A translational tile that cannot repeat periodically in three dimensions

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Family 155 · Combinatorics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

A bathroom floor can be tiled by repeating a small pattern. More complicated shapes might admit arrangements that never repeat. The periodic tiling conjecture asks a stronger question: if one shape can tile an entire space using translations alone, must there be some repeating arrangement of that shape?

The distinction is between a particular arrangement and the possibilities allowed by a tile. Showing a complicated nonrepeating tiling does not settle the problem if the same tile also admits a regular one. The manuscript in family 155 claims a finite tile in three-dimensional integer space for which every fully periodic arrangement is impossible, even though a tiling exists. It also gives a corresponding solid tile in ordinary three-dimensional space.

This would place the first possible lattice counterexample in dimension three. The result concerns moving copies without rotating or reflecting them; allowing those operations changes the problem.

2. Background and essential definitions

A tile and its translation set

Write $\mathbb Z^3$ for the triples of integers. A finite set $T\subset\mathbb Z^3$ tiles this lattice if there is a set $A\subset\mathbb Z^3$ such that every lattice point has exactly one representation

$$x=a+t,\qquad a\in A,\quad t\in T.$$

The shorthand is $A\oplus T=\mathbb Z^3$. Here $T$ describes one tile and $A$ gives its placements. Uniqueness means there are neither overlaps nor uncovered points. For example, the two-point tile $\{0,1\}$ tiles $\mathbb Z$ by translations through even integers.

A lattice tiling is fully periodic if $A$ is unchanged by every translation in some subgroup $L\subseteq\mathbb Z^3$ of finite index. Finite index means only finitely many lattice points remain after identifying points differing by elements of $L$. Geometrically, a finite repeating cell describes the whole arrangement.

Having one nonzero period is weaker. In three dimensions a pattern could repeat vertically while remaining irregular in the other directions. The theorem rules out three-dimensional periodic repetition; it does not assert the absence of every individual period.

From lattice points to a solid

Replace each point of $T$ by a closed unit cube:

$$\Omega=T+[0,1]^3 =\bigcup_{t\in T}\bigl(t+[0,1]^3\bigr).$$

Shared cube boundaries are ignored because they have volume zero. A translational tiling of $\mathbb R^3$ covers almost every point exactly once. Full periodicity now means invariance of the translation set under a full-rank lattice, generated by three linearly independent real vectors.

A tiling by integer translations immediately thickens into a Euclidean tiling. The reverse implication is harder: a Euclidean tiling may use arbitrary real displacements. Integer vertices of a tile do not automatically force every placement onto an integer grid.

3. Development history and previous partial results

In one dimension, Newman’s finite-state argument shows that a tiling of the integers by one finite tile has a periodic translation set. There are only finitely many ways a tile can meet a moving boundary, and the tiling condition imposes strong recurrence on those possibilities. The original paper is Tesselation of integers.

The two-dimensional problem was substantially more difficult. Bhattacharya’s theorem establishes that every finite tile of $\mathbb Z^2$ admits a periodic tiling. Greenfeld and Tao subsequently obtained quantitative bounds and structural information about arbitrary planar translation sets in The structure of translational tilings in $\mathbb Z^d$. These results permit disconnected finite tiles; they are not confined to familiar connected polyominoes.

Greenfeld and Tao then disproved the periodic tiling conjecture in sufficiently high dimension. Their construction encoded arithmetic rules into a single tile. An intermediate setting was $\mathbb Z^2\times G_0$, where $G_0$ is a finite abelian group. Turning the finite group into ordinary lattice coordinates raised the dimension.

The new manuscript takes that arithmetic strategy toward the smallest remaining lattice dimension. Its central improvement is to retain a cyclic finite group throughout the encoding. A cyclic group needs only one extra integer coordinate, whereas a general finite abelian group may need several.

The Euclidean history has a separate qualification. Periodic-tiling existence is known for rational polygonal sets in the plane, as in de Dios Pont, Grebík, Greenfeld and Madrid. That does not settle the question for every bounded measurable planar tile. Thus the claimed minimal dimension is justified for lattice tilings, without making an unrestricted planar Euclidean assertion.

4. What the new manuscripts claim

The single manuscript’s Theorem 1.1 asserts the existence of a finite nonempty $T\subset\mathbb Z^3$ with two properties.

First, $T$ tiles $\mathbb Z^3$, but no possible translation set $A$ in such a tiling is invariant under a finite-index subgroup. Second, its cube thickening $\Omega$ tiles $\mathbb R^3$, but no possible translation set for that solid is invariant under a full-rank lattice.

The second statement permits arbitrary real translation vectors. It is therefore stronger than merely observing that the particular integer tiling is nonperiodic. Both conclusions concern the same finite tile data.

