Image by Darren Tunnicliff.
There is an idea – strange, haunting, evocative – one of the most exquisite conjectures in science or religion. It is entirely undemonstrated; it may never be proved. But it stirs the blood. There is, we are told, an infinite hierarchy of universes, so that an elementary particle, such as an electron, in our universe would, if penetrated, reveal itself to be an entire closed universe. Within it, organized into the local equivalent of galaxies and smaller structures, are an immense number of other, much tinier elementary particles, which are themselves universe at the next level, and so on forever – an infinite downward regression, universes within universes, endlessly.
– Carl Sagan, Cosmos
Recent scientific evidence has caused a revision of traditional understanding in the fabric of space-time. Touching on the idea Carl Sagan presents in the excerpt above, more and more academics are contending that, in light of such evidence, time possesses a “fractal” structure, meaning that each and every moment literally encompasses the eternity of time.
So what does fractal mean in practical terms and what does it have to do with time?
In the late 1800s, a brilliant mathematician named Georg Cantor took on a simple question: What would happen if you took a line, divided it into thirds, took out the middle third, and repeated the process over and over again?
An example of a Cantor set. Image courtesy of the Kansas Geological Survey.
Mathematicians have known about problems like these for years. They used to call them “monsters” and “pathological equations” because they knew they had no answers. They were considered nothing more than a joke in the mathematical world, but Cantor was far too intrigued by the problem of infinity, and he wouldn’t let it go.
He spent years trying to understand this monster, until he was finally institutionalized. He ended up spending the rest of his life in and out of insane asylums.
Below is a typical “monster,” called a Koch curve. Notice how the “fractal seed” (the triangle) is repeatedly added to the sides of the next triangle, just like the lines in the Cantor set are repeatedly subtracted from the previous line.
The progressions of a “Koch Snowflake.” Image courtesy of Fredrik Jonsson.
The Koch curve demonstrates the monster’s problem, because the farther you zoom in, the more complex or “rough” the object gets. Therefore, the closer you look, the greater the circumference of the shape actually grows.
In fact, the circumference theoretically grows to become infinite, while the area somehow remains finite. Since this wasn’t understood in classical geometry, these shapes were abandoned as theoretical absurdities that could never exist in the real world. Little did these mathematicians know, they were surrounded monsters every day.
It wasn’t until 1975 that mathematician Benoit Mandelbrot took up this problem, insisting that these shapes could finally be understood. Mandelbrot spent many years contemplating the mechanics of these impossible shapes until he was hired by IBM. There he was able to plug the infinite set of numbers into newly invented computers capable of processing them. After mapping out several of these monsters, Mandelbrot came up with a simple equation for the ultimate monster. Essentially, he combined all the monster sets together and came up with the simplest equation:
Z = Z^2 + C
This equation is called a ‘mathematical feedback,’ an equation where you plug the answer back into equation again and again. With the help of IBM’s new computers, Mandelbrot was able to have the computer endlessly reiterate the equation until he came out with this:
This video is simply zooming into a graph made from the points that come out of Mandelbrot’s mathematical feedback equation. The colors you see are similar to the colors on a topographic map; they denote how fast the number is traveling away from the center. What came out of these computers was an altogether new kind of geometry, which Mandelbrot coined as “fractal” (from the Latin “fractus” or “to break”).
In the past, geometry was restricted to Euclidian shapes like, circles, lines, triangles, squares, cylinders, etc. but this geometry never came close to describing the intricacies of nature. As Mandelbrot explains, in his groundbreaking book, The Fractal Geometry of Nature, “Clouds are not spheres. Mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Nature exhibits not simply a higher degree but an altogether different level of complexity.” 
Fractal geometry became known as the geometry of nature. As Keith Devlin, “the math guy” of Stanford University, states, “classical mathematics is really only well suited to study the world that we’ve created, the things we’ve built using that classical mathematics. [But] the patterns in nature, the things that were already there before we came onto the planet, the trees, the plants, the clouds, the weather systems, those were outside of [classical] mathematics.”
