Four-Wing System

Four-Wing System equations

This three-dimensional reference model couples its coordinates through the products yz, xz and xy. The published coefficient set below produces a four-wing chaotic attractor. In the artistic process, a sampled trajectory becomes the rail for a swept cross-section.

Here, t is time; x, y and z are state variables; and a, b, c, d, e and f are fixed coefficients. These reference equations match the formulation selected by the artist for this page. The coefficients shown are the published reference values, not a recovered record of the Grasshopper settings used to create Study 11. The component’s internal parameter mapping, original initial conditions and integration settings have not yet been verified.

Formula reference: Wang et al., A 3-D four-wing attractor and its analysis (2009), system (6).

From a trajectory to Four-Wing System art

Riccardo develops the rail in Grasshopper for Rhinoceros using Chimpanzee. The system evolves over time, producing successive three-dimensional points. Connecting those samples creates a path. A cross-section is then swept along that rail, giving the line thickness and a surface. This is how a time-dependent mathematical trajectory becomes visible as sculptural geometry.

The sweep adds a visual layer to the mathematics: its width and orientation affect how the form catches light. Riccardo then uses 3ds Max to develop the material, camera and illumination. The finished work is a digital render of this geometry, rather than a photograph of a fabricated object. No generative AI is used to create the artwork.

To read the image, follow a single edge as it moves through light and shadow. Notice where the projected path overlaps, opens or disappears behind another surface. The rail comes from a rule evolving through time; the artist’s composition lets us experience that motion as a still image. The aim is to reveal the hidden beauty of chaos theory.