Benoît Mandelbrot and Fractals
Benoît Mandelbrot coined the term “fractal” in 1975 to describe the naturally occurring, never-ending, infinitely complex, [often] self-similar, geometric patterns, which look “fractured” or “broken.” Fractals are important because 1) nature creates fractals 2) They are an example of how a few simple rules can create a wildly complex system.
Big Brains. Small Films. Benoît Mandelbrot, The Father of Fractals | IBM. IBM took a chance on Mandelbrot’s revolutionary way of thinking, and this video is a quick retelling of that relationship.
FACT: The colors and shapes you see in fractals are math-based. They are representations of naturally occurring complex shapes, not artistic impressions of shapes. Color choice is meant to help people to see the complex patterns so while it’s math based, it is atheistic in this one way.
Key Terms for Understanding Fractals:
- Infinite: Fractals are infinite. Their patterns repeat forever. Learn more about the concept of infinity.
- Iteration v. Recursion (simple): In overly simple terms, recursion and iteration perform the same task, they “loop” and “repeat” to solve a complicated task one piece at a time, and combine the results. Iteration loops the task until it is done, Recursion factors in the previous result. Learn more.
- Complex and Imaginary Numbers and Fractals: A number line has positive and negative real numbers. If we intersect that line we can chart complex imaginary numbers. If we graph this with a function that creates fractals, and then plots those numbers, it creates the fractal images we see.
- Self Similar: Self similar objects are objects that look the same (or almost the same; AKA similar) when you “zoom in” on a part of the object. In other words, it is an object where the whole object has the same shape as one or more of its parts. See cool examples of how the Mandelbrot set and Koch curve are self-similar here. TIP: Although some famous fractals are self similar, not every fractal is self similar, every fractal can however be described in terms of fractal dimensions and roughness (the next two concepts we will cover).
- Roughness: One can’t accurately measure a coastline (Coastline paradox) due to its fractal nature, but one can measure its “roughness” as a number between dimensions where 0 is a point, 1 is a line, 2 is a square, 3 is a cube, 4 is a tesseract, etc (and thus a coastline can be measured to have a roughness between 1 and 2 when measuring a coastline).
- Fractal Dimensions: Shapes like circles and squares have curves one expects to see. In between these shapes are an infinite amount of odd looking curves (which we can describe in terms of roughness or as “fractal dimensions,” where a fractal dimension is a number that represents a measure of what we can describe visually as roughness). Science used to disregard fractal dimensions until Mandelbrot pointed out that they were essentially a major a key to understanding the geometry of the universe. That is one reason why we call Mandelbrot “the Father of Fractals.”
TIP: Roughness expresses the “fractal dimension” of a curve, that is any dimension between 0 and 1, 1 and 2, or 2 and 3, where the closer it is to the higher number, the rougher it is. For example, Great Britain’s coastline has been measured as D = 1.25, and the “rougher” coastline of Norway has been measured at D = 1.52 (even though a coastline is a three dimensional thing, it is being measured as two dimensional roughness, so the fractal dimension is between a line D=1 and a square D=2… not a square D=2 and a cube D=3). See calculating fractal dimensions (AKA a page on roughness).
TIP: Many computer graphics are created using fractals. Most shapes we see in nature can be created with a simple function by defining roughness. For example, a computer can generate a mountain range if given a fractal dimension between 2 and 3.
FACT: A lot of today’s technology wouldn’t be possible without fractal geometry. Most electronic devices use fractal designs for space saving; some can’t even function without employing fractals.
Why are Fractals and Benoît Mandelbrot Important?
Fractals may seem artistic, but they are based on scientific principles. Mandelbrot revolutionized science, mathematics, and technology with his Fractal Geometry while working at IBM. The “art” aspect of fractals is perhaps the least interesting thing about them. Look out your window; everything that isn’t a man made standard Euclidean shape is essentially a fractal. Fractals are an efficient and natural method of “growing” that nature employs liberally due to her frugality (natural selection is driven by energy efficiency). The human body is largely created from fractals, just look at how the human nervous system iterates in a recursive fashion. Hopefully, you are excited enough now to bare with the technical terms. Make sure to watch the mind-blowing videos below.
