White Rites

The blancmange curve is an interesting fractal formed by summing a series of zigzags. It
takes its name from its resemblance to the milk-pudding known as a blancmange
(blanc-manger in French, meaning “white eating”):

blanc_solid

Blancmange curve


In successive zigzags, the number of zags doubles as their height halves, i.e. z(i) = z(i-1) * 2, h(i) = h(i-1) / 2. If all the zigzags are represented at once, the construction looks like this:

blanc_all

Zigzags 1 to 10


blancmange_all

Zigzags 1 to 10 (animated)


Here is a step-by-step construction, with the total sum of zigzags in white, the present zigzag in red and the previous zigzag in green:

blanc1

Blancmange curve stage 1


blanc2

Stage 2


blanc3

Stage 3


blanc4

Stage 4


blanc5

Stage 5


blanc6

Stage 6


blanc7

Stage 7


blanc8

Stage 8


blanc9

Stage 9


blanc10

Stage 10


blancmange

Blancmange curve (animated)


It’s easy to think of variants on the standard blancmange curve. Suppose the number of zags triples as their height is divided by three, i.e. z(i) = z(i-1) * 3, h(i) = h(i-1) / 3:

blanc_x3_solid

Blancmange curve for z(i) = z(i-1) * 3, h(i) = h(i-1) / 3


Continue reading “White Rites”…

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