# Fractangular Frolics

Here’s an interesting shape that looks like a distorted and dissected capital S: A distorted and dissected capital S

If you look at it more closely, you can see that it’s a fractal, a shape that contains itself over and over on smaller and smaller scales. First of all, it can be divided completely into three copies of itself (each corresponding to a line of the fractangle seed, as shown below): The shape contains three smaller versions of itself

The blue sub-fractal is slightly larger than the other two (1.154700538379251…x larger, to be more exact, or √(4/3)x to be exactly exact). And because each sub-fractal can be divided into three sub-sub-fractals, the shape contains smaller and smaller copies of itself: Five more sub-fractals

But how do you create the shape? You start by selecting three lines from this divided equilateral triangle: A divided equilateral triangle

These are the three lines you need to create the shape: Fractangle seed (the three lines correspond to the three sub-fractals seen above)

Now replace each line with a half-sized set of the same three lines: Fractangle stage #2

And do that again: Fractangle stage #3

And again: Fractangle stage #4

And carry on doing it as you create what I call a fractangle, i.e. a fractal derived from a triangle: Fractangle stage #5 Fractangle stage #6 Fractangle stage #7 Fractangle stage #8 Fractangle stage #9 Fractangle stage #10 Fractangle stage #11

Here’s an animation of the process: Creating the fractangle (animated)

And here are more fractangles created in a similar way from three lines of the divided equilateral triangle: Fractangle #2 Fractangle #2 (anim)

(open in new window if distorted) Fractangle #2 (seed) Fractangle #3 Fractangle #3 (anim) Fractangle #3 (seed) Fractangle #4 Fractangle #4 (anim) Fractangle #4 (seed)

You can also use a right triangle to create fractangles: Divided right triangle for fractangles

Here are some fractangles created from three lines chosen of the divided right triangle: Fractangle #5 Fractangle #5 (anim) Fractangle #5 (seed) Fractangle #6 Fractangle #6 (anim) Fractangle #6 (seed) Fractangle #7 Fractangle #7 (anim) Fractangle #7 (seed) Fractangle #8 Fractangle #8 (anim) Fractangle #8 (seed)

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