# 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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