Bats and Butterflies

I’ve used butterfly-images to create fractals. Now I’ve found a butterfly-image in a fractal. The exciting story begins with a triabolo, or shape created from three isoceles right triangles:

The triabolo is a rep-tile, or shape that can be divided into smaller copies of itself:

In this case, it’s a rep-9 rep-tile, divisible into nine smaller copies of itself. And each copy can be divided in turn:

But what happens when you sub-divide, then discard copies? A fractal happens:

Fractal crosses (animated)

Fractal crosses (static)

That’s a simple example; here is a more complex one:

Fractal butterflies #1

Fractal butterflies #2

Fractal butterflies #3

Fractal butterflies #4

Fractal butterflies #5

Fractal butterflies (animated)

Some of the gaps in the fractal look like butterflies (or maybe large moths). And each butterfly is escorted by four smaller butterflies. Another fractal has gaps that look like bats escorted by smaller bats:

Fractal bats (animated)

Fractal bats (static)

Elsewhere other-posted:

Gif Me Lepidoptera — fractals using butterflies
Holey Trimmetry — more fractal crosses

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