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Einstein problem

The Einstein problem is a fundamental question in concerning the existence of a single , termed an "einstein" (from the ein , meaning "one stone"), that can cover the through tilings but only in an aperiodic manner, without allowing any periodic arrangements. This longstanding challenge, rooted in the study of aperiodic tilings, was first explicitly framed as the search for a monotile following earlier breakthroughs in multi-tile aperiodic sets during the and . The problem traces its origins to Hao Wang's 1961 conjecture on the decidability of tiling problems using square tiles with colored edges, which Robert Berger refuted in 1966 by constructing an aperiodic set of 20,426 such tiles, thereby proving the existence of s. Subsequent efforts reduced the size of aperiodic tile sets dramatically: Raphael Robinson achieved six tiles in 1971, while developed a two-tile pair (the and ) in 1973, establishing with minimal components. The einstein problem specifically sought to eliminate the need for multiple tiles or reflections, posing whether a single, unreflected prototile could enforce aperiodicity through its geometry alone—a question that remained open for over 50 years despite extensive computational and theoretical searches. In March 2023, the problem was resolved with the publication of a paper by amateur mathematician David Smith, software engineer Joseph Samuel Myers, computer scientist Craig S. Kaplan, and mathematician , introducing the "hat" tile—a polykite composed of 8 unit kites that tiles the plane hierarchically via substitution rules but admits no periodic tilings, as proven through geometric incommensurability and computer-assisted combinatorial analysis. The hat's tilings incorporate both reflected and unreflected copies. A follow-up paper in May 2023 introduced the strictly chiral "", a 14-sided that tiles aperiodically without reflections. This breakthrough not only answered the einstein problem but spurred further research, including the 2024 journal publication of the chiral monotile and a 2025 experimental realization using to form aperiodic structures inspired by these tiles. The solution has implications for quasicrystals, , and , highlighting the interplay between mathematical abstraction and physical applications.

Background concepts

Aperiodic tilings

An is a covering of the using a of prototiles, where the tiles fit together without gaps or overlaps, but the resulting pattern lacks any translational periodicity across the entire plane. Specifically, no non-trivial translation can map the tiling onto itself, ensuring that arbitrarily large periodic patches do not occur. This contrasts with periodic tilings, which repeat indefinitely via translations generated by two linearly independent vectors, forming a structure that allows the pattern to be superimposed on itself after finite shifts in multiple directions. Such periodic arrangements, like square or hexagonal grids, exhibit global symmetry and are common in crystalline structures, whereas aperiodic tilings introduce complexity through enforced non-repetition. The concept of aperiodic s emerged in the context of decidability problems in logic and geometry. In 1961, introduced tiles—unit squares with colored edges that must match on adjacent sides—and that any set of such tiles capable of the plane must admit a periodic . This implied an to determine tilability, but it was disproved in 1966 by Robert Berger, who constructed the first known aperiodic set of 20,426 tiles as part of his proof that the domino problem (determining whether a given set tiles the plane) is undecidable. Berger's construction demonstrated that aperiodicity could be forced through local matching rules, highlighting the intricate relationship between finite local constraints and global non-periodic behavior. Subsequent work reduced the size of aperiodic tile sets significantly. In 1971, Raphael M. Robinson developed an aperiodic set of 52 tiles, later refining it to six tiles that enforce non-periodicity via hierarchical subdivision rules. Around 1977, Robert Ammann independently discovered several compact aperiodic sets, including ones with as few as 16 prototiles, often using linear markings (Ammann bars) to impose matching conditions that prevent periodicity. These reductions underscored the possibility of minimal aperiodic systems. A key property of aperiodic tile sets is their tendency to produce tilings with hierarchical or self-similar structures, where larger patterns emerge from repeated local motifs without ever achieving full periodicity. Penrose's tilings from 1974 to 1978 exemplify this, using just two prototiles (such as kites and darts or thin and thick rhombs) with matching rules that generate infinite non-repeating patterns exhibiting fivefold symmetry. Monotiles, where a single prototile suffices, represent an extreme case of such prototile minimization.

