Lemniscate Grid Pattern

Overview

This artwork constructs a layered grid of lemniscate curves, shaped by the polar equation r = a × sqrt(|cos(2θ)|), where a is a scale constant and theta moves from 0 to 2π. There are 120 total curves, and each one is rotated by a phase of 2π × i / 120, creating a radial sweep across the canvas. I used x = r × cos(θ + phase) and y = r × sin(θ + phase) to map out the full shape in Cartesian coordinates. A sinusoidal pattern based on sin(3θ + phase) created the color variation across the pattern, where I let intensity flow between 0 and 1.

In the code, I created a loop over 120 iterations to rotate each lemniscate around the center, and for each layer I recalculated the angle phase and the color intensity. I generated five overlapping scatter plots per curve with increasing transparency, from alpha 0.02 to 0.12, to build up depth and density. The theta array has 2000 values, giving me smooth and continuous curves. I used the twilight shifted colormap to give it a subtle chromatic transition, keeping the image soft and continuous. This mathematical layering gave me full control over symmetry while still letting complexity emerge naturally.

Although visually radial, this design behaves in the opposite way of my radial line intersection piece. Instead of lines slicing through a central point and generating unpredictable collisions, here the flow follows soft curves that spiral in harmony. Both are structured around a circular frame, but one builds from consistent curvature, while the other fragments through linear chaos.