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Towards Strongly Aperiodic Monotiles in Higher Dimensions

Dmitry Kamenetsky

arXiv:2610.00916v1math.MGcs.CGmath.COmath.DS

TL;DR

Although the Hat and chiral Spectre resolved planar aperiodicity in 2023, no chair-based planar einstein is known. The paper marks each chair by its full orientation frame and derives matching rules from the substitution’s hierarchical contact language, then tests them with a finite certificate. For C_3, the certificate passes and every lattice-registered tiling obeying its facet rules is uniquely hierarchical and strongly aperiodic; the corresponding C_4 marking remains an open finite problem.

  • Problem

    Although the Hat and chiral Spectre resolved planar aperiodicity in 2023, no chair-based planar einstein is known.

    Source: §1 · p. 1

  • Method

    The paper marks each chair by its full orientation frame and derives matching rules from the substitution’s hierarchical contact language, then tests them with a finite certificate.

    Source: §6 · p. 5, §8 · p. 6

  • Results

    For C_3, the certificate passes and every lattice-registered tiling obeying its facet rules is uniquely hierarchical and strongly aperiodic; the corresponding C_4 marking remains an open finite problem.

    Source: §8 · p. 6, §14 · p. 12

  • Takeaways & Limitations

    In three dimensions, the formalism independently verifies the Chair44 mechanism at the level of matching rules, while identifying the precise computational question left for four dimensions.

    Source: §2 · p. 2, §14 · p. 12

    The certificate assumes lattice registration and a translated central child, and the paper does not claim a new polyhedron.

    Source: §10 · p. 10, §13 · p. 11

Abstract

from arXiv · show

The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\mathbb{R}^3$. This note extends the underlying mechanism---the rep-$2^N$ chair $C_N = [0,2]^N \setminus (1,2]^N$ with corner/socket markings---to $\mathbb{R}^N$. 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 $\to$ 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, $D_4$-, $Z_2\times Z_2$- 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

1 Introduction

The paper situates the chair-based monotile problem against planar and three-dimensional aperiodicity results, then introduces a finite computational framework for deriving and certifying matching rules. Its three-dimensional certificate independently recovers the Chair44 mechanism, while the four-dimensional search remains open.

1 Introduction

  • The constructive question of strongly aperiodic polyhedral monotiles in dimensions N ≥ 4 remains open, while prior planar work resolved the einstein problem with the Hat and Spectre.The three-dimensional Chair44 was announced in September 2026 as the first strongly aperiodic monotile in R^3.Source: §1 · pp. 1–2

2 Contributions and relation to the literature

  • The frame-marking formalism derives admissible contacts from a substitution, making matching-rule candidates finite and computer-testable in every dimension.The choices are the orientations of the 2^N − 1 outer children inside a supertile.Source: §2 · p. 2
  • Three finite checks imply that every rule-compliant tiling is built from supertiles at every level.The checks test one-level reproduction, face-by-face phrasing, and whether each surrounding cluster is dead or forces a specific supertile.Source: §2 · p. 2
  • In three dimensions, the certificate yields 135 facet-contact rules and independently reproduces the published Chair44 contact and enclosure statistics.The known strongly aperiodic marked chair is independently verified at the matching-rule level, not as a new polyhedron.Source: §2 · p. 2
  • Among 2187 homochiral three-dimensional frame assignments, exactly three pass the self-reproduction check and are equivalent up to coordinate-axis renaming.The paper concludes that self-similarity forces the Chair44 marking.Source: §2 · pp. 2–3
  • In four dimensions, 7712 candidates from structured families and a direct lift of the 3D solution fail; the remaining stated search has 1215 candidates.The paper does not claim a four-dimensional monotile.Source: §2 · p. 3

3 Preliminaries and definitions

The paper defines tilings and aperiodicity in terms of geometric coverage and symmetry, then specifies the N-dimensional chair as a corner-deleted cube. The chair has a simple panel structure whose size scales with dimension.

