Aperiodic congruent monotiles in three dimensions
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Can a single simply connected rational polyhedral tile force strong aperiodicity in R³ when congruent copies, including reflections, are allowed? Strong aperiodicity excludes infinite-order symmetries; a tiling may still have a finite symmetry group.
Why it matters and what counts as progress
Why it matters: Strong aperiodicity forbids screw symmetries and periodic stacking while still allowing finite stabilizers of individual tilings.
What counts as progress: Construct a tile and prove both existence of tilings and absence of every infinite-order symmetry in every tiling.
How the problem developed
- 1993 · Construction [1]
John H. Conway Described the Schmitt–Conway biprism, which tiles space without translational symmetry but can admit screw-symmetric tilings.
- 2010 · Construction [2]
Joshua Socolar and Joan Taylor Presented a three-dimensional Socolar–Taylor tile with aperiodic layer structure but a periodic stacking direction, so it is weakly rather than strongly aperiodic.
- 16 Sept 2026 · Reported claim [3]
Ioannis Tsiokos The preprint presents a single congruent three-dimensional prototile for which every tiling lacks a period, including screw or stacking periods.
AI contribution
A Strongly Aperiodic Monotile in Three Dimensions
Tsiokos reports a simply connected rational polyhedral tile that tiles R³ by congruent copies, reflections allowed, while every tiling has no infinite-order symmetry. Each tiling has a unique nested supertile hierarchy and a finite symmetry group of order at most 24.
Sources
Website timeline
Publication dates and editorial changes.
Tsiokos reports: Chair44 forces nonperiodic tilings in three dimensions