Mandelbrot, Fractal Geometry, & The Language of Creation

Absolutely every living thing and every organic compound consists of complex mathematical patterns. Mathematicians can even express some of these patterns with relatively simple mathematical formulas. The most well-known example of this phenomenon is the Mandelbrot set, named after mathematician Benoit Mandelbrot who first discovered it, and commoditized our understanding of fractals. For a basic overview and understanding of what the Mandelbrot set is, please click here.

The Mandelbrot set can be plotted and expressed in the form of a fractal. A fractal is simply defined as a never-ending pattern. According to, a fractal is defined as, “a geometrical or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions”.


A unique property of all fractals is that of self-similarity. When viewed at any zoom factor, the geometric structure of the fractal repeats itself in a consistent manner. This cycle continues indefinitely, and approaches toward infinity, as the zoom factor is increased.

From Einstein to Mandelbrot

Branching Fractals in Leaf Veins

We encounter fractals every single day in nature without even realizing it. Leaf patterns, snowflakes, and seashells are a few obvious examples. Within the scope of mathematics, we can physically observe the formation of fractal patterns by plotting formulas such as the Mandelbrot set. However, the one consistent reality in this exercise, is that there must be a ‘first cause’ of initialization for the formula to begin replicating. A mathematician, or “mathematical designer” must first plug in the initial variables into the formula, which then replicates itself on a playback loop. This means that the result of the expression is then plugged back into the beginning to become the new initial variable.

The formula for the Mandelbrot set is quite simple, all things considered. It is obtained from the quadratic recurrence equation:

Thanks to Einstein, we now know that space and time are inextricably linked, as he forever solidified in his theory of relativity. In what is perhaps the most recognized mathematical formula of all time, E=MC2, Einstein demonstrated that energy and mass are equivalent, and transmutable. Scientists agree that a ‘first cause’ had to be present in order to initialize all of the events that led to the creation of the universe and everything in it. Within the scope of secular science, this event is most commonly referred to as the Big Bang.

However, for the Big Bang to occur, and enormous amount of stored energy had to be dispelled. Remember, according to the first law of thermodynamics, energy can neither be created nor destroyed. And once again, according to Einstein’s theory of relativity, both space and time are linked in such a way that one cannot exist without the other. Furthermore, the expansion of either space or time must necessitate the other’s expansion as well. So where did the stored energy, its spatial dimension in which it existed, and the mass and matter that contained it come from? More on that in a moment.

Nature’s Geometry

The presence of fractals in nature wasn’t an observation lost on Benoit Mandelbrot. In fact, he was mesmerized by the subject. In 1977, he wrote the quintessential book on the topic, The Fractal Geometry of Nature. This book was the first of its kind to demonstrate that organic patterns found in nature could actually be replicated mathematically!

The Barnsley Fern

British mathematician Michael Barnsley further explored this topic established by Mandelbrot’s earlier work, in his 1988 book, Fractals Everywhere. In his book, Barnsley describes a fractal that he created called the Barnsley Fern, which resembled the iteration and leafing structure of a fern leaf. What’s absolutely remarkable about the Barnsley Fern, is that it not only generally or loosely appears fern-shaped – it literally mimics the intricacies and structure of a specific fern, the black spleenwort.

Whereas Mandelbrot demonstrated that fractals exists all throughout nature, Barnsley discovered the specific mathematical formula that produces a specific leaf structure of a known specimen – at least within a geometrical two-dimensional plane.

The code developed by Barnsley to produce the fern, is an example of an iterated function system (IFS). The formula is expressed in the following form, and the values are provided in Barnsley’s matrix of constants:

What’s even more fascinating, is that mathematicians have been able to use Barnsley’s equation to produce other fern varieties, such as the Cyclosorus, simply by changing the coefficients slightly. An online fern generator even exists, so that users can plug in their own values to create unique fern-like fractals.