3 Preliminaries and definitions

  • A tiling is a locally finite collection of closed topological N-balls covering R^N with pairwise disjoint interiors, each isometric to a prototile.Its automorphism group consists of isometries preserving the tiling.Source: §3 · p. 3
  • A strongly aperiodic protoset admits tilings but no admitted tiling has an infinite cyclic symmetry subgroup, excluding translations and screw motions.A single prototile with this property is a strongly aperiodic monotile.Source: §3 · p. 3

4 The N-dimensional chair polytope

  • For N ≥ 2, C_N is the cube [0,2]^N with the corner (1,2]^N removed, equivalently a union of 2^N − 1 unit cubes.The volume is 2^N − 1, and the chair is star-shaped about the origin and homeomorphic to a closed N-ball.Source: §4 · p. 3
  • The boundary of C_N has exactly N·2^N unit panels and 3N maximal flat facets.Removing the corner deletes N external panels but exposes N internal socket panels, leaving the panel count unchanged.Source: §4 · p. 3
  • C_3 has seven cubes and 24 square panels, while C_4 has 15 cells and 64 cubic panels.For N = 2, the chair is the three-cell L-tromino with eight edges.Source: §4 · p. 3

5 The rep-2N dissection (classical)

The corner-deleted chair admits a rep-2^N dissection, with a central child and reflected outer children. The paper then encodes tile orientations as frames and derives a finite contact language and local facet rules from the substitution.

5 The rep-2N dissection (classical)

  • For every N ≥ 2, the scaled chair 2C_N dissects into 2^N congruent copies of C_N with disjoint interiors.The central child is a translate; the 2^N − 1 outer children are reflected and translated into the surrounding blocks.Source: §5 · p. 4
  • The dissection subdivides 2C_N into 2^N − 1 blocks meeting at a common point, then places one central child and one reflected chair in each remaining block.The outer reflection moves each chair’s socket toward the shared point.Source: §5 · p. 4

6 Frame markings and the contact language

  • A marked tile includes its full orientation frame, and the substitution’s hierarchical contact language is the finite set of contacts occurring across all levels.The language is defined as the least fixed point generated from child contacts and contacts between supertiles.Source: §6 · pp. 4–5
  • Coarsening closure requires that admissible aligned supertile contacts reproduce the rule exactly one level up, with no offset poses becoming admissible.The closure is iterated until it stabilizes; equality, rather than inclusion, is required for the assignment to be coarsening-closed.Source: §6 · p. 5
  • The induced facet rules record both panel identities and their relative orientation, and tightness makes each contact checkable locally panel by panel.When tight, the rules can be realized as keys and locks on the panels.Source: §6 · p. 5

7 A finite certificate for unique hierarchy

A finite certificate shows that, under lattice registration and its three checks, every admitted tiling has a unique nested hierarchy of supertiles. The checks combine coarsening closure, tight facet rules, and enclosure tests that force complete supertiles.

7 A finite certificate for unique hierarchy

  • For each tile enclosure, the certificate requires exactly one compatible level-1 supertile and checks that a compatible enclosure of its central child contains the remaining children.Only live enclosures—those admitting a legal second shell—need these tests.Source: §7 · p. 5
  • The three certificate conditions imply that every lattice-registered F* tiling has a unique, nested partition into complete level-k supertiles for every k ≥ 1.The level-(k + 1) partition refines the level-k partition.Source: §7 · p. 5
  • The proof uses the actual tile enclosure to identify its unique supertile, then shows its children are present and that these supertiles partition the tiling.Coarsening closure lets the argument iterate after scaling, yielding the nested hierarchy.Source: §7 · pp. 5–6
  • Once the frame assignment is fixed, every certificate condition is a finite computation.The implementation’s exhaustive second-shell search did not reach its monitored node limit in the reported runs.Source: §7 · p. 6

8 The three-dimensional case: an independent certificate

For the 3D chair, the finite certificate passes and independently recovers the Chair44 contact and enclosure statistics, yielding uniquely hierarchical, strongly aperiodic lattice-registered tilings. A scan of homochiral frame assignments finds only the certified assignment and its two coordinate-conjugate copies among those with small coarsening closure.

