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What are Fractals? How is Maths Used to Make Art?

Updated on February 26, 2014
Fractal mathematics turned into art
Fractal mathematics turned into art | Source

As I insist on inflicting my fractals on long suffering friends I am often met with questions about how they are created. How can I turn complicate maths into art? Do I understand the mathematics?

It is actually difficult to find an exact definition of fractals. The father of fractal geometry, Benoit Mandelbrot, defined them as ""beautiful, damn hard, increasingly useful. That's fractals." That might be witty but it is not exactly helpful.

The generally accepted property of fractals is that they are "self-similar" over different scales. This means that if you zoom in on any part of the image you will see the same, or similar patterns. Small sections of the image are as complex, as the whole, which doesn't happen with non-fractal patterns. The concept of 'iterations', doing something over and over again, for example solving a formula for ever increasing values, is integral to fractals. If you plot the results of the iterations in a plane, or a computer screen, you might see a fractal pattern.

Some Fractals from my Collection

Click thumbnail to view full-size
The birth of dragons. I like the way this particular formula gives a slight 3D look to fractals.Green Spill. The same formula used to make a different fractal.Christmas Decorations.Fractal made for Valentine's Day.This type of fractals is called a ducky. I've never been able to figure out why.
The birth of dragons. I like the way this particular formula gives a slight 3D look to fractals.
The birth of dragons. I like the way this particular formula gives a slight 3D look to fractals. | Source
Green Spill. The same formula used to make a different fractal.
Green Spill. The same formula used to make a different fractal. | Source
Christmas Decorations.
Christmas Decorations. | Source
Fractal made for Valentine's Day.
Fractal made for Valentine's Day. | Source
This type of fractals is called a ducky. I've never been able to figure out why.
This type of fractals is called a ducky. I've never been able to figure out why. | Source
Broccoli is a natural fractal
Broccoli is a natural fractal | Source

Fractals and Imaginary Numbers

An added complication is that fractal maths involves dealing with complex numbers. These are numbers that have a real component, and an imaginary component. For example what is the square root of -1? Actually there is no answer since 12 and -12 both equal 1. However, that did not stop mathematicians from 'imagining' numbers that didn't exist. They just decided to represent √-1, as the letter i, and happily proceeded from there.

You might think that such mathematics is pretty useless. However, fractals don't just exist in the imagination of mathematicians. Many natural phenomena are fractal. The most famous example is probably broccoli. Fractal maths is extremely useful, from studying weather patterns, electronics, medicine and physiology to many other fields.

Benoit Mandelbrot, the father of fractal geometry
Benoit Mandelbrot, the father of fractal geometry | Source

Benoit Mandelbrot the Father of Fractal Geometry

Benoit Mandelbrot (1924-2010) is the mathematician who really invented fractal geometry. He developed the 'theory of roughness', and showed that seemingly chaotic phenomena, like clouds, the surface of rocks, the outline of a coastline, had an ordered element to them.

Mandelbrot used IBM computer graphics to visualise the result of fractal calculations. The co-ordinates of pixels on the screen corresponded to complex numbers, with the real component on the x-axis, and the imaginary component on the y-axis. The colour of each pixel is defined by the results of the fractal formulas.

The Mandelbrot fractal rendered at x1 magnification. It doesn't look very interesting, but there are amazingly complex patterns at the edges.
The Mandelbrot fractal rendered at x1 magnification. It doesn't look very interesting, but there are amazingly complex patterns at the edges. | Source

The Mandelbrot Set Fractals

Benoit Mandelbrot also discovered the Mandelbrot set, perhaps the most famous fractal formula. You don't actually need to understand the maths to use it, I wouldn't claim to understand it myself, but, if you are interested, it is a set of complex numbers c, which when plugged into the formula:

zn+1=zn2+C

And starting with z0=0, will give a 'bounded' value for z, no matter how many times the iteration is done. For example, picking C=1 gives a series of z of 1,2,5,26…….the series tends towards infinity, hence 1 is not part of the Mandelbrot set. On the other hand, when C is -1, the series for z is 1, -1, 1, -1……..so -1 is in the set.

When a computer creates an image using the Mandelbrot formula, it tests a vast number of complex numbers (in the example above the complex number component of c was null). It then colours the numbers according to whether it is in the set.

The Mandelbrot fractal might not seem very complex when viewed at normal magnification. However, if you zoom in (a lot) at its edge you discover beautiful, detailed patterns. Fractal artists are still exploring it, and discovering interesting fractals. They have named various regions of it, such as the elephant valley, the seahorse valley and various 'midgets', to make it easier to explain where the fractals were found.

You can see some of my Mandelbrot zooms in the images below.

Some of my Mandelbrot Deep Zooms

Click thumbnail to view full-size
A Mandelbrot zoom in the 'seahorse valley' region.This was created by zooming in by a factor of 2.2E16!
A Mandelbrot zoom in the 'seahorse valley' region.
A Mandelbrot zoom in the 'seahorse valley' region. | Source
Source
Source
This was created by zooming in by a factor of 2.2E16!
This was created by zooming in by a factor of 2.2E16! | Source

Mandelbrot Fractals are Easy!

Mandelbrot zooms are probably the easiest fractals to create. You don't have to try out different parameters, just pick a nice colour gradient that you like, and start zooming in. However you should be aware that they do take a long time to render, because the zoom is so extreme. The ones I show above are magnified by a factor of 1x1016 or more.

