Pinwheel triangles
The two-dimensional precursor: five-child substitution, infinitely many orientations, and the cleanest test of search modulo the full Euclidean group.
← GCTS
3D reptiles
Example 01 · Conway–Radin, 1998
Quaquaversal tiling
A fixed prism gathers an expanding world of orientations.
stage
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prisms
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Drag and release to keep panning · scroll through every face
01
The rule
Eight congruent wedges.
Two perpendicular turns.
Split the 30–60–90 prism into two half-depth slabs, then expand the result by two. Each slab begins with four half-scale copies. In one, a central pair turns by 90°; in the other, an equilateral pair turns by 120°. Repeating those noncommuting rotations sends tile orientations throughout SO(3). The scene follows a randomly chosen one of the eight child supertiles as the retained copy, so the entire existing patch stays fixed while seven congruent copies grow around it. Going back discards the latest choice; expanding again draws a new one.
π / 2
about the long axis
2π / 3
about an orthogonal axis
closure
dense in SO(3)
02
Substitution versus enforcement
Can local rules force the hierarchy?
In principle, yes—with decorated or collared tile types. Goodman–Strauss showed that broad classes of substitution tilings in dimension greater than one admit finite local matching rules that force their global hierarchy. The theorem is an existence-and-construction result; it does not turn the bare, unmarked prism above into a known one-tile aperiodic set.
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Choose an exposed face
Use the most constrained frontier face first.
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2
Mate a compatible face
Its decoration determines finitely many rigid placements.
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3
Reject or remember
Check overlap, local rules, then cache the geometric obstruction.
A non-lattice collection
What should join it?
Start with systems that each isolate a different difficulty for geometric search.
Penrose rhombs
Finite orientations but genuinely non-lattice vertices. Their explicit arrow rules make them the best control case for a face-decoration engine.
Sphinx substitution
A rep-tile whose hierarchy has an explicit Goodman–Strauss matching-rule construction—useful for comparing existence proofs with executable constraints.
SCD biprisms
Schmitt–Conway–Danzer screw layers trade substitution for irrational relative rotation, probing whether the search can learn a global helical obstruction.
3D Penrose rhombohedra
Golden prolate and oblate rhombohedra bring explicit three-dimensional matching rules without the quaquaversal tile’s dense orientation group.