branches: Smaller branches mimic the shape of the entire tree due to a pattern of dichotomous branching, where each branch divides into two smaller branches.

• River networks: The pathways of rivers and their tributaries form watersheds with fractal characteristics, appearing similar across different scales.

• Rugged coastlines: From an aerial view, the irregularly repeating coves echo the complex shape of the coastline.

• Clouds: The jagged outlines of clouds, shaped by turbulent atmospheric motion, exhibit fractal properties.

• Snowflakes: Each snowflake possesses a detailed hexagonal structure that showcases self-similarity.

• Coral: The branching patterns of coral into smaller sub-branches exemplify fractal growth.

• Romanesco: This type of cauliflower displays fractal spirals with conical tips that replicate on decreasing scales.

• Spider webs: The intricate structure of spider webs is determined by the method in which spiders weave their creations.

• Blood vessels: The branching pattern of the cardiovascular system, from arteries to capillaries, follows a fractal design.

• Bronchial trees: The air passages in the respiratory system also conform to a fractal model of dichotomous division.

  “Worlds nested within each other like the faces of a crystal multiplying to infinity through reflections.”

  —Bh?vana Upanishad

Furthermore, the Vedantic concept of Sarva-gatah ("all-pervading") aligns with the fractal perspective, both rejecting reductionism and duality, and revealing the inherent unity amid existence's diversity. Sarva-gatah signifies Brahman's omnipresence, the Absolute Reality that pervades the cosmos while also transcending it. Fractals, through recursive self-similarity across scales, mathematically illustrate the interconnection between part and whole. This mirrors how the infinite Brahman is fully manifest in every fragment of the cosmos according to Sarva-gatah, the All-Pervader. Both perspectives challenge conventional logic to unveil higher truths about the irreducible complexity of totality.

Through geometric abstraction and embodied spirituality, we encounter nature's intricate infinity. This exploration connects us to the divine mystery of the One manifesting as multiplicity. However, this multiplicity is not fragmented but infinitely interconnected, woven into a seamless tapestry — a fractal dodecahedron arising from the self-similar seed of Sarva-gatah, the omnipresent source. Thus, diversity, complexity, and infinity emerge in harmonious resonance, illustrated in fractal modeling and intuitively grasped through nondual insight. These distinct yet intuitively unified visions invite us to embrace a wisdom that transcends rationality, urging us to honor the all-encompassing fullness that resides within and radiates through every minute form.

Both Vedanta and fractal geometry prioritize direct mystical intuition over logical constructs. Their insights reveal truths that evade reductionist scrutiny, celebrating the intrinsic fullness of reality and the infinite interconnectedness of all forms.

When we recognize Visvatomukham — that divine fullness expresses itself through every aspect — the illusion of egoic separation dissolves. Ultimately, Visvatomukham and fractal cosmology evoke awe at the hidden glory of existence, teaching us that diversity emerges from and returns to a unified being. Any apparent fragmentation is temporary. At the most profound level, the universe is a single fractal organism pulsating with divine fullness in the present moment.

The parallels between Vedanta, fractals, the dodecahedral universe, and sarva-gatah suggest that diversity and unity stem from a singular Consciousness. The self-similar patterns of fractals across different scales indicate an inherent interconnectedness among all forms, potentially originating from and returning to a common source. By awakening to our infinite essence, we can transcend divisions and recognize our shared foundation of existence, uncovering the unifying Thread that weaves multiplicity into a cohesive tapestry, resonating with the One Presence that encompasses all.

Translation of “fractal” into Sanskrit

In my recent endeavor, I explored how the term "fractal" could be interpreted from the ancient Indo-European roots into Sanskrit. This exploration aimed to reflect the fractal essence inherent in Vedantic philosophy while also seeking linguistic connections.

One possible translation I devised for "fractal" in Sanskrit is:

bhrama-rupa-anubhava-prav?h

This phrase can be interpreted in two ways: 1. Illusory repeated movement of self-similar forms of unity. 2. Illusory form-experience-flow of unity.

