Fractal Patterns in Nature and Art Are Aesthetically Pleasing and Stress-Reducing | Innovation | Smithsonian Magazine

One researcher takes this finding into account when developing retinal implants that restore vision
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Fractals as an illustration in the argument for Free Will

I spend some of my spare time in discussions with a very interesting group of people who like to discuss philosophy, science, and all kinds of other things.  It’s fun for me.

One of the favourite topics in this group is the concept of free will.  It seems that many of the most vocal people in the group favour the idea that there isn’t any.  I am not one of them!  I have made my arguments for free will to them, but these arguments fall on deaf ears, most of the time.  I have, for some time now, thought my experience zooming in to the deepest parts of fractals was somehow illustrative of the (in my opinion) flawed logic that is often used as “evidence” that we have no free will.  I was wondering how I could bring this visually to the group and make them see the idea I was trying to convey.

Last night I realized I already have this illustration fairly handy.  It’s the Key, from my series entitled “The Ball Went Over the Fence”.  Some of you may remember this one from several years ago.  The Key shows what part of the large fractal image I zoomed in on to make the next smaller fractal image.

Zoom Key, The Ball Went Over the Fence series

Wikipedia defines Free Will in the following way:  “Free will is the ability to choose between different possible courses of action unimpeded.” The argument many in the group make, against the existence of free will, is that everything is caused by what went before it. Wikipedia also states that “Some conceive free will to be the capacity to make choices in which the outcome has not been determined by past events.”

My main problem with this is that while it is usually easy, from the standpoint of the outcome, to see what came before, and possibly follow a chain of causality, there is no reason to assume that means a choice made is the only choice that could have been made, given all previous conditions.  We can see into the past just fine.  The future, however, is indeterminate.

So, looking at the Key, shown here, you can see a white outlined square in each square image of the key (click on the image to make it bigger, if that helps).  From each outlined square, you can follow the white angled lines to the image found at that location if you were able to zoom in while in the fractal software.  If we start at the smallest square image (the final outcome), we can see where it came from (it’s very close to the top of the next larger image).  Likewise, if we follow to the next one, we see that it came from a very tiny place in the top middle third.  And if we keep going, we can see that the third image came from a recognizable portion of the second largest image.  And that image came from a very tiny spot in the largest image.

This is the chain of causality – it goes from the outcome back to the origin.  It would, however, be impossible to go from the largest image to the smallest one without the Key in place to guide you.  There is no chain in that direction, because it looks into the future.  The outcome would never ever be the same twice.  There is absolutely no logical reason why that particular tiny spot was the one chosen on the first image to zoom in on and make the second one.  In a fractal, while constrained by the mathematics of that fractal, the possibilities at each level of zoom are for all practical and human purposes, infinite.  I’ll grant you that maybe I would have zoomed in on an area near it, or any one of the areas where you can see the little greenish greyish balls.  Just because those areas look interesting to me.  But they all look interesting, and certainly from the perspective of the large image, equally so.  If we look at the second largest image and are choosing where to zoom in for the third… even if you make the argument that I will almost definitely choose a square featuring a ball… that square is never going to be the same exact square.  And now we get to the third image, and you can see no reason why I would have chosen to zoom in as much as I did, and in the area that I chose.

It was my free will in action, plainly and simply.  I chose, unimpeded.  The outcome was never a given.

Sacred and Forbidden

I just realized today it’s been a whole year since I posted in this blog.  I am sure you can guess what is mainly to blame.  Yes, what with exhibitions and tours canceled, and inspiration flailing for a while, there hasn’t been a lot of activity to report.  I created a few fractals early last spring, and was going to post about them right before they were supposed to be shown in April…

I’ve been keeping myself very busy in the meantime, never fear.   Just not blogging!

This would have been Welcome Back to Otterville weekend in a normal year.  I thought today would be a good time to let you all know I am still here, I do have art available, (see my home page), and if you wish to purchase any (for that extra special gift or any other reason), it can indeed be arranged.  Just use the contact page on this website.

I may as well show you a couple of these new fractals, right?

I think these two speak to our wondering ancient minds, our need for sacred rites and for forbidden places.

