Each tile has six sides and three paths connecting them in pairs. Place tiles in a spiral, each with a random rotation, and the paths stitch together across shared edges via union-find. Whenever a connection joins two edges already in the same component, a closed loop is born.
(convergence target)
(boundary law)
and size-4 loops (exact)
of loops
Watch loops/N → 0.0357
The loop density converges to c ≈ 0.035683 — one loop per
~28 tiles. The finite-size correction is c + b/√N with
b ≈ −0.225: boundary suppression keeps small spirals
below the asymptote. Grow N large to see it converge.
Component vs path-type
Component mode gives each connected component its own CIELAB-maximally-separated color. Path-type mode colors by arc geometry: red = adjacent sides (0–1), green = skip-one (2–4), blue = other skip-one (3–5). Path-type reveals the three arc families; component mode reveals the connectivity structure.
Highlight loops to see the power law
Loop sizes follow P(s) ~ s−2.11: most loops are tiny (size 3 triangles and size-4 rhombi, each at density 1/108), but the occasional giant loop winds through dozens of tiles. Size 5 is forbidden — a geometry/parity gap with zero occurrences even over hundreds of thousands of tiles.
Watch loops ignite
Use Grow to place tiles one by one in spiral order. The loop count climbs in steps — each jump corresponds to a new closed component forming. The rate at which loops appear (roughly 1 every 28 tiles) directly visualizes the c ≈ 0.0357 density.
Boundary law paths = B/2
Every open path has exactly two ends on the boundary, so the path count equals half the number of boundary edges: paths = B/2 ≈ 2√3 · √N. This is an exact analytic identity — a topological fact, not a statistical approximation. Toggle "hide open paths" to focus on the loops alone.
Select any component
Click any tile on the canvas to highlight its entire connected component. The side panel shows whether it is a closed loop or open path, and its arc count. Loops range from size-3 triangles all the way to monsters spanning the whole spiral.