The three-dimensional case: an independent certificate

  • 135 facet-contact triples on C3 yield uniquely hierarchical lattice-registered tilings that are strongly aperiodic, with |Aut(T)| ≤ 24.The result follows from the certified frame assignment and its induced facet rules.Source: §8 · pp. 6–7
  • 2388 touching poses reduce to 44 admissible contacts, 30 occurring in infinite tilings, and 33 one-shell enclosures, of which 15 are live and each forces a unique supertile.These counts reproduce the reported Chair44 statistics independently of the published matching rules.Source: §8 · pp. 6–7
  • The hierarchical language stabilizes with 30 occurring contacts; coarsening adds 14 poses and closes at 44, with no admissible offset pose.The resulting 44 poses satisfy Σ(P*) = 2P*.Source: §8 · p. 6
  • The induced rule has 135 facet triples, and exactly the 44 poses in P* pass the facet test, establishing tightness.The 2388 lattice-registered touching poses were enumerated for this test.Source: §8 · p. 6
  • Among 2187 homochiral assignments with a translated central child, exactly three close with |P*| ≤ 300: the certified assignment and two cyclic coordinate conjugates.The certified assignment is invariant under conjugation by the transposition (02).Source: §8 · p. 8
  • For the other 2184 assignments, six admit offset supertile poses and 2178 exceed 300 poses within two steps; the cap does not rule out larger closures.The three closed assignments also attain the minimum |Σ(P0)| of 44 among the scanned assignments.Source: §8 · p. 8

9 Four dimensions: what the pipeline shows

The four-dimensional pipeline tests structured frame-assignment families rather than the enormous full assignment space; none tested is coarsening-closed. The existence of a coarsening-closed C4 marking remains an open finite problem.

9 Four dimensions: what the pipeline shows

  • The C4 assignment space has 1215 ≈1.5 · 10^16 homochiral choices, making exhaustive enumeration infeasible with the authors’ resources.The code runs in seconds per assignment, so the authors tested structured families instead.Source: §9 · p. 8
  • In every examined failure, initial coarsening yields only aligned poses; offsets appear only after closure grows, indicating failed self-similarity rather than failed local rigidity.Canonical assignments admit offset supertile poses after closure grows in both N = 3 and N = 4.Source: §9 · pp. 8–9, Figure 3 · p. 9
  • All tested 4D families stay far above 2|P0|, while the certified 3D assignment has |Σ(P0)| = 44 = 2|P0| −16.Table 2 summarizes the tested C4 families and their coarsening-closure status.Source: Table 2 · p. 9
  • Large |Σ(P0)| plateaus—1566 for generic assignments and 158 for canonical-type ones—defeat local search, while the N = 3 solution is an isolated minimum.The value 158 also occurs for 693 of 2187 N = 3 assignments.Source: §9 · p. 9
  • Problem 9.1 asks whether any homochiral C4 frame assignment is coarsening-closed and, if so, whether it also satisfies (C2) and (C3).A positive answer would give a marked strongly aperiodic monotile in R4 with |Aut| ≤192.Source: §9 · p. 9
  • The tested corner-colouring analogue does not reproduce P* even in N = 3, accepting 2016 of 2388 bare contacts without corner-to-corner coincidence.The frame formalism replaces this guessed rule with one derived from the substitution.Source: §9 · p. 10

10 Polyhedral realisation via keys and locks

The proposed key-and-lock construction encodes facet matching rules as complementary reliefs, allowing only specified relative orientations to fit flush. The paper does not claim a new polyhedron or prove that a realization enforces lattice registration.

10 Polyhedral realisation via keys and locks

  • Each panel receives an asymmetric relief of height ε < 1/8, with protrusions on key panels and matching depressions on lock panels.The relief shape depends on the panel index.Source: §10 · p. 10
  • Panel a fits flush against panel b in orientation g exactly when (a, b, g) ∈ F*; the reliefs must also prevent partial off-lattice overlap.Figure 4 illustrates allowed flush fits and forbidden overlaps.Source: §10 · p. 10
  • The construction’s difficult step is realizing the exact rule set; for N = 3, Chair44 is a realization of an equivalent rule set, but no new polyhedron is claimed.Lattice registration remains an assumption that a polyhedral realization must enforce by shape.Source: §10 · p. 10

11 Conditional strong aperiodicity

The theorem gives strong aperiodicity under lattice registration and unique hierarchical decomposition, assuming arbitrarily large supertiles occur. In the N = 3 model, the paper identifies how these hypotheses are supplied.

