Most descriptions of time treat it as a simple parameter that increases in one direction and carries no internal structure. Standard models assume that events occur without any hidden ordering rules beyond basic causality. A recent mathematical proposal by Latham Boyle and Sotirios Mygdalas examines a different possibility. They explore whether spacetime could admit a form of nonperiodic order similar to the structures known in condensed matter physics as quasicrystals, but extended into a framework where time is included as a coordinate rather than treated separately. Their work investigates how precise, nonrepeating patterns could arise in spacetime from a higher dimensional construction without altering the causal structure required by relativity.
Quasicrystals in ordinary materials contain ordered atomic positions that do not repeat in space. The atomic arrangement displays long range correlation even though no finite region tiles the full pattern by repetition. This type of arrangement can be produced through a cut and projection method. A regular grid is defined in a higher dimensional Euclidean space. A lower dimensional slice is selected at a particular angle. Points of the grid lying near this slice are identified and then projected into the slice. The resulting set of positions forms a structure that contains strict order without periodicity. The discovery that nature allows these structures in ordinary materials raises a question. If an ordered but nonrepeating pattern can exist in space, can a similar mathematical construction exist for spacetime.
Boyle and Mygdalas examine that question by extending the cut and projection method to spacetimes embedded in higher dimensional lattices. Their starting point is a multidimensional lattice with exact translational symmetry. Instead of selecting a slice that represents only spatial directions, they consider slices that contain both spatial and temporal axes. Each point in the high dimensional lattice has coordinates that describe a location and a time coordinate within the larger dimensional setting. A slice of lower dimension is chosen so that its orientation relative to the lattice is irrational in a precise mathematical sense. When points lying near the slice are projected into it, the slice becomes a four dimensional spacetime containing a set of events that follow an ordered but nonrepeating structure.
The projected points are interpreted as events rather than positions. In this construction, the slice defines a spacetime with its own temporal direction. The underlying lattice determines which events appear on that spacetime. Because the slice never aligns with any lattice direction, no periodic structure appears. The arrangement of events retains deterministic rules but does not repeat at any scale. The resulting spacetime contains a nonperiodic yet ordered set of intervals between events. This type of ordering resembles the mathematical properties seen in quasicrystals, but here the pattern distributes across both spatial and temporal coordinates instead of only space.
Although the idea is general, Boyle and Mygdalas focus on a specific example motivated by string theory. They use a ten dimensional torus built by identifying opposite sides of a hypercube and equipping it with a special self dual lattice known from studies of high symmetry compactifications. A torus constructed in this way is compact but possesses exact global structure. A lower dimensional slice placed at certain angles through this torus can be dense, meaning that as one moves along the slice, it passes arbitrarily close to every point of the torus without ever closing on itself. Such a slice is not periodic and does not form cycles. It produces an embedding of a four dimensional spacetime that appears infinite from the perspective of an observer living within it even though the higher dimensional space is finite.
The interest in this construction arises partly from the scale relationships that emerge. Physics contains several large disparities among characteristic energy scales. The Planck scale is vastly larger than the electroweak scale, and the electroweak scale is vastly larger than the scale associated with vacuum energy. These gaps are known features of particle physics, and no simple mechanism explains them within standard frameworks. In the torus construction, the spacing of events along the slice is controlled by geometric factors that depend on the angle between the slice and the high dimensional lattice. The ratios of these spacings align with ratios between physical energy scales. In particular, the electroweak scale often appears as a geometric mean between the Planck scale and the vacuum energy scale. Boyle and Mygdalas point out that the numerical relationships arising from their embedding closely match this pattern.
The method does not modify the causal structure of relativity. The local light cones on the slice remain unchanged because the slice inherits its geometry from the higher dimensional space. The construction does not introduce shortcuts or closed timelike curves. It determines which events appear in the projected spacetime but does not allow signals to propagate outside the standard relativistic framework. The nonrepeating order is present only in the distribution of events and the relationships between their intervals. The dynamical laws governing particles and fields are not specified in this proposal. The authors restrict their analysis to the geometric structure.
A technical element of the construction concerns how small changes in slice orientation correspond to large changes in projected intervals. The slice angle determines which lattice points lie close enough to be included. Because the torus identification wraps each coordinate direction onto itself, shifting the slice by a slight amount produces a widely different set of lattice intersections. The resulting intervals on the four dimensional slice can vary by many orders of magnitude. This feature makes it possible for the embedding to reproduce large physical scale gaps through relatively simple geometric parameters. A long interval on the slice corresponds to a lower energy scale, and a short interval corresponds to a higher scale. The mathematical structure ties these together through the geometry rather than through dynamical assumptions.
Another aspect concerns the status of extra dimensions. Many theories that incorporate additional dimensions attempt to hide them by making them extremely small. Boyle and Mygdalas use a different approach. The extra dimensions remain large, but the slice representing our spacetime never intersects them in a simple way. Because the slice runs through the torus at an irrational orientation, it never repeats and never aligns with any compact direction. Observers confined to the slice cannot detect the compactness because they only experience the induced geometry of the slice. As a result, the extra dimensions are not physically suppressed but remain inaccessible.
One question that follows naturally is whether any observable consequences arise from this structure. The present work does not compute them. The authors identify the mathematical consistency of the embedding and the alignment between geometric scale factors and known physical energy gaps. Determining whether the nonperiodic structure influences particle behavior, field propagation, or cosmological evolution requires additional analysis. The current construction provides a framework on which such investigations could be built. It sets out which events are present in the spacetime but does not specify how physical fields populate it.
Potential future work could examine whether the nonrepeating order affects vacuum fluctuations, mass hierarchies, or field correlations. Since the arrangement of events follows precise but nonperiodic rules, it is conceivable that fields defined on the slice inherit similar relationships. Whether these relationships produce any measurable effects is unknown. Boyle and Mygdalas restrict themselves to establishing the geometry and pointing out the numerical correspondences with particle physics scales.
The embedding also raises questions for cosmology. A spacetime with built in nonperiodic structure could influence processes in the early universe, where energy scales are large and where relationships between scales can affect model behavior. The absence of exact symmetries in the spacetime structure might affect phase transitions or field configurations. These possibilities require further work and are not developed in the current analysis.
The proposal highlights that spacetime may admit forms of order that are not considered in standard cosmological or particle physics models. Conventional treatments assume that spacetime is homogeneous and isotropic at large scales and locally resembles Minkowski space when curvature is negligible. The quasicrystal inspired construction suggests that nonperiodic but ordered structures could exist without violating relativistic constraints. Whether nature uses such structures is an empirical question, but the mathematical construction itself is consistent.
Boyle and Mygdalas demonstrate that a quasicrystal like ordering of events in spacetime can arise from a straightforward extension of the cut and projection method. Their use of a ten dimensional self dual lattice provides a concrete example with scale ratios that align with known physics. The spacetime slice created in this construction is dense, nonperiodic, and infinite from the viewpoint of internal observers. Relativistic causality remains intact. The extra dimensions remain large but hidden due to the irrational orientation of the slice.
The work concludes with the recognition that the structure is mathematically viable but physically untested. It establishes a connection between geometric parameters and physical energy scales but does not provide a mechanism by which the geometry influences dynamics. The next stage would involve studying the behavior of fields on such a spacetime and determining whether any distinctive signatures arise. The question of whether the universe exhibits this type of hidden order remains open.
Source:
Boyle, Latham; Mygdalas, Sotirios. Spacetime Quasicrystals. arXiv:2601.07769 [hep-th], 2024.
Link: https://arxiv.org/abs/2601.07769
