Curlicue Fractals
1,232 views · Published 9 June 2011 · 6:29 · Indexed 11 October 2026
Channel: Paul Nathan · 2011 · Education
The curlicue fractal is made up of repeated connected Fresnel spirals with varying scale and orientation. A parameter 's' controls the characteristics of the fractal which is generated as follows: theta(n+1) = (theta(n) + 2*PI*s) MOD (2*PI) phi(n+1) = (phi(n) + theta(n)) MOD (2*PI) For visualisation in Cartesian coordinates: x(n) = cos(phi(n)) y(n) = sin(phi(n)) Initial conditions: theta(0) = 0 phi(0) = 0 In the present case an increment ds was generated randomly at the start and then used to step s from 0 to 1. The number of iterations for each value of s was limited to 40,000. For larger fractals this can be increased at the expense of computation time. In order to distinguish between the newest and oldest parts of the fractal and if there is any overlap, the earliest iterations are coloured black and the newest iterations are coloured red, with intermediate steps being linearly interpolated between the two colours. This video represent only 11,665 Curlicue fractals out of an infinite variety! The computation time was about 30mins on one core of an AMD phenom X2 @ 3.5GHz (this includes the overhead of graphical display and saving each image to disk).
