# 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**