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:
Congratulations to the European Space Agency, and to all of humanity, for boldly going where no one has gone before… for landing the human-built robot Philae on a comet!!
This is truly a historic moment and an example of how our imaginations can lead us to realities our ancestors may have considered impossibilities. It will lead to a greater understanding of our solar system and of the universe.
In honour of this event, I’m posting my piece called The Landing. I can’t think of a better day to do it! It’s purely imaginary, constructed of three different fractals, and belongs in the realm of fantasy or science fiction. But then, not so long ago, so did landing on a comet…
This series of four paintings was an experiment. Not a very scientific one but I did try to control variables and make predictions. My hypothesis was that since all pigments are different in their molecule size, shape, and hydrophilic and hydrophobic qualities, they would all move differently through the medium of water, and as they interacted with each other, and that their movement would be fractal. In other words, I expected them to appear, at the end, as if they were something like a cloud, or some other natural item that is already known to be able to be modelled using fractal geometry. A coastline, for instance.
All four pieces were executed and controlled in the same way (I won’t give away all my secrets!), the only differences being the pigments I used and the order they were used in. I tried to reduce the effects of gravity by levelling my table but it is kind of obvious there was a tiny bit of gravitational effect. I changed the orientation of the final products so that when they were hung together, it would be aesthetically pleasing. Other than that, the results you see here are basically the raw data.
I called them Negative Nebulae, because I looked at all the white space around them and imagined if it was black, and the colours were reversed, they would look a bit like those photos you see from NASA of distant nebulae. In other words, these would be like the negatives of those photos.
Here they are – what do you think?:
Here is what they looked like at the exhibit:
Curious to know what they do really look like when you invert the colours?
On the third weekend in November, every year, we hold a studio tour in my town. Otterville is a historic town located in Southwestern Ontario, Canada. This will be our 18th annual studio tour – we call it Welcome Back to Otterville – and every year the stops on the tour change slightly as the artists in town do. This year, there are eight stops on the tour, so it will be really easy to drive out, see all the stops, and return home if you live in, say, London, Kitchener/Waterloo, Stratford, or the western part of greater Toronto. It takes me less than an hour to drive to London, and about two hours to downtown Toronto, exactly an hour to Stratford. We aren’t on any major highway, but if you want directions please contact me and I’ll be happy to provide them.
If anyone reading this has been to my studio before, you will find this year quite different as I will be featuring my fractal work prominently. In fact, in the next few weeks I’ll be taking down all the art in my gallery at the rear of my house, and completely rearranging the walls to maximize the display. I always serve a lovely hot spiced cranberry punch during the tour, and I’m looking forward to the taste of it myself!
My gallery and studio are actually open all year to anyone who calls ahead or happens by on an afternoon when I’m home. I’ve just put a new sign out front (the old one suffered from weather damage) so you can easily find my location which is right on the Main Street downtown, just a few houses away from the historic mill and waterfall. Look for the yellow flags when you get here and use the map on the postcards (available at each stop) to help.
Here is some information for the tour, and a few photos.
If you find our Facebook page and “Like” us, or any of our posts, we would really appreciate the extra advertising and traffic that provides us – as you can imagine we are on a limited budget and every bit helps!
I touched earlier on the aspect of self-similarity on smaller and smaller scales in fractals. I find the ones that are exactly the same on smaller and smaller scales a bit boring – like the Koch snowflake or the Sierpinski arrowhead or the Menger sponge (though it has a nice surprise). It’s the fractals that take on the slightly chaotic characteristics of nature that are the most interesting, and which stimulate the most thought about how this whole complex universe of ours developed.
The series I am presenting to you here does not really look like something you would find in nature, strictly speaking, (except for the parts that look like peacock feathers) but I thought it lent itself well to the illustration of self-similarity while emphasizing the variation. And because it already included elements of a human game, why not present it as a game?
The series is called “The Ball Went Over the Fence”.
Up until this point, you have probably looked at the fractal images on this site and you’ve detected the self-similarity, but what you maybe haven’t seen is what happens when you travel into a fractal. You can’t properly zoom in on a fractal without the equipment to do so – i.e., the software which allows you to make the fractal in the first place, or a video or .gif someone has prepared that takes you through it. When I say travel into a fractal, I mean precisely that – it resembles exploring a new realm. You enter the realm, you set your sights on a distant object, and when you get there, your surroundings have changed – you know you’re in the same realm, because it all looks familiar, but that which was tiny is now large and detailed, and you can see off into a new distance. You set your sights on that, and continue on your journey…. and you can keep doing this over and over again for a long time, depending on how many iterations of the formula you’ve rendered.
