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Fractals - Mathematical Computer Art

Updated on August 19, 2014

What is fractal art?

Fractal art grew out of the work of Benoit Mandelbrot. Mandelbrot developed fractal geometry. This has been applied to the use of computers to make fractal art. Not all fractal art uses the Mandelbrot set, but most uses either the Mandelbrot set or the Julia set, which is derived from the Mandelbrot set. The image shows a graph of the Mandelbrot set. All computer generated fractal art is a plotting of this formula, with other mathematical material to modify it. I will explain how each fractal program I discuss produces the algorithms or otherwise modifies the basic formula to produce a unique image.

One of the frequent features of a fractal image is one or more spirals. Not all fractals have these, but most do. Sometimes it is not obvious.

One thing many people do not understand about computer art in general, and fractal art in particular, is that the art produced is as much designed and created by the artist as any other form of art. Many people think the computer does the work of design. The artist actually designs the image. Most fractal art programs have a graphical user interface (GUI) that shows what the image will look like, as it is being modified by the artist. When the artist likes what he or she sees, the program can be used to make a small set of instructions that will then tell the computer how to calculate the image by setting the color of each pixel in the image.

Fractal art has a unique characteristic. If you zoom in on the fractal, you will see the smaller and smaller detail, until you reach a point where things become "fuzzy" enough there is no point in continuing to zoom. Each formula devised by an artist has an infinite number of viewpoints, and levels of zoom.

Fractal art is a new genre of abstract art, still not widely recognized, though it is becoming more familiar as time goes on. If you become more observant, and you are familiar with fractal art, you will begin to see fractal designs in nature. Nature is full of them!

All images below made by Pat Goltz.

Ultra Fractal

Ultra Fractal is the first fractal program I used, and I started using it about ten or eleven years ago. It was developed by Frederik Slijkerman. It has an extensive library of algorithms which have been developed by volunteers. Its unique feature was the ability to make layers, which then could be used through various merge modes (some paint programs also have these) to change the appearance of the image. I have seen images that contained up to 100 layers. Masks are sometimes used to block out parts of layers. The cost for the program is, in my opinion, quite reasonable.

Below I will show you some examples of the images I have produced with Ultra Fractal.

Early fractals

This was one of the earliest with which I was pleased. It is called "Spring Snow". It shows the spiral characteristics of a fractal image.

In this one, I used an algorithm that maps a fractal onto a three dimensional egg shape. The title is "In the Beginning God", meaning God created the heavens and the earth. An egg represents beginnings.

Think of some of the strange creatures encountered by the Enterprise in Star Trek. This is called "Organic Space Ship".

I made the one below for a contest. I didn't win anything. It is also only a single layer, and I used a canned coloring algorithm. It WAS an early work.

It is called "Consider the Lilies", i.e. of the field, how they grow; they toil not, neither do they spin:

And yet I say unto you, That even Solomon in all his glory was not arrayed like one of these.

One of the other fractal artists sponsored a competition using the Barnsley formula. He would select the best to be displayed on his web site. I made three different images from the formula. The two I show here illustrate how DIFFERENT a formula can become when modified with algorithms.

The first one is entitled "Barnsley Two".

He selected the second one for his web site. It is entitled "Harbinger of Spring".

Later Work

This one is entitled "Cathedral" for what is probably an obvious reason: Gothic arches. This particular pattern is called a Sierpinski, characterized by triangular patterns.

This is another Sierpinski based fractal. It is entitled "Arcosanti". This is named after Paolo Soleri's community north of Phoenix, Arizona.

One more Sierpinski. This one is entitled "Peacock Rhombi".

The squiggles are known as gnarls in fractal art circles.

This pair was derived from the same basic formula. Notice how different they look! One of the things you will probably notice is the texture in the second image. Textures are one of the things you can produce with algorithms.

The first is entitled "Flaming Wind" and the second is "Flaming Wind Spiral".

I never came up with a good name for this one. It reminds me of some of the designs used in Danish Modern furniture, particularly for some lamps I have seen.


There is an email list for Ultra Fractal users. On this list, people exchange instruction sets for fractal images. One thing that is encouraged is called "tweaking". Tweaking is when another user plays with the instruction set to change the appearance of the fractal. Then he will post his results back to the list. This is a very educational activity for many people.

This was the result of someone tweaking my fractal and then me tweaking it back. I never came up with a good list for this one, either.

This one reminds me of an Agave plant, so I called it "Succulent".

I will be adding more to this lens in the next couple of days. Stay tuned!