I've used Fractal Explorer to make these, which you have to buy. However many of the free programs, like fractal explorer will let you achieve similar results.

A Very Deep Mandelbrot Zoom using fractal eXtreme

Have you ever tried making fractal art

See results

The Julia Fractals

Another formula that is very commonly used for making fractals derives from the Julia set, which defines a set, derived from a function in complex dynamics. Ok, I don't really know what the above means, and will definitely not bore you with the mathematics. The formula is called after Gaston Julia, a French mathematician.

Julia formulas actually have parameters, you can input different values of the real and imaginary components, which can change the shape of the fractal quite dramatically.

A Fractal Slideshow

You Can Also Pick a Colouring Algorithm to Enhance Your Fractal

The way in which the colour of pixels is defined can be altered by picking a colouring algorithm. They can significantly alter the look of the image. Different programs have different ways of achieving thing. In Ultrafractal, the program I use almost exclusively now, there is a huge list of possibilities in the the public library, and new ones are being written all the time, with exotic names such as 'trap the light fantastic' (one of my favourites, or 'enhanced polar traps'.

One very popular colouring algorithm, at least in Ultra Fractal is orbit traps. The colour of the pixels depends on how close the shape of the orbit resembles geometrical shapes, circles, lines, or asteroids, but also flowers or hearts.

Many of the algorithms have different parameters to play with. Different combinations formulas and colouring algorithms, and different parameters, give an incredible number of possible fractals to create. You can of course choose the colour palette you will be working with

Some of my Fractals Based on the Julia Formula

Click thumbnail to view full-size
A Julia fractal with trap the light fantastic colouring.A fractal for ChristmasMuted colours are sometimes nice too.
A Julia fractal with trap the light fantastic colouring.
A Julia fractal with trap the light fantastic colouring. | Source
A fractal for Christmas
A fractal for Christmas | Source
Muted colours are sometimes nice too.
Muted colours are sometimes nice too.

Unusual Fractal Formulas and Colouring Algorithms

Although we commonly associate complex patterns and spirals with fractals, some formulas give simple shapes, and patterns. One of my favourites is the gnarl type of fractal. Other formulas give fragmented-looking images.

Complicated programs such as Ultra-Fractal also allow you to add transformations, such as 3D mapping or kaleidoscope effects. Some of the fractals I've created in that program, don't actually look fractal at all.

Simple or Non-fractal Images I've Made in Ultra Fractal

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An unusual use of the gnarl mapping with Newton's decomposition colouring algorithm.Simple geometry created with the KPK Moebius Julia Barnsley formulaThe same formula and tapestry-x colouring algorithm as the one before, with different parametersThe same KPK formula with 'curves' colouring algorithmFlowers floating down the river, doesn't look much like a fractal, does it?
An unusual use of the gnarl mapping with Newton's decomposition colouring algorithm.
An unusual use of the gnarl mapping with Newton's decomposition colouring algorithm. | Source
Simple geometry created with the KPK Moebius Julia Barnsley formula
Simple geometry created with the KPK Moebius Julia Barnsley formula | Source
The same formula and tapestry-x colouring algorithm as the one before, with different parameters
The same formula and tapestry-x colouring algorithm as the one before, with different parameters | Source
The same KPK formula with 'curves' colouring algorithm
The same KPK formula with 'curves' colouring algorithm | Source
Flowers floating down the river, doesn't look much like a fractal, does it?
Flowers floating down the river, doesn't look much like a fractal, does it? | Source

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    • profile image

      Jincy 2 years ago

      To be honest, my Mandelbrot and Julia imaegs are a long distance from my current area of research, which involves something called the Assouad dimension, and has been focusing more on self-similarities. I'm also only in my first year, so haven't gotten as deep into things as I might. If you are interested in what Kawahira is doing, your best bet would probably be to read the thesis and paper linked at the top of the page you linked to.xander

    • Blackspaniel1 profile image

      Blackspaniel1 3 years ago

      Nice images here. Oh, those complex numbers with an imaginary part are not so scary, the imaginary axis id=s the y-axis, and they are two dimensional numbers. That is the simple way of looking at them. The word imaginary was the poorly chosen word that causes so much confusion.

    • profile image

      bfilipek 3 years ago

      Fractals are amazing things. From such simple equations comes so complex images. This kind of effects are great for learning maths and programming. Have you tried to code a fractal in and programming language?

    • merej99 profile image

      Meredith Loughran 3 years ago from Florida

      I LOVE this hub and I LOVE fractals. There was a story about fractals on the Science Channel and I was completely fascinated by them. I don't understand the math of it but the natural art and patterns are endless and amazing. I liken it to facing mirrors where the image of itself is reflected indefinitely.

    • Glenn Stok profile image

      Glenn Stok 3 years ago from Long Island, NY

      Displaying fractals in a visual way is extremely interesting. It brings out a whole new dimension for the mind to envision. Zooming into a Mandelbrot Fractal gives a really good interpretation of infinity. It also becomes a representation of evolution, endlessly repeating itself and reproducing similar elements as it expands into the future, which is visualized when one zooms into it.

      Congratulations on this hub being selected in the top 10 for Monday’s HubPot Challenge.

    • aa lite profile image
      Author

      aa lite 3 years ago from London

      Thank you so much Christin, I imagine you've seen most of these already, since I insist on infecting them on bubblers!

    • ChristinS profile image

      Christin Sander 3 years ago from Midwest

      Congrats on getting your hub picked today! It's a great hub and I've been a fan of your fractal art for awhile now. Voted up and sharing.