# Turbulence Revisited

Big whorls have little whorls
Which feed on their velocity,
And little whorls have lesser whorls
And so on to viscosity.
-Lewis F. Richardson

In the book I reviewed in my last post, Chaos: The Making of a New Science, by James Gleick, this quote begins one of the chapters.  And in the first paragraph of that chapter, another quote is mentioned which is in the description of this interesting video about the unexpected math in Van Gogh’s Starry Night.

James Gleick makes no reference to that painting, but goes on to describe the stories of past mathematicians and physicists trying and failing to solve the problem of turbulence.  Finally, along came Chaos Theory and Fractal Geometry, and things started to make some sense.  It is easy to understand why, when you look at the self-similarity and the complex patterns of a turbulent system.

I wonder what was going through Van Gogh’s mind when he was painting Starry Night.  According to the video, it was during one of his “periods of psychotic agitation”.  Perhaps the patterns approaching chaos happening in the electrical signals of his brain were translated to his expression with paint?  It’s an interesting point to ponder when you consider all of the systems in our bodies that involve fractal patterns.

I can assure you I was perfectly calm and sane during the painting of Turbulence and Bubbles – I was just letting my own hands and brain interpret the patterns that arose from an external fractal formula.  When I first started I had a completely different title in my mind, but then as I was painting it, I realized the black whorls reminded me of turbulence, and it looked like the yellow parts were bubbles emerging from some unknown source within it, and merging with each other when they touched.  We know turbulent systems do produce bubbles… (think boiling water)… I doubt this is how, but still!   I know I’ve introduced it before but here it is again:

Turbulence & Bubbles.
Watercolour on Gessoed Paper.
20×20″.
\$650.00.
Lianne Todd

Here is a raw fractal which, to me, looks like a cross section of a wave crashing in.  A detail, below it, shows the patterns present within.  I haven’t quite decided what I’m doing with this one yet, but thought I would show it to you as it relates to this post so well.  It’s not exactly turbulence, as there aren’t any true whorls, but you can see how fractal geometry would lend itself to the study of turbulent systems.

# Imagining the Cosmos

As you may know, if you have been reading all of my blog posts, I like to dabble a little in cosmology.  Not that I really know anything about it, but it fascinates me, and I like the kind of abstract thought it stimulates.  I was given a Great Course one Christmas on Dark Matter, Dark Energy, The Dark Side of the Universe, which I’ve enjoyed a great deal.  Dr. Sean Carroll is great at explaining cosmology in a way that I, at least, can understand.  I also read Brian Greene’s The Hidden Reality, a fascinating book.  I subscribe to many Facebook Pages which post news about the latest pieces of knowledge in this field.  The abstract thought appeals to me, and I think I’m pretty good at it, but my math and physics skills are limited to helping out my children when they were doing grade twelve homework.  Which may be nothing to sneeze at, but is less than what is required for particle physics.

One of the things that really strikes me, is that the visual components, be they illustrations or actual data translated into an image, of almost every piece of news in cosmology, are recognizable to me as being fractal in nature.  Perhaps it is because I’ve spent so much time looking at fractals, zooming in, and examining them from every angle, that I notice this.  I am always mystified when no mention of fractals is made, in these cases.  I’ve written a  whole post (and another) about fractal dimensions before, so I won’t go into that here, but that’s another part of the puzzle I like to think about.

I know these two fractals don’t really illustrate anything in particular, but they make me think along the lines of particle physics, and stardust, and the early universe.

“Stardance”. Watercolour on gessoed paper. 20×20″.
\$650.00
Lianne Todd

“Particles and Fields”.
Fractal Digital Art on Metal, single edition print. 20×20″.
\$345.00
Lianne Todd

This past summer, during my exhibit, I was told about a person in the same city (London, Ontario) who, it seemed, had a lot of the same thoughts I was having about the fractal nature of the universe.  She is a software engineer and has been studying fractals for much longer than I have!  Needless to say, her math skills are much better than mine.  I was fortunate to meet her (her name is Lori Gardi) this fall.  She has two websites, the first of which I’m going to direct you to Here, in which she has laid out some of her thoughts in a pretty clear way.  I’ll link you to the more recent one later… I think it’s a good idea to start at the beginning of her thought process. We (artists, software engineers, mathematicians, physicists, philosophers) may not all have the same thought processes or reach the same conclusions in our explorations of fractals and their role in the universe, but all avenues should be explored as long as they can exist within the rules that have been truly established by scientists and mathematicians in the past.  I am, unfortunately, not capable of judging whether anything follows those rules, but others who can, need to at least look at this work with an open mind and decide for themselves. ESPECIALLY if it may help solve any of the mysteries still out there.