Prototiles and monotiles

In the context of plane tilings, a prototile is a basic geometric shape, typically a homeomorphic to a disk, that serves as the fundamental unit from which tilings are constructed, allowing for translations, rotations, and possibly reflections unless specified otherwise. Prototiles form the building blocks of tessellations, where multiple copies are arranged to cover the without gaps or overlaps. A monotile refers to a single prototile capable of the plane on its own. An aperiodic monotile, often called an "einstein," is a monotile that admits s of the plane but only in an aperiodic manner, meaning no periodic is possible with that shape alone. The term "einstein" derives from the phrase ein , meaning "one stone" or "one tile," and was coined by Ludwig Danzer to playfully denote a singular tile solving the problem. Aperiodicity itself has nuanced definitions: weak aperiodicity for a prototile set means it allows of the but forbids any periodic ones, while strong aperiodicity demands that every valid is aperiodic, excluding even those with infinite cyclic symmetries. In the , for standard (or "normal") prototile sets without pathological features, weak and strong aperiodicity are equivalent, as established by Goodman-Strauss. The Einstein problem specifically seeks a strongly aperiodic monotile, ensuring all tilings are non-periodic. Formally, the Einstein problem asks whether there exists a single prototile in the that constitutes an aperiodic set of prototiles by itself—that is, a shape that the plane using only translations, rotations, and reflections, but admits no periodic . This quest narrows the broader study of aperiodic , which typically involve multiple prototiles, to the challenge of achieving aperiodicity with just one shape and no additional matching rules. Related variants in higher dimensions, such as , remain open for a strongly aperiodic monotile; however, a weakly aperiodic example exists in the form of the Schmitt––Danzer tile, which tiles \mathbb{R}^3 without translational periodicity but permits certain screw symmetries.

Historical context

Origins in problems

The Einstein problem originates from early 20th-century inquiries into the nature of plane , particularly those posed in Hilbert's 18th problem at the 1900 . This problem encompassed several questions about constructing space from polyhedra, including whether the Euclidean plane admits a by mutually incongruent polygons and the existence of shapes that tile monohedrally (using congruent copies) but not isohedrally (with transitive action on tiles). While partially addressed related aspects by proving in 1911–1912 that there are only finitely many crystallographic space groups in n dimensions up to , limiting periodic to 230 types in three dimensions, the broader questions about non-periodic and anisohedral configurations left significant gaps in tiling theory. Pioneering work on anisohedral tiles—shapes that permit monohedral s of the but no isohedral ones—built directly on Hilbert's framework. In 1928, Karl Reinhardt provided the first examples of such tiles in three dimensions, resolving a key part of Hilbert's 18th problem by demonstrating polyhedra that tile space without the tiling symmetries acting transitively on the tiles. Extending this to the , Heinrich Heesch identified the initial two-dimensional anisohedral prototile in 1935, a that covers the with congruent copies but lacks full transitivity in any such covering. These early constructions, however, all supported periodic tilings, highlighting a distinction from the aperiodic focus of the later Einstein problem. Heesch's investigations in further advanced tiling theory by examining the structural constraints of mosaics composed of congruent figures, laying groundwork for analyzing how arrangements influence global coverings. His later formulation of the problem, involving successive layers () around a central , quantified the extent to which a can extend without gaps or overlaps; the Heesch number denotes the maximum such layers before obstruction. This approach revealed that finite Heesch numbers preclude complete s and inspired ideas of unavoidable configurations— patterns that inevitably lead to inconsistencies in periodic extensions—paving the way for aperiodic constraints in design. The nomenclature "Einstein problem" emerged much later as a playful reference to an aperiodic monotile, derived from the phrase ein ("one stone" or "one tile"), bearing no connection to the physicist . A pivotal shift toward aperiodicity and computability occurred with 's 1961 conjecture that any finite set of square tiles ( tiles) capable of the must admit a periodic , suggesting the decidability of the problem. refuted this in 1966 by constructing an aperiodic set of 20,426 tiles—proving that tilings exist but none are periodic—and demonstrating the undecidability of the domino problem via reduction to the . This resolution transitioned geometric inquiries into , establishing aperiodic tilings as a rigorous concept emerging from these foundational origins.