So I travelled into this fractal I created, and I stopped along the way and saved some images. The game is for you to try to figure out where I zoomed in to get to the next image. Give it a shot. In a couple of days, I’ll edit this post with the key at the end so you can have the answers. Hint 1: the orientation of the image doesn’t change. Hint 2: Some of these are a lot easier to find than others. Also, the first two images will open larger if you click on them, but the rest are locked at their size. (They are all scaled relative to their actual artwork size). The key at the end will open larger so you can see more clearly.
All images are watermarked and copyrighted.
If you are having trouble with the first two, try looking at #2 and #3, this is the easiest solution of all of them.
And now, as promised, the key to where the zooms took place – a map through the series:
This is just one of my favourites. It’s only little, 6×6″, but like most fractals it took a long time to paint. So many tiny little fried eggs! It’s framed in a black lacquered shadow box frame so that it floats in the frame. The total size, frame and all, is roughly 12×12″.
Yet again, we see a natural shape. Well, natural, in that we are natural and we naturally like to fry eggs. Sometimes I find that it isn’t so much the repetition on smaller and smaller scales that makes me think of natural objects or phenomena when I look at fractals, but the shape that is being repeated.
As usual, it’s copyrighted and watermarked.
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)
Those who have been following my art for a few years may have seen these three before. They were the only pieces I had allowed the public to see, prior to holding The Fractal Nature of Our Universe exhibit this summer. I entered them, in 2011, in the Los Alamos MainStreet Science and Math-Based Art Contest. I wish I had saved what I wrote about each of them then, but while the images are still out there on the web as a result of the contest, the statements I made about each of them are gone. Perhaps it’s for the best – this way each viewer can interpret the images themselves. The wonderful thing about fractals is the way they translate pure mathematics into something that appeals to – well, it feels to me anyway – something ancient in our minds. They are often archetypal. As such, I think they can bring us all together as humans. We need something to unify us, don’t we? So I will leave interpretation out… for while inspired interpretation as an individual is wonderful, sometimes expressing that interpretation divides us from those who would interpret differently.
This was the first fractal piece I ever created – The Way.
Fire Dance and Happy Hill were the second and third pieces I created (but I can’t remember which was second and which was third!)
Here is what they looked like at the show (on the left):
And here is a detail of The Way:
Back in 2012, when I had begun my study of fractals, and had begun painting them, I wrote a blog post on my other website, entitled Fractal Dimensions. It contained my thoughts about the missing dimensions that some cosmologists would like to discover/elucidate in order to make sense of the equations describing String Theory.
While it may be a naive article to have written, given my minimal physics education, and I may be wrong in my approach, I do think there is still merit to the idea. As it turns out, there is such an area of study as Fractal Cosmology, so I am not completely off the mark. And since I wrote that post, I have read quite a bit of Mandelbrot’s The Fractal Geometry of Nature. It turns out, (as I mentioned in my first post on this blog) fractal geometry is all about dimensions – specifically, a fractal is defined as:
A set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.
i.e. every set with a non-integer D is a fractal, but a fractal may have an integer D
So, it has been established that much of nature can be interpreted or described using fractal dimensions. Can ALL of it? That is a big question. And when we refer to fractal dimensions, are we referring to the same kind of dimensions that cosmologists refer to in String Theory? I wish someone would tell me.
For now, and since I am an artist, I use what my eyes tell me. In July of 2012, the Higgs Boson was finally found at CERN. Here is a recent article referring to that discovery and its validity:
Check out the image in that article. At the time of the discovery, I was already working on a painting based on this fractal right here:
Do you see the resemblance of the black parts of this fractal to the yellow parts of the collision image? Not a proof, by any means, but it makes you wonder.
Here is the finished painting, which was beside me in the video I posted in “Publicity”:
I think this may be the first time I’ve shown both a digitally generated fractal and the painting based on it, at the same time!
More thoughts another day… it’s thundering a lot outside and I could lose power any second!