It’s one of the most famous and easily recognizable logos in the world:

The Mitsubishi diamonds (source)

Those are the three diamonds of Mitsubishi, whose name itself means “three diamonds” or “three rhombi” in Japanese (see 三菱). But if you look at the Mitsubishi diamonds with a mathematical eye, you can see how to create them in two simple steps. First, you divide an equilateral triangle into nine smaller equilateral triangles. Then you discard three of the sub-triangles, like this:

Equilateral triangle divided into nine sub-triangles

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Six sub-triangles left after three are discarded

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But why stop there? Once you’ve discarded three triangles, six triangles are left. Now do the same to the remaining six: divide each into nine sub-triangles and discard three of the sub-triangles. Then do it again and again. When you’ve reduced the diamonds to dust, you’ve got a fractal, a shape that repeats itself at smaller and smaller scales:

Diamond fractal stage #1

Diamond fractal stage #2

Diamond fractal stage #3

Diamond fractal stage #4

Diamond fractal stage #5

Diamond fractal stage #6

Diamond fractal (animated)

After that, you can convert the fractal-within-a-triangle into a fractal-within-a-circle:

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Diamond fractal, triangular to circular (animated)

You can create other fractals by dividing-and-discarding sub-triangles from a rep-9 equilaterial triangle. Here’s what I call a rep9-tri grid fractal:

Grid fractal stage #1

Grid fractal stage #2

Grid fractal stage #3

Grid fractal stage #4

Grid fractal stage #5

Grid fractal stage #6

Grid fractal stage #7

Grid fractal (animated)

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Grid fractal, triangular to circular (animated)

And here’s a rep9-tri hexagon fractal:

Hexagon fractal (initial form)

Hexagon fractal stage #1

Hexagon fractal stage #2

Hexagon fractal stage #3

Hexagon fractal stage #4

Hexagon fractal stage #5

Hexagon fractal stage #6

Hexagon fractal (animated)

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Hexagon fractal, triangular to circular (animated)