The construction is finite and can be specified by terminating finite searches for its auxiliary parameters. The source does not offer a practical small tile or a useful numerical bound on its size. Its infinite witness tiling is described arithmetically, through a last-nonzero-digit rule. The theorem is an existence and structural result rather than a compact puzzle one could immediately assemble by hand.

5. Why the advance matters

Importance: 4/5. The claimed result identifies the exact first dimension for failure of the lattice conjecture, completing a natural boundary left by the one- and two-dimensional positive theorems. The extension to arbitrary real translations also addresses a substantive geometric obstacle.

Its broader lesson is that a single finite object can impose arithmetic organization at unbounded scales. Local fitting rules need not leave any freedom to replace an irregular global arrangement by a repeating one. The tile itself is finite; the obstruction concerns every infinite arrangement it permits.

The rating reflects a major development in tiling theory and an improvement in encoding methods. The basic possibility of a translational counterexample was already established in higher dimensions, so the new claim sharpens that breakthrough rather than initiating it.

6. The main idea

Arithmetic that cannot have a vertical period

The construction begins with an infinite strip of symbols. Columns range over a fixed finite interval, while rows range over all integers. Certain samples along lines of integer slope must belong to a finite list of allowed words.

Those words come from arithmetic modulo a prime $p$. For a nonzero integer, repeatedly divide out factors of $p$ and retain the first nonzero base-$p$ digit. This creates patterns with structure at successively larger powers of $p$. The manuscript adapts the Greenfeld–Tao Sudoku rule so that an allowed array with nonconstant columns exists, but no such array has a positive vertical period.

The underlying obstruction can be pictured as an endless divisibility demand. A hypothetical period must be divisible by $p$; after looking at the rescaled exceptional positions, it must contain another factor of $p$, and so on. No fixed positive integer can satisfy every demand. This describes the role of the arithmetic mechanism, not a complete derivation of its line constraints.

Keeping all the constraints in one cyclic group

Next the manuscript translates the allowed-word conditions into several tiling equations with the same unknown translation set. Separate solutions would be useless: the eventual tile must enforce all the constraints simultaneously.

Finite functions encode which symbols are available at each position. Tests ensure that at least one symbol is active, forbid incompatible combinations, and force different values within each column. Arbitrary encoded solutions may activate multiple symbols, so the rules must remain valid for every choice they permit. The explicit constructed solution can use just one active symbol at each position.

Auxiliary finite groups are chosen with pairwise coprime orders. Their product is then cyclic, by the Chinese remainder theorem. This is the dimension-saving step: a complicated collection of finite tests still fits into a group of the form

$$\mathbb Z^2\times\mathbb Z/Q\mathbb Z.$$

Here $Q$ is a positive integer recording the combined finite state. A stacking construction merges the simultaneous equations into one tile. The quotient map from $\mathbb Z^3$ can then represent the cyclic coordinate by reduction modulo $Q$.

Preventing real translations from escaping the encoding

The final passage uses component shapes with a large common part and a displaced marked cube. In a hypothetical periodic Euclidean tiling, the common parts force component centers onto a translated grid. Coverage of the marked cubes then supplies the compatibility needed to descend to the quotient encoding.

The logic runs in both directions. An explicit arithmetic array produces an actual tiling. Conversely, any fully periodic tiling would decode to an arithmetic array with a forbidden vertical period. Both directions are necessary: a construction that prohibited all tilings would not be a counterexample.

7. Scope, limitations, and open questions

The assertion concerns one finite translational tile, which need not be connected. It does not give a connected polycube with the same property, a small optimized example, or a statement about allowing rotations and reflections.

Full periodicity is the precise target. One should not replace it by the stronger claim that every permitted translation set has no nonzero period in any direction. Likewise, the smallest-dimension conclusion refers to lattices, where the planar positive theorem applies in full generality.

The source’s finite searches establish a way to select tile data, not an efficient construction with manageable size. Natural further questions include how small such a tile can be, what connectedness or geometric regularity can be retained, and how much of the arithmetic mechanism survives under additional restrictions. None is resolved merely by the existence theorem stated here.

8. Sources and further reading

The main statement is Theorem 1.1 of M1. Its introduction explains the two directions of the encoding, Sections 2–6 develop the arithmetic and cyclic-group constructions, and Section 7 treats arbitrary real translations. The earlier papers linked in Section 3 provide the one-dimensional, planar and high-dimensional milestones.

For optional terminology, Wikipedia’s tiling and Chinese remainder theorem articles offer background; the tiling and periodicity conventions used here are defined above.

Source manuscripts

• M1. OpenAI, A translational tile with no fully periodic tiling in dimension three (September 23, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.