building a fractal - episode 1

Let’s try to construct some fractals!

These aren't static images - they're computed on the fly!

It’s no secret that I love fractals. Today we’re going to try to build our a fractal with nothing more than some regular polygons you might have lying around. However, there’s a slight twist. I’m not just interested in drawing this fractal, but instead finding a general representation for a class of fractals. Let’s start with defining the fractals I’m trying to find.

the problem

The basic construction for this fractal is that we construct a regular polygon, then on every vertex of that polygon, We construct a smaller version of the same polygon, and then on the verticies of the smaller polygon, I want even smaller polygons and so on. The catch is that no polygon should intersect a polygon of the same size. While this defines a class of fractals, what I want is a slight refinement. I want to find, for a given $$n$$ where $$n$$ is the number of sides of the polygon, what is the largest scaling factor $$f$$, that the size of the polygon needs to reduce by such that the resulting fractal has the non-intersection property.

We know that for $$n = 4$$, $$f = \frac{1}{2}$$. Here’s an animation showing the construction of this fractal.

We can convince ourselves that a factor of $$0.5$$ is in fact the largest possible factor by noting that $$\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n = 1$$ , so anything larger that $$0.5$$ would intersect before infinite iterations.

so what is the factor for an arbitrary $$n$$?

I don’t know yet - a future post will cover my work so far. But, to help approach this problem, I made a visualization that we can play with different values of $$n$$ and $$f$$ for to see what happens.

This tool has helped me explore some different factors and I’ve found some good upper/lower bounds on $$f$$ for various values of $$n$$ (see the art at the top of the post). The tools has also just been a good way to find some aesthetic patterns like this one:

what’s next?

So I’ve done some math and come up with various hypothesis that turned out to be false. In my next post, I hope to detail those approaches. For now, I’m probably just going to let the problem set in my head, and maybe this will all turn out to be some painfully obvious thing.

The rest of this post will talk about how I rendered the fractals above.

visualization tricks

So my first approach to visualizing this was painfully slow. It was basically the following:

for i in range(0, lvl):
draw_level(i)

def draw_level(i, location):
if i == 0:
draw a n-gon at location
return

for each vertex v of a n-gon centered at location:
draw_level(i - 1, v)

this is $$O(n^{lvl})$$ which gets slow fast! So I changed it to this:

for i in range(0, lvl):
draw_level(i)

def draw_level(i, location):
if i == 0:
create a offscreen buffer b
s = 1 / (1 - f)
resize b to (3s, 3s) # 2 * s is probably enough
draw a n-gon at (1.5s, 1.5s) on b
return b

b = draw_level(i - 1, v)
for each vertex v of a n-gon centered at location:
draw b centered at v

which is $$O(n * lvl)$$. This makes it fast enough to draw hundreds of levels in milliseconds!

It might seem strange that I redo a lot of work because I only draw one level at a time, instead of drawing all levels while iterating, but that was so that I could view a single level and inspect it to see if it had the properties I was looking for without as much noise.

Written on December 29, 2021