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

Forbidden Forest. Digital Fractal Art printed on metal, single print. 18×24″. $400.00. Artist Lianne Todd


Welcome Back to Otterville this weekend

One of my favourite things about opening my gallery for studio tours is the interaction with people who are seeing my fractal art for the first time.  It’s definitely not the same as viewing it online, partly because of the image resolution and detail and size, and partly because of the metal and acrylic surfaces I usually print on.  And for the paintings, the size and the detail provide even more of a difference from what you’ll see here on my website.

But the best part is introducing some to the entire concept of fractals.  If you missed the fun of reading my blog from the very start, back in 2014, and you don’t have a way to come this weekend to see them, I am linking you back to that first post here.

I would love for you to come and visit my gallery this weekend so I can share with you the joy and wonder I feel about creating vivid reflections of nature with both paint and with mathematics.

Just come to Otterville and look for the yellow flags – we welcome you!

Welcome Back to Otterville 2019

Click on the image above to be directed to more information about what you’ll find at each studio or shop.



I’ve never been to Utah, unless you count a layover at the airport on the way to California.  It was snowing in Salt Lake City at the time.

However, I have seen some photos of the landscape there.  Somehow, when I created this image, I was reminded of those beautiful rock formations on a clear hot sunny day.  I imagined hiking through this fantasy landscape, and the name for this image came to me and stuck.

‘Utaopiah’ will be available for viewing (along with many other pieces) in person on May 4 & 5, from 10 to 5 p.m. during the Oxford Studio Tour, at my home gallery, which is Location #4 on the tour.  It is the only one of my metal prints that actually has a frame (the frame is black) – something new I tried out this time!

Fantasy fractal landscape reminiscent of Utah.

Utaopiah. Digital fractal art. Single metal print. 12×16″. Artist Lianne Todd

Time for Tea

Every once in a while I discover and develop an image that is just uncannily familiar.  From the similarity of “Pollinator” to an actual pollinator, to the “Ocean Floor” that isn’t actually an ocean floor, we see how nature follows the patterns dictated by the geometry we know as fractals.   However, it is not strictly ‘natural’ things which follow fractal patterns.  We can see fractal patterns in things like architecture, art, music, and… fancy china dishes!

I absolutely love this image which I’ve entitled ‘Time for Tea”.  To me it seems vaguely reminiscent of Alice in Wonderland and the interior of my Mom’s china cabinet.  I recently printed it on metal.  Here is what the digital image I created looks like.  In the metal print, the white parts are silver, so it does change the look of it a little.  It would be perfect in a super modern dining room, I think.

To see the 24×32″ metal print, you’ll have to come out to my gallery, as it is just very difficult to photograph the reflective surface.  Lucky for you, the perfect opportunity to do that is right around the corner, during the Oxford Studio Tour, May 4 & 5, 10 a.m. to 5 p.m.!

Time for Tea. Digital Fractal Art. Single metal print is 24×32″. Artist Lianne Todd.

New work

While I haven’t posted here in a while (been busy on my other blog though), I have indeed been making fractals!  This is one of a couple of new ones I have recently had printed on metal.

Spring is coming (yay!) and so is the Oxford Studio Tour, a perfect opportunity for you to visit the gallery and check out new work.  If you’d like to visit before then, just let me know.

I won’t say too much about this piece, as the title sort of speaks for itself!

Break Out of the Mold. Fractal Digital Art. Single edition print on metal. 18×24″. Artist Lianne Todd. $400.00

Not-so-alien Vegetation

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!



I cannot wait to show you this latest piece in person.  Some images conjure up abundance, richness, and decadence, and this is one of those images. There is plenty here for your own imagination to work on, but you won’t be able to properly see it unless you come to my studio!  Lucky for you, my studio is often open, all you need to do is contact me.  But even luckier, there are a whole bunch of studios open on May 5 & 6 in Oxford County, and mine is one of them!  I am at Location #4. 

This image you see here is a mere fraction of the size I have printed it – 24 x 24 inches.  As usual, I only have one print made of each fractal piece, on metal (or sometimes acrylic), so each is an original.  Although, I think this would make a great metal album cover don’t you?

Feast. Digital Fractal Art printed on metal, single edition print. 24×24″. Artist Lianne Todd. $450.00



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 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