Chaos Theory

The study of fractal geometry is considered part and parcel of a broader branch of mathematics called chaos theory. Chaos theory focuses on the behavior of dynamical systems that are highly influenced by, and sensitive to their initial conditions. Chaos theory states that within chaotic complex systems that feature behavior that appears random, there are underlying patterns, feedback loops, repetition, self-similarity, self-organization, and of course, fractals.

In common parlance, the word ‘chaos’ typically means “and absence of order”. However, in complex mathematics, there is indeed an order within the chaos – though it consists of extremely complex and complicated information.

The Double-Rod Pendulum

A simple and well-known chaos theory illustrative exercise is that of the double-rod pendulum. This is a pendulum that features a secondary freely swinging pendulum attached at its end. The adjacent graphic shows a representation of the double pendulum if the movement of the secondary pendulum were plotted graphically. The result is considered mathematically chaotic, since although it is a closed dynamical system, the initial conditions influence its behavior. Remember, chaos as a mathematical property means order actually does exist in its movement, but the calculations are incredibly complex.

The second law of thermodynamics states that the entropy of a isolated system never decreases. Entropy is a complex phenomenon that can be expressed in various ways. Generally, it is the quest from order to disorder, or thermodynamic equilibrium. Maximum entropy is considered to be in a state of equilibrium, which is equivalent to maximum randomness, or chaos. Non-isolated systems may lose entropy, provided that their environment’s entropy increases by at least that amount, so that overall entropy increases. A practical example of this phenomenon is a refrigerator. Entropy is reduced inside of the appliance, but increased in its environment. This is easily observable by the increased amount of heat dispelled from its power source.

It is, and has been the position of this blog to reject any notion that science and creationism cannot coexist. In fact, the understanding of one is greatly amplified by the embracing of the other

This article barely scratches the surface of complex topics within the scope of math and physics, and yet we’ve already established that constraints, laws, behaviors, and constants within our physical universe lead to a paradox of initial cause. Even systems considered mathematically chaotic, adhere to the boundaries of physical law. It is an observable reality of our physical universe that ‘something’ cannot arise from ‘nothing’.

The Entropy Paradox

The most intelligent and ardent adherents to the idea of a Creator-less Big Bang cannot account for the very laws of physics and reality being suspended in every imaginable way, to allow for the initial sudden existence of energy, mass, and matter. These mysterious ingredients then somehow exploded into a state of maximum entropy, or disorder – and inexplicably reverse the second law of thermodynamics, moving from disorder to order; and eventually creating an orderly universe with living creatures and lush green flora. At some point between the Big Bang and thriving rainforests, the second law of thermodynamics is righted, and everything begins evolving toward a state of maximum entropy. When did this incredible transition and reversal take place exactly?

The mysteries and amazing features of our universe are carefully constrained within the boundaries of the laws of physics and mathematical properties that undergird our existence. 2 + 2 cannot and will not ever equal anything but 4. Likewise, the laws of physics cannot be fundamentally altered or ignored. However, a mathematician or physicist can experiment with and test these phenomenon. In this regard, scientists can be considered ‘designers’ in a certain sense; utilizing the existing universal laws and constants to create certain outcomes, such as the case with Barnsley’s fern.

The question is, if a mathematical designer is required to produce the relatively simplistic two-dimensional representation of an organic geometric pattern as in the case of Barnsley, why is it beyond the pale to ask scientists and physicists to at least consider the idea of an intelligent designer to be behind the creation of infinitely more complex three-dimensional, living organisms?

Creation Through Mathematics?

Fractal Tree

As a tree grows, it is fascinating to think that it is slowly replicating a fractal equation as it branches outward. The main trunk slowly grows vertically, splits into two main branches, these in turn split into multiple branches, and so on. The self-similarity of the pattern is evident no matter what zoom factor the structure is viewed through.

