024 — The Fractal Does Not Care Where You Started

The agylövés

3 pont, aztán kezdünk bárhol egy random helyen elugrunk a fele távolságig, onnan megint és megint mindig a következő ponthoz a háromból. és hogy mi van ha más irányba megyünk nem a következő ponthoz vagy nem a három pont belsejéből indulunk?

“Three points, then we start anywhere at random and jump half the distance, then again and again towards the next of the three points. And what if we go in another direction, not towards the next point, or do not start inside the three points?”

What the mathematics said

For chosen vertex v and step fraction λ, one jump is:

Tᵥ(x) = (1−λ)x + λv

Give two starting points the same sequence of chosen vertices. Their difference after one step is (1−λ)(x−y), and after n steps:

|Tⁿ(x) − Tⁿ(y)| = |1−λ|ⁿ |x−y|

Thus every initial position is forgotten exactly when 0<λ<2. Starting outside changes the transient, not the attractor. At λ=½ with independent random vertices, the three image triangles meet only at their boundaries and the invariant set is the Sierpiński gasket.

What changes when the rule changes

A cyclic vertex sequence does not explore every address: contraction collapses it onto a period-three orbit. A negative λ means moving away from the selected vertex, giving |1−λ|>1. For 1<λ<2, the point overshoots and reverses its displacement direction, but the maps still contract.

The handles

verify.py checks the exact separation law across random vertex sequences, confirms seed-independence inside the contraction interval, and confirms the predicted period-three orbit for cyclic choice.

python verify.py

The symbolic steps can be submitted independently to WolframAlpha: cancel the shared vertex term, then solve the contraction condition. These are external re-runnable handles, not recorded confirmations.

— agylövés: Lysarith
mathematics, verification, prose & build: GPT-5.6 Sol (Codex terminal)
2026-08-08