Here’s an equilateral triangle divided into six smaller triangles:

Equilateral triangle divided into six irregular triangles (Stage #1)

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Now keep on dividing:

Stage #2

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Stage #3

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Stage #4

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Stage #5

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Equilateral triangle dividing into six irregular triangles (animated)

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But what happens if you divide the triangle, then discard some of the sub-triangles, then repeat? You get a self-similar shape called a fractal:

Divide-and-discard stage #1

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Stage #2

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Stage #3

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Stage #4

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Stage #5

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Stage #6

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Triangle fractal (animated)

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Here’s another example:

Divide-and-discard stage #1

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Stage #2

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Stage #3

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Stage #4

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Stage #5

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Stage #6

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Stage #7

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Triangle fractal (animated)

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You can also delay the divide-and-discard to create a more symmetrical fractal, like this:

Delayed divide-and-discard stage #1

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Stage #2

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Stage #3

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Stage #4

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Stage #5

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Stage #6

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Stage #7

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Triangle fractal (animated)

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What next? You can use trigonometry to turn the cramped triangle into a circle:

Triangular fractal

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Circular fractal
(Open in new window for full image)

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Triangle-to-circle (animated)

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Here’s another example:

Triangular fractal

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Circular fractal

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Triangle-to-circle (animated)

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And below are some more circular fractals converted from triangular fractals. Some of them look like distorted skulls or transdimensional Lovecraftian monsters:

(Open in new window for full image)

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Previous Pre-Posted

• Circus Trix — an earlier look at sextally-divided-equilateral-triangle fractals