Early aperiodic constructions

The first explicit construction of an aperiodic tile set was provided by Robert Berger in 1966, who demonstrated the existence of a of 20,426 capable of tiling the plane only in non-periodic ways. This set, consisting of unit squares with colored edges that must match adjacently, encoded simulations of computations to enforce aperiodicity, simultaneously proving the undecidability of the domino problem—the question of whether a given of can tile the plane at all. Subsequent efforts focused on reducing the number of tiles while preserving aperiodicity. In 1971, Raphael M. Robinson constructed a smaller aperiodic set of six prototiles, introducing a hierarchical structure based on larger "supertiles" that recursively subdivide into smaller versions of themselves. Further reductions followed in the 1970s and 1980s; for instance, Robert Ammann developed an aperiodic set of 16 tiles around 1977, and work by Branko Grünbaum and Geoffrey C. Shephard in their 1987 book Tilings and Patterns explored additional families of aperiodic prototiles, contributing to progressive minimizations that eventually reached sets as small as 13 tiles by the mid-1990s. These constructions typically relied on edge-matching rules or decorations to prevent periodic arrangements. These early sets were primarily tiles, but later constructions shifted to more geometrically diverse prototiles. Parallel developments shifted away from square Wang tiles toward more geometrically diverse prototiles. In 1974, Roger Penrose introduced an aperiodic set comprising a "dart" and a "kite"—two rhombi derived from a 72°-108° rhombus, with ratios governed by the golden ratio \phi = \frac{1 + \sqrt{5}}{2} to ensure matching constraints. By 1976, Penrose refined this to a pair of thin and thick rhombi, also scaled by \phi, which tile the plane aperiodically through forced hierarchical subdivisions. These sets exemplified how irrational ratios in tile dimensions or angles could inherently prohibit translational periodicity. Aperiodicity in these early constructions was often enforced through rules, where basic prototiles are replaced by larger supertiles composed of multiple copies, scaled by an inflation factor (such as \phi in Penrose tilings). This process generates a self-similar hierarchy: each supertile level mimics the overall structure at a larger scale, but the irrational scaling prevents the pattern from repeating periodically across the infinite plane. In Robinson's framework, for example, macro-tiles emerge that must align in ways that propagate defects or shifts, ensuring no global periodicity is possible despite local validity. Grünbaum and Shephard analyzed such mechanisms in their comprehensive , highlighting how local matching rules cascade into global non-repetitive order. Despite these advances, all known aperiodic tile sets from this era required multiple prototiles—at minimum two, as in Penrose's constructions—and no single (monotile) capable of aperiodic s of the had been identified, setting the stage for the longstanding Einstein problem.

Proposed solutions

Pre-2023 attempts

Early efforts to find an aperiodic monotile in the before focused on shapes that imposed non-periodicity through additional constraints, higher dimensions, or non-standard tiling rules, but none satisfied the strict criteria of a single connected prototile forming only aperiodic edge-to-edge tilings without markings, overlaps, or reflections. These attempts highlighted persistent challenges, such as the need for supplementary rules or deviations from planar , underscoring the difficulty of achieving strong aperiodicity through shape alone. In 1988, Peter Schmitt constructed a convex polyhedron prototile for three-dimensional that tiles aperiodically via screw symmetries, producing tilings invariant under translations but not the full group of Euclidean motions; however, this solution operates outside the and thus does not address the two-dimensional Einstein problem. The tile's inspired later work but failed to yield a planar monotile, as its aperiodicity relies on 3D structure. Petra Gummelt proposed a star-shaped in 1996 that covers the plane aperiodically through substitutions and overlaps in five specific ways, corresponding to Penrose-like patterns; this approach deviates from standard edge-to-edge tilings by permitting overlaps, disqualifying it as a true monotile for the Einstein problem. The construction demonstrates how overlap rules can enforce aperiodicity but highlights the need for non-overlapping alternatives. In 2010, Joshua Socolar and introduced a hexagonal prototile with and color markings to enforce local matching rules that force aperiodicity, drawing from Penrose tilings; when realized without markings via shape alone, the effective tile consists of disconnected components and necessitates reflections (mirror images) to complete tilings, violating connectivity and requirements for a strong planar monotile. This near-miss advanced understanding of rule-based enforcement but underscored limitations in geometric realization. The Voderberg tile, a non-convex enneagon discovered by Heinz Voderberg in and revisited in later analyses, admits both periodic and spiral (non-periodic) s of the plane in an edge-to-edge manner without inherent overlaps or rules to exclude periodicity; its ability to form repeating patterns prevents it from qualifying as aperiodic, serving instead as an early example of a monotile with flexible behaviors. Weak aperiodic candidates, such as certain shapes or those relying on restrictions, can produce only non-periodic s when reflections are forbidden by fiat, but they admit periodic arrangements if mirrors are allowed, failing the strong aperiodicity criterion that prohibits all periodic s under the full . Examples include modifications of earlier prototiles where enforces non-repetition, yet these depend on imposed constraints rather than intrinsic . Computational searches prior to 2023 exhaustively enumerated polygons up to , confirming that no , , or serves as an aperiodic monotile, as all known tilers admit periodic arrangements; for instance, the complete of that the revealed 15 families, none inherently aperiodic. These efforts established that simple shapes cannot solve the problem, shifting focus to non-convex forms.

