Blancmange butterfly. Is that a ’60s psychedelic band? No, it’s one of the shapes you can get by playing with blancmange curves. As I described in “White Rites”, a blancmange curve is a fractal created by summing the heights of successively smaller and more numerous zigzags, like this:

Zigzags 1 to 10

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Zigzags 1 to 10 (animated)

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Blancmange curve

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In the blancmange curves below, the height (i.e., the y co-ordinate) has been normalized so that all the images are the same height:

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Construction of a normalized blancmange curve (animated)

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This is the solid version:

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Solid normalized blancmange curve (animated)

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I wondered what happens when you wrap a blancmange curve around a circle. Well, this happens:

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Construction of a blancmange circle (animated)

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You get what might be called a blancmange butterfly. The solid version looks like this (patterns in the circles are artefacts of the graphics program I used):

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Solid blancmange circle (animated)

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Next I tried using arcs rather zigzags to construct the blancmange curves and blancmange circles:

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Arching blancmange curve (i.e., constructed with arcs) (animated)

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And below is the circular version of a blancmange curve constructed with arcs. The arching circular blancmanges look even more like buttocks and then intestinal villi (the fingerlike projections lining our intestines):

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Arching blancmange circle (animated)

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The variations on blancmange curves don’t stop there — in fact, they’re infinite. Below is a negative arching blancmange curve, where the heights of the original arching blancmange curve are subtracted from the (normalized) maximum height:

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Negative arching blancmange curve (animated)

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And here’s an arching blancmange curve that’s alternately negative and positive:

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Negative-positive arching blancmange curve (animated)

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The circular version looks like this:

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Negative-positive arching blancmange circle (animated)

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Finally, here’s an arching blancmange curve that’s alternately positive and negative:

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Positive-negative arching blancmange curve (animated)

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And the circular version:

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Positive-negative arching blancmange circle (animated)

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Elsewhere Other-Accessible…

• White Rites — more variations on blancmange curves