A Matter of Scale

There exists a very old phrase, ‘as above, so below’.  Its meaning is interpreted in various ways, depending on where you look.  Its source is generally attributed to Hermes, though according to some, it is probably even older than that.

According to Wikipedia, the full quote translated from Hermes ‘ The Emerald Tablet of Hermes Trismegistus, as translated by Dennis W. Hauck, is “That which is Below corresponds to that which is Above, and that which is Above corresponds to that which is Below, to accomplish the miracle of the One Thing.”

Isaac Newton translated the Emerald Tablet’s passage as follows: ‘That which is below is like that which is above & that which is above is like that which is below to do the miracles of one only thing’  (according to Quora).

I don’t know exactly where I first heard the phrase, but it certainly popped into my head a lot as I began to explore fractal geometry.  The more I learn about fractals and about the cosmos, the more I see similarities between large scales, like the universe, and small scales, like an atom.  Perhaps an easier example to envision is the similarity between say, a river drainage pattern and the venation in a leaf.  After all, fractals are often self-similar on smaller and smaller scales.  It is one of the ways in which fractal geometry was discovered by Benoit Mandelbrot.  My cursory understanding of such things, as an artist whose education was mainly in biology, does not diminish my enthusiasm for humanity’s search to find a Theory of Everything.  Whenever I see a Physics article in my various news feeds, I am struck by either their use of illustrative images which I recognize from experience as being generated fractals, or how much the actual images generated by their physics experiments resemble generated fractals.  Maybe someday the ideas will all fit together.  Until then, I will continue to happily make my art and notice how in reality, sometimes it is tricky to know what the scale of an image is.

This piece will be on display in my gallery this weekend during ‘Welcome Back to Otterville’, our town’s 21st annual studio tour.  Please visit www.WelcomeBackToOtterville.ca for details of the tour, including maps and times.

microcosm or macrocosm?

A Matter of Scale. Digital Fractal Art, printed on metal. 20×20″. Single edition print. Artist Lianne Todd. $325.00

 

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New Fractals

I’m excited to have some new fractal art to show you in a week at the Oxford Studio Tour.

We have thirty-one artists at seventeen locations throughout Oxford County, Ontario, Canada in the heart of the southwestern part of this province.  It will be a fun day trip for anyone in the region – even those coming out from Toronto! (Wouldn’t it be nice to get out of the city for a day or two?)

Here is a preview of one of the fractals.  I have printed it (using Posterjack) on metal, 20×20″ and that is the only print I will do, so it is an original piece.  It is created digitally using the Mandelbulb 3D software.  As I find usual and striking for fractals, it looks very natural. It is pretty obvious what I thought it resembled!  I hope you’ll come and see it.  I think it would make a great piece to gaze at from anywhere in your house, while you consider nature and its mysteries, and it draws you in close, as well – as all fractals do with their self-similarity on smaller and smaller scales.

I call this one Ocean Floor:

Ocean Floor. Digital Fractal Art. Lianne Todd.

Ocean Floor. Digital Fractal Art. Lianne Todd.

 

November News

I am very pleased to announce that some of my fractal art will now be available at the Art Gallery of Lambeth!  I grew up in that area so it is really nice to be able to display my art there.

I still have many pieces at my home studio/gallery though, and that’s a good thing, because this weekend is Welcome Back to Otterville, our 19th annual Studio Tour.

For details, please see my previous post. As always, I have a few things left to do before morning so I will keep this short!

You could also visit my other site, liannetodd.wordpress.com, to see a couple of new pieces I will have on display.

KONICA MINOLTA DIGITAL CAMERAWBTOCardBack2015sm

Following the Patterns of Nature

It is an absolutely beautiful day today in Otterville, full of colour and the patterns of nature, so I plan to spend some time outside.  It was during another beautiful day a few years back, hiking in the woods at Awenda Provincial Park, that I came across many kinds of fungus.  I took a number of photos, and an edited version of one of them ended up as part of this image I am presenting to you today.

On another completely separate occasion, I was creating fractal images and found that, as is often the case, there were distinctly natural and vegetative features recognizable in one.  I saved it, and later on when looking through all of my photos, I noticed how well the features in it mimicked and extrapolated the patterns of growth I had noticed in the fungal photo.  I had even just happened, by whim, to have edited the photo so that its colours matched the ones I had, by chance, used in the fractal creation.

What you see below is a digital collage of the natural and the generated fractal patterns, printed on metal.  Once again nature shows how it is a manifestation of the fractal patterns of the universe.

Following the Patterns.  Digital Fractal Art printed on metal, single edition.   16x16".  Lianne Todd.  $225.00.

Following the Patterns. Digital Fractal Art printed on metal, single edition. 16×16″. Lianne Todd. $225.00.

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″.
$625.00

 

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″.
$625.00.
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″ $125.00
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″
$325.00 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

 

Butterflies and Moths

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″
$325.00

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″
$325.00

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

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″
$225.00

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.

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.