The hat tile

The hat tile, also known as an "einstein" in reference to the Einstein problem, was discovered in November 2022 by amateur David Smith during an exploratory search for polyforms capable of the , using software such as the PolyForm Puzzle Solver developed by Jaap Scherphuis. Smith, a retired technician from East , , identified the shape while experimenting with irregular polygons, initially suspecting its aperiodic properties after generating numerous tilings that appeared non-periodic. He shared his findings with Craig S. Kaplan at the , leading to collaborative verification. Geometrically, the hat tile is a 13-sided polygon composed of eight congruent kites, each with angles of 60°, 90°, 120°, and 90°. Its side lengths incorporate rational multiples of 1 and \sqrt{3}, along with irrational values derived from these to ensure incommensurability, which contributes to the enforced aperiodicity; specifically, it is denoted as Tile(1, \sqrt{3}) in the foundational construction. The overall shape vaguely resembles a fedora hat, hence its name, and spans a total area that allows for dense packing without gaps. Tilings using the hat tile are edge-to-edge, covering the without gaps or overlaps, and require both the tile and its mirror reflection to achieve full aperiodicity, resulting in an achiral overall. The matching rules enforce a hierarchical where edges align precisely based on the kite subdivisions, preventing periodic arrangements through forced irregularities in orientation and positioning. The proof of aperiodicity, established in 2023 by Smith, Joseph Samuel Myers, Kaplan, and , demonstrates that every valid of the by the hat tile must contain arbitrarily large finite supertiles exhibiting 12-fold , which inherently disrupts translational periodicity across the infinite . This geometric forcing arises from rules that build larger metatiles iteratively, ensuring no global lattice can emerge. The serves as the prototype for an infinite family of aperiodic monotiles, generalized through hierarchical meta-tile constructions and systems; notable variants include the "" , denoted Tile(\sqrt{3}, 1), and the "" , all derived by parameterizing the side length ratios and angles while preserving the core aperiodic mechanism. The discovery and proof were first detailed in an preprint released in March 2023, with the formal publication appearing in Combinatorial Theory in July 2024.

The spectre tile

The spectre tile, introduced in May 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and , represents a strictly chiral aperiodic monotile that tiles the plane only through non-periodic arrangements without requiring reflections. This development builds directly on their earlier "hat" tile by enforcing , using either left-handed or right-handed enantiomers exclusively to address whether a single chiral shape can force aperiodicity independently of its . Geometrically, the is a 14-sided derived from an equilateral variant in the , specifically Tile(1,1), but with adjusted kite-like proportions and edges modified by smooth, non-straight S-curves to prevent mirror-image matches. These modifications ensure that tilings rely solely on translations and rotations, maintaining consistent across the plane while allowing mixtures of orientations within that . The proof of aperiodicity demonstrates that all valid tilings of the spectre generate hierarchical clusters through substitution rules, inevitably producing supertiles with irrational scaling ratios, such as $4 + \sqrt{15}, that preclude any periodic lattice structure. Central to this are nine marked clusters—labeled with the Greek letters \Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Phi, and \Psi—which form supertiles combinatorially equivalent to marked hexagons; two substitution rules apply: one replaces a single spectre with a cluster containing one "mystic" (a reflected auxiliary shape) and seven spectres, while the other replaces a mystic with one mystic and six spectres, forcing an infinite hierarchy of increasingly complex, non-repeating supertiles. As a meta-tile within the hat family, the spectre shares the foundational continuum of the hat but refines it to eliminate reflections entirely, enabling purely chiral tilings that the hat's achiral design cannot achieve without mirrors. This relation highlights the spectre's role in extending the hat's weakly chiral properties to strict chirality. A formal proof confirming strong aperiodicity for a single enantiomer of the spectre was published in 2024 under the title "A chiral aperiodic monotile" in Combinatorial Theory. The key innovation lies in resolving the longstanding question of whether a lone chiral prototile can enforce aperiodicity without its mirror counterpart, advancing the understanding of chiral constraints in aperiodic tilings.

