009 — The Pole Is in the Map

agylövés: Lysarith · mathematics, parameterisation & execution: Codex (desktop) · GPT-5.6 Sol

The agylövés, verbatim

Lysarith, 2026-07-22:

„módosítjuk egy barnsley ifs értékeit úgy hogy egy mandelbrot c ből kapja őket, áthúzzuk 4Dbe, behajtjuk egy gömbbe szóval vissza 3D és figyelsz a pólusoknál.”

English: We modify the values of a Barnsley IFS so that it gets them from a Mandelbrot c, pull it through 4D, fold it into a sphere, so back to 3D — and watch at the poles.

This is not a revision of 001. The Barnsley fern returns as one coordinate system in a different construction.

What the mathematics said, before building

A complex Mandelbrot parameter supplies two real numbers. They drive the scale and rotation of each of the four Barnsley affine maps, with a different phase for each map. The drive changes the coefficients themselves, not only the probabilities, so it changes the attractor's geometry. Every linear part is kept below σmax = 0.965; the fern remains an IFS rather than losing its attractor.

The second pair of coordinates comes from the orbit z(n+1) = z(n)^2 + c. An orbit may escape, so its complex value is compactified as z/(1+|z|): direction survives while infinity approaches the unit circle. A fern point and one compactified orbit value make

p = (x, y, Re ẑ, Im ẑ) ∈ R⁴, q = p / |p| ∈ S³.

The page then performs a genuine orthogonal rotation in four dimensions. It carries one selected point of the cloud toward the north pole of S³ without stretching the sphere. The return to three dimensions is stereographic projection:

P₊(q) = (q₁,q₂,q₃) / (1−q₄).

At the north pole the denominator vanishes. For a unit q,

|P₊(q)|² = (1+q₄)/(1−q₄).

The apparent explosion is therefore exact, but it is not an explosion of the object. On S³ every point remains at radius one. Switch to the south-pole chart,

P₋(q) = (q₁,q₂,q₃) / (1+q₄),

and the same point is ordinary; away from the equator the two projected radii are reciprocal. The singularity belongs to the chart.

What was built

A clickable Mandelbrot plane chooses c. The selected parameter reshapes a Barnsley IFS while its live maximum singular value reports the contractivity leash. Each fern point is paired with the Mandelbrot orbit, normalised onto S³, and drawn through a rotating three-dimensional view.

The approach the pole control applies the four-dimensional rotation. At its far end the selected anchor reaches the north pole and the north chart opens without bound. Change pole replaces that chart with the south one: the explosion closes although the four-dimensional object has not moved. Two instruments keep the distinction visible — distance to the projection pole and radius in the three-dimensional chart.

What went wrong first

The first verbal reading was that a Mandelbrot value would pour chaos into a fern and the resulting attractor would blow up at a pole. Neither statement is earned. A c can sit in a stable component, on a cycle, or outside the set; and a non-contractive IFS does not become an exciting fern, it stops having the promised attractor.

More importantly, the pole is not a singularity of S³. Calling the projected burst a broken attractor would confuse an object with one coordinate chart on it. The opposite-pole control is the necessary witness: what is infinite from one pole is finite from the other.

Handles

Machine-checked on every push — see verify.py and ../../HANDLES.md:

• changing c changes the affine coefficients, not merely the sampling measure;
• every driven map stays at or below σmax = 0.965 across the displayed Mandelbrot plane;
• the four-dimensional pole approach preserves unit radius;
• the selected point reaches the north pole at the end of the approach;
• stereographic radius obeys |P₊(q)|² = (1+q₄)/(1−q₄);
• near that pole the north chart diverges while the south chart stays finite, with reciprocal radii.

— Codex (desktop) · GPT-5.6 Sol, 2026-07-22