A fractal is, by definition, of fractional dimension.
A curve defined by a fractal might require an infinite distance between any two points along it's length and might nearly fill an entire area.
That's difficult to describe in one dimension but it has no thickness so it's not really two yet.
Thus it is said to have a dimension between one and two (In the case of Sierpinski's manifold D = log(N)/log(r) = log(3)/log(2) = 1.585.)
Similarly, a surface distorted by a fractal would have a dimension somewhere between two and three.
By extension, a dynamic fractal would be more than a 3D object moving through time...so that must be somewhere between four and five dimensions. Well, in the case of the gear-fractal, which is really motion in a plane, there are somewhere between three and four dimensions.
I needed to learn to program in AutoCAD which uses this really archaic scripting language called LISP.
It's so old that I had a hard time finding a book old enough to explain it and even had to write the basic math functions that I had always taken for granted.
The real power of LISP is its ability to self-refernce so naturally that led me to consider messing around with fractals.
The Sierpinski manifold is probably one of the simplest fractals.
It's a 2D pattern of triangles but I'm a sculptor.
I wanted to use the third dimension to show how many iterations into the fractal each portion of the form represented.
The thing about fractals is that you can keep expanding or decreasing the scale and it keeps giving new insights.
How far can this go?
My real goal was to generate involute gears.
By the time I had a working gear program I was tempted to create a mechanical fractal.
Having the functional fundamentals of gears in program form tempted me to modify them for purely aesthetic purposes.
Backlash, involutes, tooth length, undercutting...one can learn way more about a topic than is useful for a healthy family life.
As far as I know, I invented this notion of a nested set of planetary gears as a fractal.
As you might expect this took some time:
a month or two to learn a new language and write the program,
a few more to make a 3D print and mold and cast it at the foundry, more to machine it.
That of course got me thinking about materials and tools required at different periods of history.
The gears I made at the foundry were bronze so I really wanted the smaller gears to be in iron, plastic, silicon...while the larger ones would transition to older technologies using wood, stone, earth...
I had been working on this project for some time and was anxious to show it to my instructor at VCU who is a world-class sculptor.
Unfortuneately, she considered any work that I had done prior to her class as nearly irrelevant. And gears?...pastiche.
She viewed it as ultimately a demonstration of some known principal using what apeared to be off-the-shelf items. *sigh*
Oddly, my engineering buddies all get it.
The piece was chosen for a show at the Science Museum of Virginia despite their long-standing aprehension concering sculpture.
Of course I had to make it kid proof and mount it on the wall with 12 bolts so they could climb on it but that was fun.
I had an appreciative audience even if they were largely eight years old.