Tag Archives: fractal geometry

The Landing

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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

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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

 

A Game of Ball

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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.

The Ball Went Over the Fence 1, Lianne Todd. Original Fractal Digital Art, single edition print on metal, 28x28" $525.00

The Ball Went Over the Fence 1, Lianne Todd. Original Fractal Digital Art, single edition print on metal, 28×28″ $550.00

 

The Ball Went Over the Fence 2, Lianne Todd. Original Fractal Digital Art, single edition print on metal, 20x20" $325.00

The Ball Went Over the Fence 2, Lianne Todd. Original Fractal Digital Art, single edition print on metal, 20×20″ $345.00

If you are having trouble with the first two, try looking at #2 and #3, this is the easiest solution of all of them.

The Ball Went Over the Fence 3, Lianne Todd. Original Fractal Digital Art, single edition print on metal, 16x16". $225.00

The Ball Went Over the Fence 3, Lianne Todd. Original Fractal Digital Art, single edition print on metal, 16×16″. SOLD.  Private Collection.

 

The Ball Went Over the Fence 4, Lianne Todd. Original fractal digital art, single edition print on metal. 12x12". $145.00

The Ball Went Over the Fence 4, Lianne Todd. Original fractal digital art, single edition print on metal. 12×12″. SOLD.  Private Collection.

 

The Ball Went Over the Fence 5, Lianne Todd. Original Fractal Digital Art, single edition print on metal, 8x8". $110.00

The Ball Went Over the Fence 5, Lianne Todd. Original Fractal Digital Art, single edition print on metal, 8×8″. SOLD.  Private Collection.

And now, as promised, the key to where the zooms took place – a map through the series:

Zoom Key, The Ball Went Over the Fence series

Zoom Key, The Ball Went Over the Fence series

Fried Eggs

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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.

Fried Eggs Lianne Todd Watercolour on Aquabord 6x6" $175.00

Fried Eggs
Collection the Artist Lianne Todd
Watercolour on Aquabord
6×6″

Patterns

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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 n 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

 

The Photographs

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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. 10x10" Prints, framed $125.00

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.

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

Butterflies and Moths

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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
20×20″
$345.00

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
20×20″
$345.00

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
12×12″
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
16×16″
$240.00

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.

Some of the early ones.

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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

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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!

Publicity!

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I made the news today – and while an actual review would have been nice, I will take free advertising without complaint!  This article appeared on the front page of the Today section in The London Free Press, and I am grateful to them!  Nice to be featured with these other excellent artists as well.

I have decided to post my own watermarked photo of the featured painting they chose for the article – it will give you a better idea of the colour.  This is one of the five watercolours on gesso that are in the exhibit, and it’s called Colourfest.  Like the others, it began as a digitally generated fractal, and developed from there.

Colourfest, 20x20", Watercolour on Gessoed Paper. Lianne Todd

Colourfest, 20×20″, Watercolour on Gessoed Paper. Lianne Todd

Another item of publicity is this video made by The ARTS Project.  It was the end of the day we hung the show, so I hope I am not rambling too much in it.  You can see Colourfest hanging behind me!