Hat walks

Step from hat to hat by leaving through a chosen side.

rule — step — hats seen — drift —

Click any hat to start there.

drag to pan · scroll to zoom · space to play · → to step

Exit side

Violet long (√3), teal short (1). Click to append to the rule. e2 is two grid edges, so it splits into 2a/2b. A chip reading e0/e12 means both sides face the same neighbour.

Rule sequence

The walk cycles through this list, one side per step.

Run

Speed 20 / s Keep walker centred Shade reflected hats

Run to end jumps to the cycle or the patch edge. The chips over the tiling name the metatile and supertile the walker is in.

Heat maps

Basins confined to

Not computed yet.

One run from every hat, ending where it leaves the patch or first repeats a state; log scale, level 5 and below. No preimage (Garden of Eden) and exit tiles and in-degree are scans of the rule, so they work at level 6 too. In-degree counts how many states step onto each hat: 0 is a Garden of Eden hat, 1 a pipe, 2 or more a merge. Program cost ignores the rule: from the clicked hat it runs every program of length 1, 2, … and colours each hat by the shortest program whose walk ends on a loop through it (or merely passes it). Hover a hat to see that program and its walk.

Hierarchy

H T P F

Paper-style metatile outlines Trace the walker's ancestry Show outflows

Metatiles hold 4, 1, 2 and 2 hats; each order substitutes H → 3H+T+3P+3F, T → H, P → 2H+P+2F, F → 2H+P+3F. Paper-style outlines draws the straight-edged polygons with their Fig 5.1 edge charges (zoom in to see them); hats overhang these edges. Untick for each group's true jagged boundary.

Patch

Root metatile Substitution level Go to hat #

Current hat: —
Handedness: —
Next exit: —
Build: —
Start address: —

Walks stop at the patch edge. Level 6 takes a few seconds to build.

Information

Exact Shannon accounting over a uniform prior on starts, level 5 and below. Kept is the entropy of the final record (position while in transit, basin once captured), capacity log₂ of the sink count, packing kept/capacity; the address line is raw MI with the supertile address, uncorrected. Bits burned paints each hat by the bits it erases.