CONGRUENT TILES. EXPONENTIALLY GROWING SPACE.
See the tiling. Then watch a search find its way.
Switch to Live search to inspect actual branches, not a prerecorded animation.
In the upper half-plane, take the vertices (0,1), (1,1), …, (m,1), (m,m), (0,m), and connect them by hyperbolic geodesics. The binary case m = 2 is the five-sided tile discussed here. Each of the m lower edges is congruent to the upper edge.
These apparent dilations are hyperbolic isometries. “Known tiling” displays a finite window of this explicit infinite tiling; its animation only reveals tiles. It does not perform a search.
“Live search” starts from one tile. It selects the exposed side nearest the origin that meets the target disk, then tries every length-compatible side of a fresh congruent tile. It computes disk isometries, detects interior overlaps in the Klein model, rejects T-junctions, and recursively grows the patch. An exhausted branch is actually removed.
The goal is to cover a finite hyperbolic disk, allowing tiles to extend beyond it. This version searches edge-to-edge tilings only. It uses floating-point geometry with tolerances, not a formal certificate or a decision procedure for infinite tilability. The presets have reflection symmetry, so orientation-preserving copies already include their reflected shapes.
Write α = arctan(1/2). The corner types are a = π/2 − α, b = π − 2α, and c = π/2 + α. Because α/π is irrational, an interior vertex must have either a+a+c+c or b+c+c. With the optional pruning enabled, the search rejects a partial vertex multiset that cannot be extended to either. With it disabled, the same mismatch may surface only after more placements and backtracking.
The tile area is 2(m−1)α, an irrational multiple of π. Gauss–Bonnet therefore rules out a cocompact symmetry group in any tiling by this tile. The explicit tiling nevertheless has the infinite cyclic symmetry z ↦ mz: this is weak aperiodicity, not the absence of every infinite-order symmetry.
The binary construction and its convex geodesic-edge modification are described by Dolbilin & Frettlöh, Properties of Böröczky tilings…, Figure 1 and the following paragraph. The m = 3 and m = 4 presets here are explicit generalizations derived from that construction, not claimed to be specific Margulis–Mozes tiles.
Margulis & Mozes (1998) construct other convex hyperbolic monotiles. Ahara et al. (2018) study a rhombus they found. Those exact vertex specifications have not yet been verified for this app and are deliberately not included. The geometry engine and tile definitions are separated to support their later addition.
All animation events are produced locally in your browser. Nothing is sent to a server. Trace recording pauses at 50,000 events rather than silently discarding backtracks. A 600-tile safety limit reports “limit”, never “impossible”.