Fractals 1 of 5: Beauty, fun, and more

Jitze Couperus

A fractal is any shape that repeats itself over and over and over again at different sizes and scales.You can zoom in and find the same shapes forever.

Besides being beautiful and fun, fractals are very useful! For example, they can be used to predict or analyze parts of the human body, such as the inside of our lungs or the pattern of nerve dendrites. Another important use of fractals is in image compressing, so that some of our photos don't take up too much space in our computers! It’s mostly used if an image has patterns that are repeated (fractals!) .

Drawing, weaving, beading, building…

Indigenous people have used fractals since long before modern-day mathematicians:

Dreamcatchers can have fractal designs, like

https://shop.slcc.ca/learn/the-dreamcatcher/

Look at the fractal nature of parts of our universe in artist Margaret Nazon's beading work:
http://robertthirsk.ca/2018/10/19/the-spirit-of-innovation/
https://www.nwtarts.com/artist-profile/margaret-nazon

Many African cultures have long used fractals in their architecture, art, and other aspects of their culture, as explained in Ron Eglash’s book “African Fractals - Modern Computing and Indigenous Design” https://monoskop.org/images/f/fc/Eglash_Ron_African_Fractals_Modern_Computing_and_Indigenous_Design.pdf

Fractals 2 of 5: Drawing Fractals

We can draw fractals!

https://www.youtube.com/watch?v=PU1dwwB_Im0

Try drawing these... then, see if you can come up with your own fractal 'seeds' and draw them out to make your own original fractals... and share them with us in the comments!

www.study.com

   

https://en.wikipedia.org/wiki/Pythagoras_tree_(fractal)

Can you find some fractals outside? Or maybe even inside your fridge?

More references:

https://www.wikihow.com/Draw-the-Koch-Snowflake
https://www.es.com/Shows/FantasticFractals/FantasticFractals_EducatorsGuide.pdf

Fractals 3 of 5: Fractals in plants

I went outside fractal-hunting today. I was looking at plants, and here are a few of my findings:

A red cedar branch. It's Y's all the way down!

A camelia flower, its petals going round and round.

Each branch of this shrub comes from another (thicker) branch and so on until the trunk. Below ground, its root system is like a branch system that we could draw as a fractal.

Look at this really cool photograph (done by an artist) of an autumn leaf CLOSE UP:

PAUL OOMEN / FINEARTAMERICA.COM

Are there fractals in the plants around where you live, too? 

I wonder why plants grow with repeating patterns like this...does Nature just find fractals pretty or is there more to it? What do you think?

Fractals 4 of 5: What is the shape of a cloud?

Cloud photo credits: Susan Gerofsky

 Here's a lovely thing to do together, especially on a day when the sky is blue and those fluffy cumulus clouds are moving around in the wind. Try looking up (or better yet, lying on your back in the grass) and watching the clouds.

You might try photographing or drawing them, as I've done here. But how to draw a cloud? What shape is a cloud?

We have learned the names of shapes with straight-line edges like triangles, rectangles and octagons, and curved shapes with continuous line edges like circles, ellipses and parabolas. But none of these really describes the ragged-edged (and ever-changing) shapes of clouds.

You might notice that some clouds look a lot like the maps of islands, continents and shorelines. (The bottom photo here reminds me of a map of Europe and the Mediterranean...) Shorelines are also ragged-edged and changing, more slowly, because of forces of erosion and tectonic shift. So if we can figure out what shape clouds are, we might also figure out what shape islands, continents and shorelines are!

Try looking at a small bit of a cloud you have photographed. Is there a small section of the cloud that looks almost like a miniature version of the whole cloud? If so, then you may have discovered a fractal 'seed' that can be copied bigger and bigger (and/or smaller and smaller...) to create the shape of the whole cloud! That is what is meant by a fractal: a shape that repeats itself in more or less the same way on different scales ( really tiny, small, medium, large, really large...) to create a ragged-edged and evolving shape without straight-edged boundaries.

Here is a good explanation by a meteorologist at the University of Bonn, Germany about the ways that clouds are at least partly fractals (and not always completely so, as the big central parts of clouds are not as affected by wind turbulence as the edges).

 And here is a very nice film about natural fractals, featuring the mathematician who 'discovered' fractal geometry, the late Benoît Mandelbrot!

Try drawing the shape of bumpy or ragged edge of a cloud. It might be easiest to draw it fairly large on your page. Then, on each of the 'bumps' try drawing smaller bumps that have the same shape only smaller. Do that again, three or four times. Are you getting a shape that looks a bit like the edge of a cloud -- or a shoreline?