Here again is the famous Sierpiński triangle, a fractal named after the Polish mathematician Wacław Sierpiński (1882-1969):

A Sierpiński triangle

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And here’s a Sierpiński ciangle — my name for a Sierpiński triangle stretched into a circle:

A Sierpiński ciangle or Sierpiński pircle

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More generally, you could call that shape a pircle, a polygon turned into a circle. But you can also squeeze the Sierpiński triangle and turn it into this shape:

A Sierpiński squircle

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You could call that a squircle, a squeezed circle. You use the same trigonometry to create both the Sierpiński ciangle and the squeezed Sierpiński triangle. Here’s an animated gif showing the Sierpiński triangle cycling between polygon, pircle and squircle:

Sierpiński triangle, pircle and squircle (animated at EZgif)

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Now for the Sierpiński carpet, the square analogue of the Sierpiński triangle (you can create it with a point jumping 2/3rds of the way towards the vertices and midpoints of a square, as marked with green dots):

A Sierpiński carpet

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And here’s the carpet as a pircle:

Pircle from Sierpiński carpet

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Like the Sierpiński triangle, you can both pircularize and squeeze the Sierpiński carpet. Here’s the cycle as an animated gif:

Sierpiński carpet ⇔ pircle and squircle (animated at EZgif)

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But you can also pervert the pircle, as it were (in Latin, pervertere means “to thoroughly alter”). For example, what if you flip the radius used in pircularizing the Sierpiński carpet, so that points on the perimeter of the pircle move to the center, and points at the center move to the perimeter? One variant of that perverted pircle looks like this:

Perverted pircle from Sierpiński carpet

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Here are some more polygonal fractals turned into pircular fractals:

Fractal from point jumping 1/2 way to vertices of square (but not same vertex twice in a row)

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Pircle from square fractal

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If you double each point’s distance from the centre, then fold any resulting distance beyond the circle’s radius back inside by halving it, you get this perverted pircle:

Perverted pircle from square fractal

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Here’s another polygonal fractal, the T-square fractal:

T-square fractal from point jumping 1/2 way to vertices of square (but not towards vertex directly opposite vertex just jumped towards)

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Pircle from T-square fractal

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Again, you can pervert the pircle:

Perverted pircle #1 from T-square fractal

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Perverted pircle #2 from T-square fractal

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Perverted pircle #3 from T-square fractal

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Perverted pircles from T-square fractal (animated at EZgif)

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Finally, another polygonal fractal turned into a pircle and perverted pircle:

Another square fractal

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Pircle from square fractal

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Perverted pircle from square fractal

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