Tag Archives: Fractal Cosmology

Not-so-alien Vegetation

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Have you ever wondered if there is any other life out there in the universe?  I think there must be.  The universe we can see contains around 100 – 200 billion galaxies and it is estimated that is only ten percent of what it actually contains.  That’s just galaxies – each galaxy contains perhaps 100 billion stars.  I don’t know about you but I have a hard time wrapping my brain around numbers that large.  The odds against this being the only planet to support life of some kind are huge.  So, then one wonders, what would alien life look like?  We’ve seen so many science fiction movies, now, where others have imagined it, and they’ve done a really convincing job.  It’s amazing, really, what our minds can come up with.

But what if we leave it all to the mathematics?  If fractal geometry is the geometry of nature, let’s assume that all nature, not just that of planet Earth, is governed by it.  I find it really fun to create a little ‘planet’ using Mandelbulb 3D, and then explore it, finding scenes that remind me of Earth.  I’ve done this before with ‘Climb it, Change’, ‘There is Always a Bridge’, ‘No Port in Sight’, ‘Ocean Floor’, and many more that I haven’t shown anyone yet. This one I call ‘Coniferous Tree on Planet B’. Obviously the ‘coniferous tree’ isn’t the only bit of vegetation on this particular ‘planet’.  I see vines, and cabbage-shaped types of vegetation, and driftwood – but you might have to come see the piece in person to appreciate all that.  Next weekend is the 11th annual Oxford Studio Tour and I am at Location #4.  Why not come out and see it for yourself?

Coniferous Tree on Planet B. Digital fractal art printed on metal, single print. 24×32″. Artist Lianne Todd. $550.00

This is the first piece of fractal art I’ve created and shown that isn’t square, but rather a landscape format.  I hope that doesn’t throw anyone off!

 

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A Matter of Scale

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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. $345.00

 

Chaos

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In the mind’s eye, a fractal is a way of seeing infinity. – James Gleick

I just finished reading a really good book called “Chaos – Making a New Science” by James Gleick.  It was recommended to me by the London Free Press photographer who took photos at my The Fractal Nature of Our Universe exhibit last summer.  (Don’t forget the reprisal of that show, A Fractal Universe, is currently at the Station Arts Centre in Tillsonburg until April 7!)

It was a really interesting read, full of insight into the difficulties scientists and mathematicians have had in the past, with certain problems they encountered.  Most of them involved non-linear dynamical systems – the kind you often find in nature.  They were so troublesome that these problems would be put aside, ignored, deemed unsolvable.  So many different kinds of scientists and mathematicians in the late 1960s and 1970s were separately converging on the same theories to solve these problems at the same time, while the tools (computers) to more freely explore these theories were also developing, one can truly say it was a science whose time had come.  That didn’t mean that it didn’t meet with resistance!  Sometimes even those who were essentially promoting the same ideas refused to acknowledge each other.

Fractals are a large part of Chaos Theory.

Wherever chaos led, Mandelbrot had some basis to claim that he had been there first. – James Gleick

However, there was much to discover even after Mandelbrot had provided this language for describing nature.  Scientists wanted to know the “why” – and they still do.  I am not sure how many scientists today are attempting to use chaos theory and the language of fractals to interpret systems from the smallest to the largest of scales.  Certainly many ecologists, medical researchers, economists, meteorologists, and some astronomers are.  But there is still some resistance.

Will those who are looking to complete a Grand Unified Theory give full consideration to Chaos Theory and Fractal Geometry?  I hope so.  Time will tell, and these are exciting times indeed.

 

 

Physical Phenomena

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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

Imagining the Cosmos

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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. 20x20". $625.00 Lianne Todd

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

"Particles and Fields". Fractal Digital Art on Metal, single edition print. 20x20". $325.00 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 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

 

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.