Origin: The Zoetrope Intuition

An invariant structure can appear to come alive - realigned to the observer, one transverse phase τ at a time. Nothing moves - only the perspective: projection, not dynamics.

This construction emerged while attempting to create a perfect 3D illusion - inspired by a computer-generated zoetrope of the running Catbus from My Neighbor Totoro by Hayao Miyazaki (watch on YouTube).

My intent was to build a virtual 3D zoetrope - one that unifies form, colour, and motion into a coherent visual phenomenon observable directly, without relying on rendered CGI.

What first appeared to be a "simple" resonance algorithm turned out to be something much deeper - the model that became ResonanT⁴.

Gallery

Explore the 4D model gallery. Click to toggle the zoetrope illusion.

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A swarm of multicoloured butterflies frozen mid-flight, arranged on looping bands in a toroidal cloud against a dark background — A swarm of 161 butterflies against a dark background, arranged along curved, looping bands that spiral inward into an oval, toroidal cloud - dense at the centre, thinning to trailing arcs at the edges. Each butterfly is frozen at a different instant of its wing-flap: some fully spread and clearly butterfly-shaped, many caught edge-on as thin slivers, so the entire flap cycle is scattered across the swarm at once. Colours run the full spectrum - lemon, amber, coral, rose, magenta, violet, indigo, cobalt, cyan, mint - keyed to each butterfly's position in the sequence, so neighbours shade smoothly into one another. Nothing is a solid surface in motion: sweeping the transverse phase τ realigns the same invariant structure to the observer one step at a time, and the sequence reads as a single butterfly taking flight. Controls below offer Previous/Next navigation, a Vibrate button to sweep τ, fullscreen, download, and a copy-link button to share the exact model on screen.

What This Gallery Contains

4D zoetrope models that demonstrate how an invariant structure can appear to rotate or invert purely by varying the transverse phase τ. The effect is stroboscopic - projection, not dynamics.

Controls

Previous / Next: Navigate between the zoetrope variants.
Vibrate: Toggle animation that sweeps the transverse phase τ, producing apparent motion.
Fullscreen: View the model in fullscreen mode for detailed inspection.
Download: Download the current 3D model in GLB format.
Camera controls: Click and drag to rotate the view. Scroll to zoom.

To understand the mathematics behind these structures:

Read Papers