Clifford attractors

datashaderpanel

Jim Bednar

Published: September 17, 2018 · Updated: November 26, 2024

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Clifford attractors are a type of iterative equation that traces the path of a particle through a 2D space using functions of sine and cosine terms that make interesting “attractor” patterns (covering only some portions of the possible space, in certain shapes).

Here we use Numpy and Pandas to calculate a dataframe consisting of millions of such locations, using Numba to make generating them 50X faster than bare Python. We’ll then plot the results as a static image using Datashader, which renders arbitrarily large data into fixed-sized images.

import numpy as np
from numba import jit

@jit
def clifford_trajectory(a, b, c, d, x0, y0, n):
    xs, ys = np.zeros(n), np.zeros(n)
    xs[0], ys[0] = x0, y0
    for i in np.arange(n-1):
        xs[i+1] = np.sin(a * ys[i]) + c * np.cos(a * xs[i])
        ys[i+1] = np.sin(b * xs[i]) + d * np.cos(b * ys[i])
    return xs, ys

We can visualize the resulting dataframe using Datashader, with colormaps from Colorcet:

import datashader as ds, pandas as pd
from colorcet import palette_n as ps

def clifford_plot(a=1.9, b=1.9, c=1.9, d=0.8, n=1000000, colormap=ps['kbc']):
    cvs = ds.Canvas(plot_width=600, plot_height=600)
    xs, ys = clifford_trajectory(a, b, c, d, 0, 0, n)
    agg = cvs.points(pd.DataFrame({'x':xs, 'y':ys}), 'x', 'y')
    return ds.tf.shade(agg, cmap=colormap[::-1])

/home/runner/work/examples/examples/attractors/envs/default/lib/python3.11/site-packages/dask/dataframe/__init__.py:31: FutureWarning: 
Dask dataframe query planning is disabled because dask-expr is not installed.

You can install it with `pip install dask[dataframe]` or `conda install dask`.
This will raise in a future version.

  warnings.warn(msg, FutureWarning)

clifford_plot(a=1.7, b=1.7, c=0.6, d=1.2, n=20000000, colormap=ps['dimgray'])

Twenty million data points from an attractor clearly makes an interesting shape! The shapes depend on the parameters provided, and we can now easily build a control panel for exploring the effect of those parameters, using interactive widgets from Panel:

import panel as pn
pn.extension()

widgets = {
    'a': pn.widgets.FloatSlider(value=1.9, end=2.0, step=0.1, name='a'),
    'b': pn.widgets.FloatSlider(value=1.9, end=2.0, step=0.1, name='b'),
    'c': pn.widgets.FloatSlider(value=1.9, end=2.0, step=0.1, name='c'),
    'd': pn.widgets.FloatSlider(value=0.8, end=1.0, step=0.1, name='d'),
    'n': pn.widgets.IntSlider(value=10000000, start=1000, end=20000000, step=100, name='n'),
    'colormap': pn.widgets.Select(value=ps['bmw'], options=ps, name='colormap'),
}

bound_clifford_plot = pn.bind(clifford_plot, **widgets)
pn.Column(*widgets.values(), bound_clifford_plot)