Fractal geometry and Vedanta: Seeking the connection

Sudeep Adhikari


The great eccentric polish mathematician Benoit Mandelbrot started his seminal book The fractal geometry of nature with the following: “Why is geometry often described as “cold” and “dry”? One reason lies in its inability to describe the shape of real nature. If we look around, what we see is that nature is not smooth or Euclidian, it is intricately structured in a complex way. It is most probably fractal. The world of perfect spheres, cylinders, and circles which the ancient Greek geometricians/medieval astronomers dreamed of is a platonic one. When Kepler found that the planets move around the sun in an elliptic orbit with sun as one of its foci, it was taken as the ultimate triumph of Euclidian geometry.  However, it was definitely not the scheme that was used to structure it. Here we need to discuss a bit on the difference between form and structure.

Form implies a static category, hence it can be Euclidian. Structure is a dynamic category, and it implies evolution. We can see a tree-trunk as a structure which evolves out of its fundamental cylindrical form, and mountain as a fractal structure which evolves from its basic conical outline. Though the fundamental form is Euclidian, the structure is a complex emergent phenomenon, which further implies dynamism. We can loosely say that anything that evolves (dynamic), tends to develop a complex and enriched structure which might be fundamentally fractal. A small pile of sand tends to retain an approximately conical shape under the right amount of moisture and temperature. With the increase in temperature and size, it tends to evolve as a dynamic system which shows a very intricate structuring pattern; more preferably fractal. If we observe closely, we can see avalanches of various scales distributed all over its mass.

When Benoit Mandelbrot posed the question “How long is the coast of Britain?” it was initially perceived as an innocuous geometrical enquiry. However, mathematical length of the coast is somewhat a fluid category, since coastlines are known for their monstrous irregularities. However, there must be a limiting magnification under which all the irregularities are visible and hence its length can be measured with reasonable accuracy. But a fractal object has a different idea; it is scale invariant. No matter how much we magnify, the irregularities will keep popping up. Assuming the coast of Britain to be fractal, it implies that it is of infinite length but it is contrary to our empirical observation because it encloses a finite area within it. The length of common border between Spain and Portugal or Belgium and the Netherlands differ by 20 percent as reported in their corresponding encyclopaedias. Depending on the scale of accuracy as different countries adopt, this difference can be arbitrarily infinite; however, the area of the country it encloses is always finite.

Thus fractal is the strange case of the co-existence between finite and the infinite. A part of a fractal is exactly the same of which it is part! Fractal is an irreducible multiplicity, a strange mixture of finitude and infinity. As a distinctly existing geometrical object, a fractal is a complete unity in itself but at the meantime, it is a complex juxtaposition of innumerable self-similar multiplicities. A fractal is singular and multiple, and finite and infinite at the same time. This we will term as fractal identity, which purports simultaneity of singular and plural, finite and infinite, static and dynamic, and form and structure. Fractal identity thus allows the co-existence of finite-infinite, singular-multiple and unity-multiplicity in utter compatibility without digressing into various metaphysical pitfalls.


Fractal identity thus doesn’t require the multiplicity to be a Vedantic illusion to admit the static unity of all. Fractal identity posits a field of all-pervading totality which assures the complete identity between the singular and the multiple without compromising their own individual validities. Multiplicity is the active principle behind manifestation and structure, and it is fractally identical with its non-dual core; and this is basic tenet of Dynamic Monism. Dynamic Monism is the acceptance of the fact that everything is one indivisible whole, but composed of innumerable multiplicities which form the active principle of its manifestation. One is singular, when “it is”, but a non-linear juxtaposition of irreducible multiplicities when “it acts, sings and dances”. Such an example of dynamic monism is Biosphere, which arises solely due to the dynamics of its associated multiplicities, that is, Rhizosphere, Geosphere and Atmosphere. And the dynamic monism is established on the basis of the fractal identities between all the constituent elements, for instance, Rhizosphere itself being a miniature Biosphere.

Vedanta in the similar way can be viewed under different light based on the concept of Dynamic Monism. “Bramhan”, as per Vedanta is the ultimate substratum of our consciousness, a non-dual and singular category. It is a purely monistic field of archaic psychic inertia, it “just is”. But the principle of its manifestation and action is also tripartite, that is, Sat-Chit-Ananda or Truth-Consciousness-Bliss. Bramhan is the singular totality of three different irreducible multiplicities; truth, consciousness and bliss which all are fractally identical to each other, and also to the singular field of Bramhan. Multiplicity is not the veil or illusion covering the Brahman-nucleus, but the active principle of its action, structuring and manifestation. When “one” acts, it becomes “multiple”, proliferating infinitely to create exotic structures; nonetheless, preserving its fractal identity with the whole.

Multiplicity or becoming is neither an illusion nor an original sin; it is the shape and pattern of the “becoming” of the inseparable whole called Brahman. A finite “jivatma” is always fractally identical to its ultimate truth, that is, infinite “Paramatman”.

The writer is a structural engineer by profession. Also a poet, he writes regularly on philosophy, religion, science, psychology and popular culture.