Let’s delve into the Indo-European roots:

• bhrama (????):
  • Syllables: bhra + ma
  • Meaning: illusion, error, confusion
  • Indo-European root: bher, meaning "to wander," "to stray," "to be confused."
• rupa (???):
  • Syllables: ru + pa
  • Meaning: form, appearance
  • Indo-European root: reu, meaning "to see" or "to perceive."
• anubhava (?????):
  • Syllables: anu + bhav
  • Meaning: experience, perception, understanding
  • Indo-European root: anbheue, meaning "to become aware of" or "to perceive."
• prav?h (??????):
  • Syllables: pra + v?h
  • Meaning: flow, movement
  • Indo-European root: wegh, meaning "to carry," "to transport."

Explanation.

Fractals embody the paradox of infinity contained within the finite. They are self-similar structures, where each part reflects the whole in an endless cascade.

First, we have “bhrama” — illusion, a trick of perception. Fractals give the impression of infinite shapes while occupying no physical space. They lead observers into a maze with no exit, a waking dream from which one cannot escape, much like a mirage that suggests water in a desert.

Next is “rupa” — form or shape. While fractals continually morph like smoke, prolonged observation reveals emerging patterns. Spires, snowflakes, and ferns gradually crystallize from chaos, showcasing intricate webs where once only haze existed. Defying logic, order takes shape without any guiding boundaries. Mandelbrot sets bloom in imaginary realms, where beauty manifests without reason.

Following this is “anubhava” — experience or revelation. Only by delving into the fractal flux can we unlock their secrets. The journey through the maze leads to profound realizations as barriers dissolve to reveal worlds within worlds. Knowledge transforms into wisdom as we comprehend the templates governing endless iterations. Clarity emerges from confusion as we discern the symmetries across scales, revealing the oracle's mysteries through observation.

Lastly, we encounter “prav?h” — flow and dynamism. Fractals lack fixed forms, instead undergoing perpetual rebirth through continuous generation. They shape-shift endlessly, birthing new possibilities free from constraints. Their paths remain unwritten, transcending rigidity. Each recreation unveils unexpected vistas that emerge from the void at both micro and macro levels. Unrestricted by time and space, evolution flourishes in fractal domains, where motion is omnipresent among countless iterations.

In the interplay between illusion and form, experience and flow, fractal paradoxes unfold. Deceptions yield to revelations, mazes lead to meanings, and aimlessness aligns with order as space becomes filled. By pursuing bhrama-rupa-anubhava-prav?h, we unearth insights concealed in the blink of an eye.

Infinite Richness

Ultimately, a surprising convergence arises between the Vedantic perspective and fractals: both celebrate the infinite richness stemming from the One, unveiling worlds within worlds in a continuous vortex. Fractal geometry appears to provide an abstract, mathematical representation of reality, as perceived by ancient seer-sages millennia ago — a vast web of interconnected forms, concealing universes within every grain of sand. This reality is stratified into increasingly profound levels of cosmic order.

These two approaches harmonize and reflect one another, remaining conceptually distinct yet converging on the intuitive plane. They convey, in different languages, the dazzling light that permeates all things while simultaneously transcending them, reflecting the eternal dance of being that overflows from the blissful stillness of non-being.

They remind us that ultimate beauty and truth elude rational thought, yet unfold to the pure heart that, like an infinite fractal or the vast Brahman, embraces everything within its boundless gaze.

In conclusion, what remains? “Bhrama” dissipates as the veils lift to reveal the radiant face of truth. “Rupa” unfolds in the patterns of nature, spiraling out from the microscopic realms. “Anubhava” awakens, as the enlightened mind grasps the enduring truth of unity. And “prav?h” carries all existence within its currents, swirling together toward a shared ocean.

In that gaze, the boundaries of the self dissolve, and separation yields to unity. Here, all searching concludes; the One is all that exists, eternally.