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Archive for July, 2013

These are a few of my favorite things: #29 – Fractals

I have long been fascinated by number sequences and finding the patterns within them.  This brings me to a fun topic . . . Fractals!

What are Fractals? – According to the Fractal Foundation, a fractal is a never-ending pattern that repeats itself at different scales.  This property is call “Self-Similarity”. This means that as you look closer and closer into the details of a fractal, you can see a replica of the whole.  What you may not know is that fractals of been of interest to people for over 2,000 years when Euclid, a Greek mathematician, claimed a straight line segment can be prolonged indefinitely . . . although, Mandelbrot stated the new fractal geometry is different . . . Oh, those crazy mathematicians!  (But, now I’m starting to sound like an episode of The Big Bang Theory . . . so, so I’ll move on)

Frankly, I think it’s easier to understand fractals by seeing an example.

Step 1 – Take Triangle

Triangle - step 1

Step 2 – Subdivide by three

Triangle - step 2

Step 3 – Subdivide again by three

Triangle - step 3

Step 4 and beyond  – Keep subdividing by three

Triangle - step 4  Triangle - step 5  Triangle - step 6

This can literally go on and on

Let’s see with a simple line

Fractal line

We can do this with a number of elements

Fractal graphic

Okay, so this is possible in graphics, but you may be wondering . . .

Are there naturally occurring fractals? – You may not know it, but we’re surrounded by fractals every day.  From the veins in a leaf, to the branches on a tree, the structure of a snowflake, mountain ranges, rivers, lightning bolts, and much more.  We can’t go a day without fractals.  Below are a few examples of naturally occurring fractals.

Fractal snowflake Leaf fractal #2 Leaf fractal Lightning fractal

Graphical fractals

Fractal #1 Fractal #2

As a design approach, fractals can be very inspirational and there are some easy on-line resources for generating your own fractals

Illuminations – http://illuminations.nctm.org/ActivityDetail.aspx?ID=17

Illuminations

Easy Fractal Generator – http://www.easyfractalgenerator.com/

Easy Fractal Generator

Just looking at fractals can be inspirational and there are on-line galleries and other resources available for viewing

Fantastic Fractals – http://www.fantastic-fractals.com/index.html

Fantastic Fractals

Fractal Foundationhttp://fractalfoundation.org/

Fractal Foundation

Even googling the word “fractal” will generate many examples.  Enjoy exploring fractals and discover ways to use them as inspiration.

Update . . .

Here are a couple of simple fractal-like drafts . . . a sis-shaft Bronson lace and a 16-shaft summer & winter

6-shaft Bronson.dtx

 

16-shaft fractal draft as summer & winter.dtx

 

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