Applications and implications

In mathematics

The resolution of the Einstein problem in 2023 with the discovery of the hat tile, a 13-sided polykite, confirmed the existence of a strong aperiodic monotile capable of the solely in non-periodic ways, though requiring reflected copies and hierarchical rules. This breakthrough directly addresses a posed by mathematicians like , whose 1970s work established aperiodic s using two prototiles, by demonstrating that a single tile suffices to enforce aperiodicity through geometric incommensurability and hierarchical rules. The proof combines geometric arguments showing the absence of translational symmetries with exhaustive case analysis of local patch configurations, marking a significant advancement in and theory. Subsequent work introduced the spectre tile, a chiral variant that tiles the plane aperiodically without needing mirror images, further solidifying the existence of such monotiles and expanding the theoretical framework for non-periodic structures. These tiles serve as discrete mathematical analogs to quasicrystals, the aperiodic atomic arrangements first observed by Dan Shechtman in 1982 via electron diffraction patterns in aluminum-manganese alloys, which challenged crystallographic periodicity and earned Shechtman the 2011 Nobel Prize in Chemistry. The hat and spectre tilings, with their substitution-based hierarchies and diffraction-like properties, provide idealized models for analyzing the long-range order in quasicrystals, bridging pure geometry with the mathematical description of physical aperiodic order. In January 2025, researchers demonstrated an experimental realization of aperiodic chiral tilings through molecular self-assembly of tris(tetrahelicenebenzene) on silver surfaces, forming quasicrystal-like structures. Computationally, the proofs for these monotiles relied on software for enumerating thousands of local configurations and verifying rules, positioning the Einstein problem as a for problems in , where tiles must satisfy adjacency rules without global periodicity. This approach highlights connections to , as tools like those used by Kaplan systematically checked finite cases to establish global aperiodicity. The work also bears on decidability in : while the general domino problem of whether a tile set admits a is undecidable (as shown by in using aperiodic sets), the monotile solutions raise refined questions about algorithmic verifiability for single-tile aperiodicity, with implications for higher-dimensional or non-Euclidean settings where undecidability persists. The hat tile generalizes to an infinite continuum of aperiodic monotiles, parameterized by side-length ratios that preserve combinatorial equivalence and tunable symmetries while maintaining strict aperiodicity, allowing for families with varying geometric properties. Extensions to other spaces, such as aperiodic monotilings of the hyperbolic plane or three-dimensional Euclidean space, remain open challenges, with current results limited to the planar case and higher-dimensional undecidability results suggesting significant obstacles. To foster broader mathematical engagement, the National Museum of Mathematics hosted the 2023 Hat Tile Art Contest, which drew over 245 entries from 30 countries and awarded three winners—Evan Brock and Shiying Dong in the global category, and Devi Kuscer in the scholastic category—for creative artworks inspired by the tiles, thereby promoting public understanding of aperiodic structures.

In engineering and materials science

Aperiodic monotiles, such as and spectre tiles, have inspired designs for analogs in , where mimics hat-like patterns to create structures with forbidden rotational symmetries. These assemblies enable photonic and electronic materials exhibiting unique optical and conductive properties due to their aperiodic order, as demonstrated by the formation of chiral tilings using tris(tetrahelicenebenzene) molecules on silver surfaces, achieving dense packing with topological defects that promote entropy-driven aperiodicity. Such self-assembled offer potential for advanced devices like chiral metamaterials for light manipulation, where the lack of prevents in conventional crystals. In metamaterials engineering, aperiodic honeycombs derived from hat tilings have been developed to achieve zero or negative , resulting in auxetic behavior that enhances impact resistance by expanding laterally under compression. For instance, hat-based lattice metamaterials exhibit anisotropic ranging from -0.064 to 0.215, with directional auxetic effects that improve energy absorption compared to periodic counterparts. Similarly, isotropic zero- honeycombs constructed from the hat monotile maintain consistent deformation across orientations at relative densities above 0.225, making them suitable for applications requiring uniform shock absorption without lateral strain transfer, such as protective coatings. Aperiodic monotiles facilitate the design of composites with tunable , leveraging infinite families of variants to optimize , , and . These structures promote zigzag crack propagation, increasing energy dissipation and defect tolerance over periodic ; for example, hat-based composites with 80% VeroClear show 103% higher , 34.5% greater strength, and 15.9% improved . The chiral , in particular, enables directional in multi-phase composites, where curved edges enhance through curvature-induced distribution, as explored in recent designs that integrate soft and hard phases for superior load-bearing capacity. Prototypes of aperiodic structures have been fabricated via for applications, including lightweight components and vibration-damping surfaces. Polyjet-printed hat-derived honeycombs demonstrate enhanced under varying temperatures, reducing weight while maintaining structural integrity in high-stress environments like panels. These additively manufactured metamaterials also exhibit superior due to their irregular geometries, which disrupt wave propagation more effectively than periodic lattices, as validated in prototypes for and control in assemblies. Despite these advances, challenges persist in scaling aperiodic tiles to volumes, including geometric complexity in boundary and assemblability issues that lead to collisions during topological . Ongoing addresses these through multi-phase chiral composites, with 2024 studies on aperiodic monotile-patterned structures exploring deformation-tolerant methods to enable practical fabrication.