Polynomial attractor

Written by Paul Bourke
September 2020

I rediscovered the original article by J.C.Sprott (Sprott, J. C. "Automatic Generation of Strange Attractors." Comput. & Graphics 17, 325-332, 1993c) that was one of the inspirations for my fractal explorations. The images in that publication are rather simplistic, not a complaint but a fact of graphics in that era. I didn't readily find other example and the ones on Wolfram Mathworld were equally uninspiring. So I decided to produce some myself using my 2D histogram method along with a collection of colour maps.

The attractors are based upon the equations below.

x_n+1 = a₀ + a₁x_n + a₂x_n² + a₃x_ny_n + a₄y_n + a₅y_n²
y_n+1 = a₆ + a₇x_n + a₈x_n² + a₉x_ny_n + a₁₀y_n + a₁₁y_n²

GLXOESFTTPSV

What is this "GLXOESFTTPSV"? A nice innovation in the paper was to identify the attractors by their series a₀... a₁₁. This was done by choosing values for each between -1.2 to 1.2 in steps of 0.1 and mapping those to uppercase letter A through to Y. S, GLXOESFTTPSV represents a₀ = -0.6, a₁ = -0.1, a₂ = 1.1, a₃ = 0.2, a₄ = -0.8, a₅ = 0.6, a₆ = -0.7, a₇ = 0.7, a₈ = 0.7, a₉ = 0.3, a₁₀ = 0.6, a₁₁ = 0.9.

MCRBIPOPHTBN

VBWNBDELYHUL

FIRCDERRPVLD

QFFVSLMJJCCR

LUFBBFISGJYS

EJYDREGLYQPV

HIYIWHOKNVCG

MSSSRRPADDSO

OIHVGHAHGYRK

RALLTIOBDULT

WJJFXGXHTRPG

Extension to 3D

The paper extends the notion to polynomials of 3 variables (x,y,z), this results in 30 parameters for a general 2 degree polynomial. This is written here as follows, noting the convention is different to that used in a subsquent republising of the paper.

x_n+1 = a₀ + a₁x_n + a₂y_n + a₃z_n + a₄x_n² + a₅y_n² + a₆z_n² + a₇x_ny_n + a₈x_nz_n + a₉y_nz_n
y_n+1 = a₁₀ + a₁₁x_n + a₁₂y_n + a₁₃z_n + a₁₄x_n² + a₁₅y_n² + a₁₆z_n² + a₁₇x_ny_n + a₁₈x_nz_n + a₁₉y_nz_n
z_n+1 = a₂₀ + a₂₁x_n + a₂₂y_n + a₂₃z_n + a₂₄x_n² + a₂₅y_n² + a₂₆z_n² + a₂₇x_ny_n + a₂₈x_nz_n + a₂₉y_nz_n

While the previous papers presented 2D projections onto a coordinate plane, here I populate a volume as a histogram and then render that volumetrically using ray casting. The volumes used in the examples here are 1000x1000x1000 voxels.