11 Conditional strong aperiodicity

  • Under the theorem’s three assumptions, every admitted tiling has no nonzero translation or infinite-order symmetry, and its symmetry group has size at most 2^N N!, or 2^(N−1)N! for tiles of one handedness.The assumptions are arbitrarily large supertiles, lattice registration, and a unique nested hierarchy with inflation factor 2.Source: §11 · p. 10
  • The theorem’s proof uses hierarchy preservation and finite-order lattice symmetries to rule out nonzero translations, then embeds the symmetry group into the signed-permutation group.The resulting group-size bounds follow from that embedding.Source: §11 · pp. 10–11
  • For N = 3, Proposition 7.1 supplies arbitrarily large supertiles and the unique hierarchy, while lattice registration is part of the model.The theorem therefore applies conditionally to the marked tile of Theorem A.Source: §11 · p. 11

12 Dimensional summary

The dimensional summary organizes the status by dimension, while the conclusion describes a finite open problem for C4 and identifies routes for further work.

12 Dimensional summary

  • Table 3 summarizes the situation dimension by dimension.Source: §12 · p. 11

13 Future work

  • The decisive C4 question is whether any coarsening-closed frame assignment exists, approachable through conceptual marking or a pruned assignment search.Suggested search reductions include closure on each three-dimensional supertile face or a SAT/SMT encoding of condition (C1).Source: §13 · p. 11
  • The certificate assumes lattice registration and a translated central child; relaxing the latter enlarges the search 24-fold, while relaxing the former requires off-lattice contacts.Source: §13 · p. 11
  • An explicit key-and-lock polyhedron for the rules F* remains to be constructed, with a proof that its shape forces lattice registration.Source: §13 · p. 11
  • Whether the 135 facet triples can be reduced, or the 14 admitted but never-occurring poses excluded by a local rule, remains open.Source: §13 · p. 11
  • The frame-marking formalism may also apply to other rep-tiles, including notched cubes, or reveal that self-similar markings fail for them.Source: §13 · p. 12

14 Conclusion

  • The frame-marking formalism makes chair matching-rule searches finite in every dimension; the conclusion reports a successful certification in three dimensions and an open problem in four.The C4 marking question is whether a coarsening-closed frame assignment exists.Source: §14 · p. 12

Appendix A: the 44 admissible relative poses for N = 3

Appendix A describes the relative-pose notation and distinguishes poses that occur in hierarchical tilings from those admitted by the rules but never occurring.

Appendix A: the 44 admissible relative poses for N = 3

  • Relative poses use (g, t), with g written as signed coordinate images; “H” marks the 30 poses occurring in hierarchical tilings, while a dot marks 14 rule-admitted poses that never occur.Source: Appendix A · p. 13

Appendix B: human inquiry record

The inquiry record traces the work from the initial higher-dimensional research prompt through review-driven revisions and later presentation and search updates.

Appendix B: human inquiry record

  • The initial prompts launched a higher-dimensional study of Chair44, followed by a preprint request, authorship and diagram work, citation refinements, and tiling visualizations with proof verification.Source: Appendix B · p. 13
  • External review prompted corrections to the proof and a reframing of N ≥ 4 as a research programme; later revisions added the frame-marking formalism, finite certificate, N = 3 results, N = 4 computations, and code.Source: Appendix B · pp. 13–14
  • Presentation revisions included proof write-ups, repository and figure updates, and completion of the Z4-symmetric search with a revised table.Source: Appendix B · p. 14
  • Later figure edits removed a confusing tilings figure, restored opaque tiles, added the chair and hat comparison, and combined the exploded view, supertile, and frame table.Source: Appendix B · p. 14
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