TIP: Consider buying this amazing Fractal Documentary from NOVA “Hunting the Hidden Dimension” (Amazon).
Fractals are typically not self-similar. The excellent fractal explainer video describes fractal dimensions and explains non-self-similar fractals.
TIP: The fractal documentary above mentions Kleiber’s law. Kleiber’s Law is based on his observation that, for the vast majority of animals, an animal’s metabolic rate scales to the ¾ power of the animal’s mass. Or, E=m¾.
Who Discovered Fractals?
Mandelbrot wasn’t the first to discover fractals. That credit should probably go to Pierre Fatou and Gaston Julia in 1918 (although Gottfried Leibniz pondered recursive self-similarity as early as the 17th century). Mandelbrot coined the term “fractal” and claimed to be the first to discover the mother of all fractals, the Benoît-named Mandelbrot set. For this reason, Benoît Mandelbrot is called “the father of fractals.” See a more complete history of fractals from WikiPedia.
TIP: The Mandelbrot set contains an infinite amount of Julia sets. Every point in a Mandlebrot set relates back to a Julia set, given some mathematical manipulation, which is part of the reason the Mandelbrot set is so famous. According to Scientific American, “Julia sets had been described as early as 1906 by the French mathematician Pierre Fatou. They were named later for Gaston Julia, who successfully claimed that his work on the sets some dozen years later had greater significance than Fatou’s. Mandelbrot, who was born [in 1924] in Poland, had read the work of both men and studied under Julia in the 1940’s. Later he realized he a shape could be created that contained within it all Julia sets, the Mandlebrot set.”
Benoit Mandelbrot: Fractals and the art of roughness. Here is a Mandelbrot set, bounded to a YouTube video for 21 minutes, why not learn about fractals from the master himself?
FACT: When talking math, fractals, and systems, the word “set” means, “that which is bound within the limits of the system being considered,” for our purposes we are saying everything within the bounds of the limits of the function that creates the fractal. So in the on-page image, everything on the XY axis within the limits of our coordinates (as noted on the image). Learn more about systems.
FACT: It is impossible, for all practical purposes, to accurately measure a coastline in real life. This is because nature seems to employ recursive fractals in its design process. The closer you zoom in, the more surface area you will measure. Oh, the art of roughness, you are complex. I guess that is why you are so popular with math nerds.
What is a Fractal?
A Fractal is a never-ending self-similar pattern driven by a recursive algorithm and complex numbers.
It is a simple algorithm that, when complex numbers are introduced and plotted on a graph, creates an infinite amount of patterns that resemble the growth of branches on a tree, or the way one might think the galaxies and stars appear from a birds eye view. The shapes are self-similar so, although are infinite, they are recognizable in the same way that every snowflake is different.
The simplest example of a fractal is a Sierpinski Triangle, as featured in the video below.
How to Draw The Sierpinski Triangle: A Fractal Zoom.
TIP: Fractals are Recursive. Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other, the nested images that occur are a form of infinite recursion. So a recursive algorithm is a function that loops back to the start each time and results in recursion.
What is a Mandelbrot Set?
The famous Mandelbrot set is an example of using a specific recursive function to create a specific bounded fractal set that contains an infinite number of self-similar math-based patterns that are beautiful to look at. There are many different types of fractals, but a Mandelbrot set is special because every single point on a Mandelbrot set, of which there are infinite points, relates to a Julia set, which itself is infinitely recursive.
Now is probably a good time to mention that there are different types of infinity. Below is a spectacular full-length educational film on Fractals and computer graphics.
The Mandelbrot Set – Numberphile.
TIP: Iteration is a math term, but it means just what it sounds like, a repetition of a mathematical or computational procedure applied to the result of a previous application, typically as a means of obtaining successively closer approximations to the solution of a problem. For instance, you may try to get to the number 2 by saying: 1 + 1/2 + 1/4 + ⅓, etc. You will never get to two, but you’ll keep getting closer with each iteration as you add the new number to the last result.
TIP: Learn more about Fractals from the very cool Fractal Foundation.