Fractals as an illustration in the argument for Free Will

I spend some of my spare time in discussions with a very interesting group of people who like to discuss philosophy, science, and all kinds of other things.  It’s fun for me.

One of the favourite topics in this group is the concept of free will.  It seems that many of the most vocal people in the group favour the idea that there isn’t any.  I am not one of them!  I have made my arguments for free will to them, but these arguments fall on deaf ears, most of the time.  I have, for some time now, thought my experience zooming in to the deepest parts of fractals was somehow illustrative of the (in my opinion) flawed logic that is often used as “evidence” that we have no free will.  I was wondering how I could bring this visually to the group and make them see the idea I was trying to convey.

Last night I realized I already have this illustration fairly handy.  It’s the Key, from my series entitled “The Ball Went Over the Fence”.  Some of you may remember this one from several years ago.  The Key shows what part of the large fractal image I zoomed in on to make the next smaller fractal image.

Zoom Key, The Ball Went Over the Fence series

Wikipedia defines Free Will in the following way:  “Free will is the ability to choose between different possible courses of action unimpeded.” The argument many in the group make, against the existence of free will, is that everything is caused by what went before it. Wikipedia also states that “Some conceive free will to be the capacity to make choices in which the outcome has not been determined by past events.”

My main problem with this is that while it is usually easy, from the standpoint of the outcome, to see what came before, and possibly follow a chain of causality, there is no reason to assume that means a choice made is the only choice that could have been made, given all previous conditions.  We can see into the past just fine.  The future, however, is indeterminate.

So, looking at the Key, shown here, you can see a white outlined square in each square image of the key (click on the image to make it bigger, if that helps).  From each outlined square, you can follow the white angled lines to the image found at that location if you were able to zoom in while in the fractal software.  If we start at the smallest square image (the final outcome), we can see where it came from (it’s very close to the top of the next larger image).  Likewise, if we follow to the next one, we see that it came from a very tiny place in the top middle third.  And if we keep going, we can see that the third image came from a recognizable portion of the second largest image.  And that image came from a very tiny spot in the largest image.

This is the chain of causality – it goes from the outcome back to the origin.  It would, however, be impossible to go from the largest image to the smallest one without the Key in place to guide you.  There is no chain in that direction, because it looks into the future.  The outcome would never ever be the same twice.  There is absolutely no logical reason why that particular tiny spot was the one chosen on the first image to zoom in on and make the second one.  In a fractal, while constrained by the mathematics of that fractal, the possibilities at each level of zoom are for all practical and human purposes, infinite.  I’ll grant you that maybe I would have zoomed in on an area near it, or any one of the areas where you can see the little greenish greyish balls.  Just because those areas look interesting to me.  But they all look interesting, and certainly from the perspective of the large image, equally so.  If we look at the second largest image and are choosing where to zoom in for the third… even if you make the argument that I will almost definitely choose a square featuring a ball… that square is never going to be the same exact square.  And now we get to the third image, and you can see no reason why I would have chosen to zoom in as much as I did, and in the area that I chose.

It was my free will in action, plainly and simply.  I chose, unimpeded.  The outcome was never a given.


Biological Forms

Examples of fractals in biology are not difficult to find, and indeed if the universe is fractal, there should be a fractal component to all biological forms.  In the post entitled The Photographs, in which I have captured some natural fractal forms, there are at least five forms which are biological.  In the post entitled Butterflies and Moths, there were several digitally generated fractals which just happened to look biological.  Anyone who has looked up the word fractal has probably been given the example of the fern, or the romanesco, or even the tree.  In fact, people can create extremely realistic looking plants using software that takes advantage of fractal geometry.  Our lungs, and our vascular systems are obviously fractal in nature.  Ever looked at a sea slug?  Beautiful little fractals.

When I create my fractal digital art, and sometimes watercolours, I don’t try to make things that are biological, but I recognize natural forms when I see them and they pop up on their own all the time.  The fact that I’m not making them on purpose somehow speaks to my scientific side, and relates them to evolutionary theory.  I talked about this a little bit in The rose and the creation process as well.

These two pieces are examples which are maybe not as obvious as the butterflies but do remind me of biology just the same.

Cell Division. Lianne Todd. Watercolour on Aquabord. 6x6". $175.00

Cell Division.
Lianne Todd.
Watercolour on Aquabord.

Triad. Lianne Todd. Watercolour on Paper. 20x20". $625.00

Lianne Todd. Watercolour on Paper.


