Towards Strongly Aperiodic Monotiles in Higher Dimensions
Abstract
The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in ℝ3. This note extends the underlying mechanism---the rep-2N chair CN = [0,2]N ∖ (1,2]N with corner/socket markings---to ℝN. Besides expository material (the rep-2N dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For N=3 the certificate passes: it yields explicit facet matching rules on the 24 panels of C3 (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 → 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For N=4 the same pipeline is run on several structured families of frame assignments (canonical, D4-, Z2× Z2- and Z4-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of C4 thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN
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BibTeX
@misc{kamenetsky2026towards,
title = {{Towards Strongly Aperiodic Monotiles in Higher Dimensions}},
author = {Dmitry Kamenetsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00916},
eprint = {2610.00916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00916},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Towards Strongly Aperiodic Monotiles in Higher Dimensions
AU - Dmitry Kamenetsky
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00916
UR - https://arxiv.org/abs/2610.00916
AB - The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in ℝ^3. This note extends the underlying mechanism---the rep-2^N chair C_N = [0,2]^N ∖ (1,2]^N with corner/socket markings---to ℝ^N. Besides expository material (the rep-2^N dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For N=3 the certificate passes: it yields explicit facet matching rules on the 24 panels of C_3 (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 → 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For N=4 the same pipeline is run on several structured families of frame assignments (canonical, D_4-, Z_2× Z_2- and Z_4-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of C_4 thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN
ER -