How does the tree ‘know’ how to produce this structure? A computer programmer can enter certain combinations of data input and receive specific output based on what the computer is being commanded to do with the input. However, most mainstream scientists and physicists deny the existence of an omniscient ‘programmer’ that wrote all of the unfathomably complex code for our universe that produced the output we observe around us every day. It appears that as an observable fact, extremely complex mathematical formulas contained within the DNA of living organisms serves as the source code for organic geometry. We can only replicate very crude geometric similarities with computers – but only with the provided input of a programmer.

Why is a designer/programmer required for the simplistic, but not the complex?

In order for one to argue an ordered universe that adheres to unchanging physical laws and mathematical constants without a Creator, one faces a number of difficult paradoxes and conundrums that have to somehow be addressed. Below are just a few of perhaps hundreds that could be presented:

  • How did the initial order contained in universal laws and constants become established from the disorder of the Big Bang aftermath?
  • Where did the Big Bang source energy and material come from? How about the very fabric of spacetime for it to exist within?
  • If matter and energy were spontaneously created, what suspended that phenomenon within our universe?
  • After the Big Bang, an ultimate state of disorder and entropy existed – how did then order proceed from that to lead to the universe we observe today?

It is, and has been the position of this blog to reject any notion that science and creationism cannot coexist. In fact, the understanding of one is greatly amplified by the embracing of the other! To at least accept the hypothesis that a Creator may in fact be responsible for the origins of our universe does NOT necessitate the embracing of any particular religious dogma or theology. In this manner, secularism and creationism can simultaneously coexist as well.

One thing is certain, it isn’t a lack of faith that precludes one from accepting at least the possibility of a Creator; it is a misplaced faith. The faith and dogmatic devotion to an atheistic version of scientific exploration is palpable among the world’s leading science scholars and researchers.

In 2016, Chinese researchers wrote an exceptional paper for a peer-reviewed journal called PLoS ONE that explored the intricacies and functions of the human hand as a well designed precision instrument. In the paper’s abstract, the piece stated, “Hand coordination should indicate the mystery of the Creator’s invention”. Almost instantly, the science and research community went ballistic. Researchers threatened to boycott PLoS ONE, and the journal was forced to retract the piece. Scientists and researchers took to Twitter to voice their disdain. The incident caused an international stir across the research community. Lead author Ming-Jin Liu went on the defensive and issued a statement that revealed that he and his team were fellow atheist travelers themselves, and the inclusion of the word “Creator” was simply due to a mistranslation of the word “Nature” between Chinese and English. However, the paper was never reinstated, and the research went down as an abject failure due to the incident.

The issue isn’t that the idea of a Creator falls outside of the scope of legitimate science – it is the fear that if there is a Creator, then we are all inferior to that higher power.

This incident, and others like it, reveal an underlying dogma – not an open-minded exploration of scientific endeavor. The presumption that there is no Creator, and everything is chance is established first and foremost with almost religious devotion, and all scientific exercise is conducted from that framework. However, this isn’t utilizing the true spirit of the scientific method. Scientists are quick to retort that a Creator can’t be a valid hypothesis, because the idea cannot be tested or replicated. However, there are recent strange discoveries in quantum mechanics and mysteries such as anti matter that cannot be tested or replicated either. Yet these are still explored as legitimate phenomenon – as they should be.


The issue isn’t that the idea of a Creator falls outside of the scope of legitimate science – it is the fear that if there is a Creator, then we are all inferior to that higher power. And that is ground that most scientists are not willing to cede. In their pride, they continue to plow forward while rejecting any notion of a Creator, and invariably wind up at the same dead ends where there is no natural explanation. How rich and fulfilling science would be if only the possibility of a Creator was considered legitimate – even without any particular religious tenet attached. To arrive at the inevitable conclusion of a Creator, there is no need for one to adopt any religious belief.

Religious expression, belief, and dogma can be addressed completely outside of science – but conceding the probability of a Creator isn’t religion. It’s a legitimate possibility, when all others fail to satisfy answers to the most profound mysteries of the universe.