An equilateral triangle is a rep-tile, because it can be tiled completely with smaller copies of itself. Here it is as a rep-4 rep-tile, tiled with four smaller copies of itself:

Equilateral triangle as rep-4 rep-tile

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If you divide and discard one of the sub-copies, then carry on dividing-and-discarding with the sub-copies and sub-sub-copies and sub-sub-sub-copies, you get the fractal seen below. Alas, it’s not a very attractive or interesting fractal:

Divide-and-discard fractal 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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Stage #8

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Stage #9

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Divide-and-discard fractal (animated)

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You can create more attractive and interesting fractals by rotating the sub-triangles clockwise or anticlockwise. Here are some examples:

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Now try dividing a square into four right triangles, then turning each of the four triangles into a divide-and-discard fractal. The resulting four-fractal shape is variously called a swastika, a gammadion, a cross cramponnée, a Hakenkreuz and a fylfot. I’m calling it a fylfy fractal:

Divide-and-discard fractals in the four triangles of a divided square stage #1

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Fylfy fractal #2

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Fylfy fractal #3

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Fylfy fractal #4

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Fylfy fractal #5

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Fylfy fractal #6

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Fylfy fractal #7

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Fylfy fractal #8

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Fylfy fractal (animated)

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Finally, you can adjust the fylfy fractals so that each point in the square becomes the equivalent point in a circle:

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