One reason that fractal geometry works so cohesively in nature is because fractals have what is known as a “self-similar” structure. This means that the more you zoom in, the more you see the same image, hence, ”the structure of every piece holds the key to the whole structure.” 
Photo by Joe Shlabotnik.
If you cut off one flower of this cauliflower, it would resemble the whole, and if you cut off one piece of that flower, it would also resemble the whole.
One good example of this self-similar structure that we all know is the tree-like structure of the bronchial tubes in our lungs. A recent study on fractal physiology by William Deering and Bruce J. West of the Department of Physics University of North Texas states that, lungs have a “branching structure that attempts to put end points in the neighborhood of every point in space. In this way, the one-dimensional line fills the two-dimensional area in an optimal way in order to perform the gas exchange function for which this lung is designed.”
Photo courtesy of Yale University, demonstrating how the fractal patterns in the structure of human lungs allows for optimal efficiency.
Suddenly the efficiency of nature’s fractal structure seems so basic, and simple, and at the same time it is a totally revolutionary concept to us. James Brown, a mathematician from the University of New Mexico explains “If you think about it for a minute; it would be incredibly inefficient to have a set of blue-prints for every single stage of increasing size, but if you have a fractal seed, a code that says ‘when to branch’ as you get bigger and bigger, then a very simple genetic code can produce what looks like a complicated organism.”
This diagram by Ed Mortimer shows how the fractal seed (a) repeats itself in a self-similar way in nature.
This self-similar fractal structure can be found everywhere in nature, from the branches on trees, to the neural structure of the brain, to weather patterns, and the structure of DNA. It has even been shown that the movement of animals follows a fractal pattern.
In fact, a fractal structure can be recognized from microscopic atomic structures all the way to the macroscopic structure of the universe as a whole. As physicist Francesco Sylos Labini of the Enrico Fermi Centre in Rome declares, “the universe continues to look fractal as far out as our telescopes can see.”
Since space and time are really the same, it makes sense that if all space in the universe follows a fractal structure, then time must have a fractal structure as well; or maybe it’s the other way around. Recent studies have shown that “space-time is regarded as a Cantor set with a very high dimensionality. In fact its dimensionality is infinite.” If this were the case, space-time would also be like the Mandelbrot set, you could zoom in to any moment forever and you would see the whole of time repeated within.
As Carl Sagan says, “this is the only religious idea I know that surpasses the endless number of infinitely old cycling universes in Hindu cosmology.”  This thought alone must give us pause. If time were fractal, it would be infinite and self-similar, and if time is self-similar it means that each moment literally contains every moment from the future and the past.
Even more astounding though, is that scientists are now finding evidence the fractal structure of time could be the reason behind nature’s fractal structure.
The Chinese Institute of Applied Physics and Computational Mathematics recently published a study, which theorizes that “the fractal metric space-time influences physical behaviors.” In more explicit terms, a study from the Indian Institute of Tropical Meteorology traced the presence of this phenomena to even the smallest realms of existence, “Dynamical systems in nature exhibit self-similar space-time fractal fluctuations of all scales down to the microscopic scales of the subatomic world, i.e. the vacuum zero point energy fluctuations. Self-similarity implies long-range correlations, i.e. nonlocal connections in space and time.”
Since life on this earth began with elements featuring fractal structures, and mirrored foundations are found at the base of nearly every living thing, further study in fractal mathematics and it’s presence in nature could explain the how life started, and furthermore, could unlock secrets behind our evolution and even the development of our consciousness.
1. Mandelbrot, Benoit B.. The fractal geometry of nature. San Francisco: W.H. Freeman, 1982. 1. Print.
2. Mandelbrot, Benoit B.. The colours of infinity: the beauty, the power of fractals. Bath: Clear, 2004. 44. Print.
3. Sagan, Carl. Cosmos. New York: Random House, 1980. 162. Print.