Patterns of humanity

I have mentioned a few times in previous posts about fractal images often being archetypal.  I’m not really sure I was correct in describing them that way, but what I will say is that they often look familiar in a way that transcends their immediate translation into a real object.  For instance, when I first looked at this fractal image, I immediately thought of the Aztecs and the patterns they incorporated into their art forms.  Having now looked at some Aztec art in more depth, I’m not even sure why I thought of them, other than the  feather motif and the colours.  That’s just what immediately came to mind.  And I have other fractal images created (not shown here) which really speak to me of Native American blanket patterns.  Others might look at this piece and be more immediately aware of the hourglass shape.  Maybe you will look at it and see something else entirely.

My point is, from near the beginning of humanity, we have been making patterns, whether or not they were drawn with a stick in the sand, sculpted along the edge of a stone building’s rooftop, painted on a cathedral ceiling, or digitally on a tablet.  Possibly for 60,000 years, we have been making patterns!  And if you look at all the patterns we have been making, you may notice that many of them are self-similar on smaller and smaller scales.  We were making fractals and we didn’t have a name for what was common to them all.  And we didn’t have a concept of the way fractals were involved in the geometry of nature – not consciously, anyway.  Maybe we were consciously inspired by nature, but didn’t recognize that specific aspect of it.  Only for the last 30-40 years have we, thanks to Mandelbrot, come to an awareness of this common denominator.  I like the way fractals connect all of humanity over time, and the way they connect us to nature.   I’m really looking forward to exploring this with future pieces!

Aztec Gold. Watercolour on Gessoed Paper. 20x20". $625.00. Lianne Todd

Aztec Gold.
Watercolour on Gessoed Paper.
20×20″.
$650.00.
Artist Lianne Todd

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.
6×6″.
$175.00

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

Triad.
Lianne Todd. Watercolour on Paper.
20×20″.
$650.00

 

Happy New Year!

I would like to thank everyone who has visited this site in its first calendar year of existence.  And if you’ve shared anything from it, even better!

Today I saw a post by an acquaintance who had begun, a few years ago, a tradition that I think is a wonderful idea.  And I’m telling you about it here because it’s kind of like a fractal.  I hope she doesn’t mind.

She and her husband take a photo of themselves on New Year’s Eve each year, holding a printed photo of themselves they took the year before, holding the photo they took the previous year.  So in each photo, as you zoom in, there is the couple holding the photo… and they get smaller, and smaller….

I wish I had thought of this 29 years ago.

Now, what can you look forward to on the site this year?  More fractals of course.  The ones I haven’t introduced to you yet, and new ones I will be creating.  I hope to do some more experimentation as well.  I’ve got lots of ideas floating around my head and once I settle into the routine after the kids go back to school, there will be plenty of creation happening!  Also, I will be keeping you posted with any news in the world of physics that might pertain to my ideas about the role of fractals in the structure of our universe.

I hope you and yours have a very good year.  And while I’ve got your attention… have a look around you.  Is there enough art on your walls?  😉

Physical Phenomena

If I am postulating that the universe is fractal in nature, it makes sense that structures formed by molecules behaving in their natural way should be recognizable as fractals.  Such is the case with frost, turbulence, and bubbles.  Just do a little Google search with each of those terms alongside fractal, and you’ll see what I mean.  Mandelbrot made groundbreaking progress modelling turbulence, which had confounded mathematicians before him, using fractal geometry.

It also makes sense that in my random wanderings through the fractal universes I create, I encounter images that remind me of these phenomena.  Such is the case with these two fractals which I chose to paint.  I especially like the way the bubbles in Turbulence & Bubbles look like they are in the process of being blown, sometimes from multiple locations, and melding together when they meet, just like real bubbles would.

Hot Frost. Watercolour on Aquabord. 6x6". $175.00 Lianne Todd

Hot Frost.
Watercolour on Aquabord.
6×6″.
$175.00
Lianne Todd

Turbulence & Bubbles. Watercolour on Gessoed Paper. 20x20". $625.00. Lianne Todd

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

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

 

A Game of Ball

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

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