The Landing

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…

The Landing Digital Art printed on metal, single edition 20x20" Lianne Todd $325.00

The Landing
Digital Art printed on metal, single edition
20×20″
Artist Lianne Todd
SOLD.  Private Collection.

 

The Experiment

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?:

Nebula Negative I Watercolour on Yupo 10x10" Lianne Todd

Nebula Negative I
Watercolour on Yupo
10×10″ (sold)
Lianne Todd

Nebula Negative II Watercolour on Yupo 10x10" Lianne Todd

Nebula Negative II
Watercolour on Yupo
10×10″ $125.00
Lianne Todd

Negative Nebula III Watercolour on Yupo 10x10" Lianne Todd

Negative Nebula III
Watercolour on Yupo
10×10″ (sold)
Lianne Todd

Negative Nebula IV Watercolour on Yupo 10x10" Lianne Todd

Negative Nebula IV
Watercolour on Yupo
10×10″ (sold)
Lianne Todd

Here is what they looked like at the exhibit:

KONICA MINOLTA DIGITAL CAMERA

 

 

Curious to know what they do really look like when you invert the colours?

Non Negative Nebula I (the inversion) Lianne Todd

Non Negative Nebula I
(the inversion)
Lianne Todd

Non Negative Nebula II (the Inversion) Lianne Todd

Non Negative Nebula II
(the Inversion)
Lianne Todd

Non Negative Nebula III (the inversion) Lianne Todd

Non Negative Nebula III
(the inversion)
Lianne Todd

Non Negative Nebula IV (the inversion) Lianne Todd

Non Negative Nebula IV
(the inversion)
Lianne Todd

 

Patterns

This piece is called Looking Through.

"Looking Through" Digital Art printed on metal, single edition 20x20" $325.00 Artist: Lianne Todd

“Looking Through”
Digital Art printed on metal, single edition
20×20″
$345.00 Artist: Lianne Todd

Looking through what?  A microscope?  A telescope? A porthole?

In a fractal universe, it doesn’t really matter.  Similar patterns are present on multiple scales.  Use your imagination!

This image is actually a combination of fractals – one for the thing we are looking through (the self-similarity on smaller scales provides the illusion of perspective and depth here), and one for what we are looking at (this is a flame fractal – more about them later).

As illustrated here, fractal geometry is quite versatile.  I’ve seen some discussion on ‘true’ fractals versus ‘near’ fractals and I would like to address that here for a moment.  There seems to be an opinion out there that for a fractal to be ‘true’ it must be a)infinite and b)exactly the same no matter what scale you look at.  Having read most of Benoit Mandelbrot’s Fractal Geometry of Nature, I have a problem with these stipulations.  First of all, the equation for Mandelbrot’s set is z_{n+1}=z_n^2+c, with as the number of iterations, where c is a complex parameter.  

There is more to explaining the Mandelbrot set than that, of course, but that is the equation, and if n is a given number, then it’s not infinite, is it?  Perhaps the possibility of an infinite number of iterations exists, but that’s an argument for another day.

And even Mandelbrot’s set is not exactly the same on multiple scales. The PATTERN is there, it’s just slightly altered at different scales.  It is self-similar.  This is one of the things which makes fractal geometry so suitable for modelling the universe.

In my understanding, there was never a suggestion by Mandelbrot, the founder of fractal geometry, that a “true” fractal had to be infinite OR exactly the same on multiple scales.  Rather, a fractal is strictly defined as “a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.”

So, perhaps I’m getting it all wrong, but if you would like to argue I would welcome your discussion.

And now, because I was once a biologist and if you’re anything like me you need a more highly magnified look at that thing, here is a zoom of what you were “looking through” at:

sea creature zoomed in

 

Some of the early ones.

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.

The Way Watercolour on Paper, 20x20" Lianne Todd $625.00 framed

The Way
Watercolour on Paper, 20×20″
Lianne Todd
$650.00 framed

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!)

Fire Dance Watercolour on Paper 20x20" Lianne Todd $625.00, framed

Fire Dance
Watercolour on Paper
20×20″
Lianne Todd
$650.00, framed

Happy Hill Watercolour on Paper 20x20" Lianne Todd $625.00, framed

Happy Hill
Watercolour on Paper
20×20″
Lianne Todd
$650.00, framed

 

Here is what they looked like at the show (on the left):

The Fractal Nature of Our Universe exhibit, East wall. Lianne Todd artist.

The Fractal Nature of Our Universe exhibit, East wall.
Lianne Todd artist.

 

And here is a detail of The Way:

Detail of The Way Lianne Todd

Detail of The Way
Lianne Todd

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:

Happy3croppedwm

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”:

The Higgs Boson Collision

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!

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.