Xenodream is another fractal program that is reasonably priced. It was developed by Garth Thornton. It is used to make three dimensional fractal images. I like to make images I call "fractal sculptures" and I hope someday I can get a few of them "printed". The technology to make actual physical sculptures from an image such as these exists.

Xenodream is charactized by having both coloring and lighting algorithms. The program has one or more holons. The perameters of the holons are set individually, and the coloring also can be modified at will. Lightings are like shining spotlights on the finished render, but have more characteristics than mere spotlights. Parts of the "sculpture" will actually cast shadows on other parts, and as the "spotlights" are moved around, the shadows shift.

I have designed quite a few of my own lightings, and in fact, marketed a set of over 80 of them on the site 3DCommune, which has since disappeared.

You will observe below how the SAME render can take on a very different appearance, depending on the lighting used. In this next pair and several others, I will show this.

As I will show you, this is TRULY a three dimensional object generator! These are cross-eyed stereograms I made in Xenodream. Cross your eyes and merge the two images. Observe that not only can you see them in three dimensions, but because of their transparency, you can see INTO them, to see things on the interior!

The first one is called "Ribbonwrapped Stereogram" and the second one is called "Tiger Ribbon Stereogram".

I think that it would be possible to obtain actual sculptures by using 3D printing, and I hope to be able to afford to test this someday. I would like to have one of these in my living room.


This one is called Orchid Vase. I tweaked it at the request of another fractal artist, who gave me the original file.

In this pair, you can see that there is a texture in places on each of them. The texture is a result of "incomplete" rendering. I left it there on purpose. Sometimes you can't get rid of this entirely, and other times you can. The rendering mechanism is something like a progressive jpeg. It first makes a very coarse version of the entire image, and then refines it so that it is half as coarse as it was before. It keeps doing this until the artist stops the process.

Again, observe how different lightings change the appearance of the fractal.

The first one is called "Curled Glass" and the second one is called "Curled Rainbow".

These two use my own lightings. African Dance was a modified version of a lighting that comes with the program.

The first one is called "Flower - African Dance" and the second one is called "Flower - Sharp Dramatic".

One more pair.

The first is called "Mod 3" and the second one is called "Fractal Pottery". The pattern on the second one is a coloring rather than a lighting.

Using the same lighting...

The first one is called "Spaceship" and the second one is called "Corrugated Technopipe". I enjoyed thinking up the name for that one! :)

This one is called "Curved into Frame".

Eat your vegetables! This one is called "Artichoke Sculpture".

This one is called "Convoluted".

This one is called "Toris". Notice the fine swirled texture appearance on each of the blades. If you scroll it slightly up and down on your screen, you will see movement.

Finally, I give you "Swirled Gimble". Some of these are named after the holons I used. In Xenodream, a holon is one of the elements used to design the image.

Do you make fractals? - Do you like to look at them? What are your experiences?

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    • profile image

      Guy1982 4 years ago

      I'm not into maths, but this was very interesting! Thank you

    • Pat Goltz profile image

      Pat Goltz 5 years ago

      @SteveKaye: Thank you! I hope to put up some from Apophysis very soon. I have a huge collection of these. And thank you for all the blessings!

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      SteveKaye 5 years ago

      These designs are beautiful. Thank you for showing this artistic side of math.

    • Pat Goltz profile image

      Pat Goltz 5 years ago

      @ohcaroline: Thank you! Learning the programs takes awhile, but the community of fractal artists is very helpful. I hope to put up some Apophysis images soon. I have many of those.

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      ohcaroline 5 years ago

      These designs are awesome. I would like to try making some designs like these.

    • Pat Goltz profile image

      Pat Goltz 5 years ago

      @angharad: Thank you! I will post some from Apophysis soon.

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      angharad 5 years ago

      Wow, these are amazing!

    • Pat Goltz profile image

      Pat Goltz 5 years ago

      @sheilamarie78: Thank you! I will be posting some flame fractals soon. Please drop by again. :)

    • sheilamarie78 profile image

      sheilamarie78 5 years ago

      I really enjoy looking for fractals in nature. These examples of fractal art are stunning! Congratulations on such fine work!

    • Pat Goltz profile image

      Pat Goltz 5 years ago

      @darko45: Thank you very much! I will be adding more real soon; I have Apophysis to do.

    • profile image

      darko45 5 years ago

      Great stuff, the graphical representations of fractals is amazingly good. Thanks for shared

    • Pat Goltz profile image

      Pat Goltz 5 years ago

      @anonymous: That would be neat! I am happy that you found my lens inspirational!

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      anonymous 5 years ago

      I love to see the graphical representations of fractals. I think I have at least two fractal generators on my hard drive. I haven't played with them in ages though. Maybe I'll start up again.