Busy Beaver Basins

Turing machines on a cyclic tape · 3-D state graphs

Does it stop?

Unbounded tape · the real question —

This ring · what you are flying through —

Spacetime

Ring

tape cells 12

orbit flow edges split halts

Every halting configuration is merged into one sink, so the halting basin is a single cone. Split halts gives each final tape its own fixed point — and shatters the picture into thousands of specks, which is exactly why merging is the right call.

The graph

configurations—

basins—

halting share—

loop basins—

tallest transient—

garden of eden—

Reading it

Amber cone — the halting basin. Every configuration in it eventually stops. The bright core is the halt state itself.
Cyan cones — basins that never halt. The glowing ring at the bottom of each is the cycle it is trapped in; a period-9 orbit is drawn as a literal 9-gon.
Magenta node — the blank tape, where a busy beaver starts. The thread hanging off it is its whole run; its height is the machine's score.
Height — steps left before the machine's fate is sealed. Every edge points strictly downhill, so time runs like gravity.

Click any node to run it in the panel above and hang its trajectory off it here. Drag to orbit, scroll to zoom.

The catch

A tape of N cells has finitely many configurations, so on this ring halting is always decidable — that is why there is a picture at all. The real busy beaver problem is not. Where the ring is wide enough to hold the whole computation the two agree exactly; where it is not, the tape wraps and the machine answers a different question. BB(5) needs 12,289 cells, so no ring you can draw will ever reach it.

← → to move

building state graph