Lenia · True Fourth Dimension
A continuous automaton integrated on a four-torus
Chan (2020) · Sebastian & Claude (mmxxvi)
A creature that lives in four dimensions
Not a hypersphere drawn on a flat sheet. This is a continuous field A(x, y, z, w) defined on a four-torus — every one of its L⁴ cells a real degree of freedom, integrated each tick by a genuine four-dimensional Lenia rule. Nothing is projected in advance. You are looking at a three-dimensional cross-section of a living 4D organism, and when you turn the XW, YW, ZW planes you sweep the hidden fourth axis straight through the slice you can see. Drag the canvas to rotate; watch a shape appear from, and vanish into, a direction that isn't there.
drag ↔ YW · drag ↕ XW — both turn through the 4th axis
Genesis — A single 4-ball dropped into the homeostatic band. It neither dies nor floods — it settles into a self-maintaining lump of four-dimensional structure and drifts. The proof the engine is alive.
the planes with no 3D analogue — each sweeps w through the slice
K : radial kernel over the unit 4-ball
G : Gaussian growth, μ=0.300 σ=0.060
the living band sits near μ≈0.30, σ≈0.06 — push μ up to starve the field, σ up to flood it. 2D Orbium values (μ≈0.15) simply kill it.
R=4 in 4D is a 1,064-tap sparse convolution per cell. Raising R or L makes the organism finer and the frame heavier; the SIM shader recompiles when the tap count changes.
Load Hidden Twin and leave every rotation at zero: one lump. Now nudge YW and the single lump splits — the second was always there, sitting a short distance away along an axis you had no line of sight to. Load Hopf Link and turn slowly: the two rings pass clean through one another and never touch, because in four dimensions they are linked in a way no three-dimensional link can be. Load Glome and spin any fourth-axis plane: its cross-section breathes — a 2-sphere whose radius is really the height of your slice through the 3-sphere. The mass trace holds roughly steady because the rule is tuned to the homeostatic band; drive μ or σ out of it and you can watch a whole four-dimensional ecology starve or drown.