024 — The Fractal Does Not Care Where You Started

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

Put the first point anywhere — especially outside. Choose a vertex, move by a fraction of the way towards it, and repeat. The familiar triangle appears only for one fraction and one way of choosing. Change either rule and watch what was essential separate from what merely looked essential.

This frame holds a drawing your browser makes as you watch; with JavaScript off it stays empty. Nothing the page argues depends on it — the idea, the mathematics and the paths that failed are all in the words.

DRAG OR CLICK · START ANYWHERE

next vertex step fraction λ · 0.50

What you are looking at

The three coloured dots are the only destinations. The small yellow square is where the journey began; put it anywhere by clicking or dragging. The white dot is the traveller now. At every turn it chooses one coloured destination and makes part of the journey towards it.

The pale yellow path shows the first few jumps, while the traveller still remembers where it started.
The growing turquoise dust keeps the later landing places. Given enough jumps, those places reveal the long-term shape.
Next vertex decides how destinations are chosen. Random choice fills a fractal; visiting 1, 2, 3 in order collapses onto a repeating three-stop route.
Step fraction says how much of the distance to travel. At 0.50 it goes halfway. Below zero it moves away instead; above one it passes the destination and lands beyond it.

If the picture suddenly becomes tiny, it has not vanished. The point is escaping, and the view is pulling back to keep it visible.

Outside is only a transient

Every step has the form x′=(1−λ)x+λv. Two runs receiving the same vertex choices keep only |1−λ| of their previous separation at each step. For 0<λ<2, that factor is below one: the system forgets every starting point, including those outside the triangle. At λ≤0 or λ≥2, the forgetting stops and escape becomes possible.

Four experiments

Begin far outside at λ=½

The pale line is the approach that will eventually stop mattering. Every jump cuts the influence of the starting point in half; the familiar triangle still appears.

Change random to in order

The movement still settles down, but it can now visit only three repeating places. Randomness is not dirt on the fractal: it is what lets the traveller reach every part of it.

Move λ below zero

“Towards” becomes “away from”. Each jump magnifies the distance left by the previous one, so the camera must retreat to keep the traveller in view.

Overshoot without escape

Between 1 and 2, each jump passes through the chosen destination and lands beyond it. Surprisingly, the journey can still forget where it began; the short formula above explains why.

origin & verification ↗