3D Hypnotic Patterns: Spirals, Tunnels, Moiré and Recursion
The strongest illusion of infinity usually comes from a small system of equations repeated with the right relationship between angle, radius, scale, and time.
Spirals combine two kinds of motion
A spiral couples angular rotation with radial change. Archimedean spirals grow by a roughly constant radial increment; logarithmic spirals change scale multiplicatively. In animation, logarithmic-style mapping is particularly useful because rotation can feel like simultaneous forward movement into a deeper structure.
Log-polar space turns radius into depth
When radial distance is transformed with a logarithm, equal changes of scale become equal changes in the transformed coordinate. Combining that transformed radius with angle creates a compact coordinate system for endless zooms, coiling tunnels, repeating bands, and nested structures that appear to continue far beyond the screen.
Infinite tunnels do not need infinite geometry
A tunnel can be generated by repeating cross-sections along a closed curve, or entirely in a fragment shader by making depth periodic. Time moves the sample position through that periodic field; modulo-style repetition returns the coordinate without exposing a visible reset.
Moiré creates structure that was never drawn
Two nearby spatial frequencies can interfere to create larger apparent bands, waves, and moving contours. Slight changes in scale or rotation continuously reorganize those interference regions, producing detail that is visually richer than either source pattern alone.
Tunnels, spirals, lattices and webs recur for a reason
Geometric families such as tunnels, funnels, spirals, lattices, checkerboards, and cobweb-like forms have long appeared in research on visual pattern perception. They also map naturally onto polar, radial, and retino-cortical style coordinate systems used in computational graphics.