Further into the realm of Fantasy

Often, when I’m voyaging through the through the little fractal universes I have generated using the software which I am so thankful exists, I encounter ‘places’ that look like they belong in an illustration for a book I’ve read somewhere along the way. I also encounter characters that look like they belong in those places.  Such was the case for this piece:

The Mage Emerges Digital Art Printed on Metal, single edition 24x24" $425.00 Lianne Todd

The Mage Emerges
Digital Art Printed on Metal, single edition
SOLD.  Private collection.
Artist Lianne Todd

Butterflies and Moths

Insects, and particularly butterflies and moths, are recurring motifs that I often encounter when I’m creating fractals.  Sometimes, it’s just the simple shape, and other times it seems to be a whole detailed creature.  Sometimes it’s done with what I call the ‘regular’ fractal generator and other times with the flame fractal generator (more on those differences later).  If a mathematical formula iterated over and over by a computer can randomly generate images like these in a matter of minutes or hours, imagine what the physical forces of nature and a few billion years of evolution can do with a periodic table of elements (and, shall I say, an underlying fractal structure?).  Oh wait, you don’t have to imagine.  You can go outside!


(All images are watermarked and copyrighted)

Butterfly Hub Digital Art printed on metal, single edition 20x20" $325.00

Butterfly Hub – Artist Lianne Todd
Digital Art printed on metal, single edition

Detail of Butterfly Hub

Detail of Butterfly Hub

Butterflire - Artist Lianne Todd Digital Art printed on metal, single edition 20x20" $325.00

Butterflire – Artist Lianne Todd
Digital Art printed on metal, single edition

Detail of Butterflire

Detail of Butterflire

Mother of Moths - Artist Lianne Todd Digital Art printed on metal, single edition 12x12" SOLD

Mother of Moths – Artist Lianne Todd
Digital Art printed on metal, single edition
SOLD. Private Collection.

Pollinator - Artist Lianne Todd Digital Art printed on metal, single edition 16x16" $225.00

Pollinator – Artist Lianne Todd
Digital Art printed on metal, single edition

A thumbnail of the raw generated fractal - just to illustrate part of the process.

A thumbnail of the raw generated fractal – just to illustrate part of the process.

The ‘rose’ and the creation process.

See the “rose” in the header of this blog?  It’s a selected portion of one of the first fractals I ever generated, using one of the many programs available for such purposes.  It was a completely random occurrence, really.  I was playing around with formulas, and voila!  One of nature’s most recognizable shapes, noted for its beauty, appeared before me.  A little colour tweaking, some removal of extraneous image parts, and there it was.

This is the wonderful thing about working with fractals.  It becomes readily apparent that mathematics is truly the language of the universe.  The fractal rose is not one of the pieces of art I’ll be showing at the upcoming exhibit in London, but it symbolizes the exhibit very well, which is why I have chosen it for my promotional materials.

Mathematicians have spent a good deal of time and effort to demonstrate the fractal geometry of various parts of nature, tweaking formulas for the very purpose of modelling it.  This has (in most cases) involved an analysis of natural shapes and distributions prior to the effort of coming up with a formula.  I, however, am not a mathematician.

Most of my images start on a whim. I should qualify this with the statement that I am standing on the shoulders of the people who have created the software I use.  Without their brilliance I wouldn’t be able to do any of this.  So… my images start on a whim, and they continue with further whims (what happens if I change this?), and even further whims.  The possibilities really are endless.  If the image strikes me, I render it in high resolution and save it.  Sometimes I save the parameters as well, sometimes I don’t. So, in a way, I am the natural selector, deciding which image survives, which parameters get passed along to the next selection process.  It never ceases to amaze me how often I am confronted with an image that triggers recognition of something that exists in our universe – or at least, the universe within my imagination.  These are the ones that are most likely to be selected for the creation of my art.  The next step is the editing that occurs before I consider a digital piece finished (sometimes several images are combined into one piece), or, the painting of the image that I was inspired by.  The paintings require a great deal of patience to execute.  I draw them on the paper (or gessoed paper, or aquabord) freehand, but I start with very precise measurement of the positions of the largest features.  I decide which pigments are best to represent what I like about the digital image, and if there is any element I don’t like and wish to change or omit.  Then comes the sorting out in my brain of the pattern, and how it repeats on smaller and smaller scales, and exactly how small of a scale it is possible for me to keep painting this pattern.  It is like a puzzle and I’m drawing the pieces and fitting them inside each other, to the limit of my brush size and my eyesight (and my resolve).  The results are very satisfying but I am usually at the end of my rope by that point and have to switch to my traditional paintings for a while just to retain my sanity!  This is one reason why this upcoming show is the culmination of three years of work.

Ultimately, all of my fractal art, digital or paintings, or photographs of nature, comes from the place in my brain where reality meets imagination.  A place where the universe seems to reveal itself to the part of my brain that can imagine both its most vast and its most infinitesimal features, and how they relate to each other.

I hope my fractal art will trigger your imagination as well!  Stay tuned (follow my blog  please!) and don’t forget to save the date:  July 8, 7-9 pm for the opening, and the exhibit runs until July 19, at The ARTS Project in London, ON.  (See previous post for more info)