# The Photographs

It isn’t difficult to spot natural fractals all around you, if you know what you’re looking for.  It’s quite probable that you just don’t recognize them because you haven’t looked at enough computer generated fractals, at enough scales, to realize that even if something in nature doesn’t look like a whole fractal, i.e., you don’t really see the repetition of a pattern on smaller and smaller scales,  it will look like part of one.

Before Benoit Mandelbrot came along, nature was regarded as a rather chaotically influenced version of Euclidian geometry.  The artist Paul Cezanne said as instruction to young painters: “Everything in Nature can be viewed in terms of cones, cylinders, and spheres.”  But Mandelbrot’s famous quote “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line” contradicts this, and rings very true.

Some mathematical experts are able to generate entire landscapes, or, small parts of nature like a fern leaf, just using fractal formulas.  Benoit Mandelbrot gave a few examples of these in his book, The Fractal Geometry of Nature.  Indeed, the generation of natural looking landscapes, textures, etc. using fractals is quite common, in video games.  Ever wondered how the game manages to keep up the appearance of the surrounding landscape the character is travelling through?  That’s how.  They are also used in movies, creating alien landscapes.

Some examples of fractals in nature are depicted in the photographs that were part of my exhibit this summer.  The patterns of ice crystal formation, mountain ranges, clouds, branching patterns of trees, growth patterns of mosses and lichens, flower structure, butterfly wings and their coloration, fur growth and patterning… on every scale you can recognize fractals, not just on earth but in the entire solar system and universe.  (More about that later).  For now, have a look at these, and look at the world in a new way the next time you go outside.

Top to Bottom, Left to Right:
Freezing; Mountains & Vapour; Mossy Branches; Reiterated Beauty; Nature’s Drapery; All the Markings of a Bandit. Digital Photography. 10×10″ Prints, framed \$125.00

As they appeared in the exhibit at The ARTS Project.  Thanks again to the Ontario Arts Council!

# Thoughts on fractals and dimensions

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!

# 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)

# My Fractal Art: An Introduction

A few years ago, in the fall of 2010, a famous piece of art (The Great Wave, by Katsushika Hokusai, 1760-1849) kept re-appearing in my life.  Though it contains fractal shapes, the artist, who lived more than 100 years ago, wouldn’t have called them that as the term hadn’t been invented.  However, it led me back to this fascinating topic which I had briefly been introduced to about 20 years ago.  I remembered my initial excitement, and the more I explored it, the more I felt this was the direction I needed to take to satisfy my artistic and my scientific nature all at once.  (I have been a watercolour painter for about fifteen years, and prior to that I worked in biology).  While the mathematical definition and the name (credit to Benoit Mandelbrot) are relatively new (circa 1975), the concept and patterning involved in them is hauntingly familiar.  The idea of dimensions as in 1-D, 2-D and 3-D is fairly straightforward.  A dot is 0-D, a curve or a line is 1-D, a surface or plane is 2-D, and a sphere or cube is 3-D, for example.  But as Benoit Mandelbrot said, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”¹  So, what if the best way to define something’s dimensionality is to use a non-integer, like, for instance, 0.6309, or, say, 1.2618?  Without going too far into the mathematics, which I myself barely understand, these are the situations where we use the term fractal.  Fractals are generally self-similar, on smaller and smaller scales.  There are many examples in nature, and also, many practical applications for fractal geometry in our lives.

Mandelbrot, sadly, passed away that same fall.  His legacy is a novel way to observe, appreciate, and replicate the natural world.  In fact, when I look at fractals, I often see everyday things.  And when I look at everyday things, I often see fractals.   My fractal art is an attempt to bring this idea of mine home:  Fractals may not be just a model for, but may be the underlying structure of our universe.

I hope you will join me in discovering a new way to look at the world.  Soon, I will add photos of my creations, but for now, let me invite you to the opening reception of my solo show, The Fractal Nature of Our Universe, which introduces my fractal art to the world and is to take place on July 8, 7-9 pm in London, Ontario, Canada.  Mark your calendar and follow this blog for more details at a later date!

1. MANDELBROT, B. B. 1977.  The Fractal Geometry of Nature.  New York: W.H. Freeman and Company.