Self-assembly of nanostructures into quasicrystals and aperiodic arrangements represents a fascinating departure from periodic crystallinity, governed by complex theoretical frameworks. Unlike conventional crystals, quasicrystals exhibit long-range order without translational symmetry, displaying rotational symmetries forbidden in periodic systems, such as fivefold or tenfold axes. Aperiodic nanostructures further challenge classical crystallography, requiring advanced mathematical and computational models to explain their formation and stability. Key theoretical concepts include matching rules, phason strain, and entropy stabilization, each contributing to a deeper understanding of these non-periodic systems.

Matching rules serve as a foundational framework for describing how local atomic or molecular interactions enforce global quasicrystalline order. These rules define the allowed configurations of tiles or clusters in a quasicrystal, ensuring compatibility between neighboring units while preventing periodic arrangements. For instance, Penrose tilings, a well-studied 2D quasicrystal model, rely on two rhombus tiles with specific edge-matching constraints to produce an aperiodic pattern. Computational studies have extended these rules to 3D systems, such as icosahedral quasicrystals, using polyhedral clusters like the Ammann-Kramer-Neri tiling. Molecular dynamics simulations demonstrate that energetically favorable local interactions can propagate these rules across larger scales, leading to stable quasicrystalline phases. The enforcement of matching rules often involves energetic penalties for deviations, creating a rugged free-energy landscape that favors quasiperiodicity over competing periodic or disordered states.

Phason strain emerges as a critical concept in describing deviations from ideal quasicrystalline order. Phasons represent additional degrees of freedom unique to quasicrystals, corresponding to rearrangements that preserve local connectivity but alter long-range correlations. Unlike phonons, which describe atomic displacements, phasons involve configurational changes in the tiling or cluster arrangement. Theoretical models treat phason strain as a continuous field, with gradients indicating distortions from perfection. Density functional theory and Monte Carlo simulations reveal that phason strain affects thermodynamic stability, often lowering the free energy of quasicrystals compared to periodic approximants. Phason fluctuations play a dual role: at low temperatures, they may destabilize quasicrystals, while at higher temperatures, entropy from phason modes can enhance stability. Phase-field models incorporate phason strain to predict defect formation and grain boundaries in quasicrystalline nanostructures, showing how strain fields mediate transitions between different quasiperiodic phases.

Entropy stabilization provides a counterintuitive mechanism for quasicrystal formation, where high degeneracy of low-energy states drives self-assembly. While traditional crystals minimize enthalpy, certain quasicrystals benefit from configurational entropy arising from phason flips and tile rearrangements. Statistical mechanical models, such as the random tiling hypothesis, propose that entropic contributions dominate in systems with many degenerate configurations. Computational studies on binary nanoparticle systems demonstrate that entropy alone can stabilize quasicrystals when particles interact via soft potentials, allowing for numerous nearly equivalent packing arrangements. Molecular dynamics simulations of patchy colloids reveal that entropy-driven self-assembly produces dodecagonal or icosahedral order when the rotational entropy of particles compensates for slight enthalpic penalties. The interplay between energy and entropy becomes particularly pronounced in finite-temperature simulations, where entropic effects smooth free-energy barriers between competing phases.

Theoretical frameworks often combine these concepts to explain the dynamic pathways of self-assembly. Phase-field crystal models, for example, incorporate density-wave descriptions to track the evolution of quasicrystalline order from disordered precursors. These models show how initial nucleation events obey matching rules, followed by phason-mediated relaxation toward equilibrium. Machine learning approaches have recently augmented traditional methods, identifying hidden order parameters that predict quasicrystal formation from particle interaction potentials. Neural networks trained on simulation data can classify local motifs as precursors to periodic or aperiodic order, providing insights into early-stage self-assembly.

Mathematical tools like projection methods from higher-dimensional spaces remain central to quasicrystal theory. By treating quasicrystals as irrational slices of periodic lattices in 4D or 5D spaces, researchers derive analytical expressions for diffraction patterns and phason dynamics. This approach enables the computation of elastic properties and defect energies purely from symmetry considerations. Concurrently, group-theoretical analyses classify quasicrystals based on their space-group-like symmetries, revealing universal features across material systems.

Challenges persist in modeling the kinetic arrest of quasicrystalline phases during self-assembly. Kinetic Monte Carlo simulations suggest that growth faults accumulate when phason relaxation cannot keep pace with attachment events, leading to metastable states. Advanced sampling techniques, such as parallel tempering, help map the complex free-energy surfaces governing these processes. Additionally, coarse-grained models bridge atomic-scale interactions with mesoscale patterning, showing how competing interactions—like short-range attraction versus long-range repulsion—favor aperiodicity.

Theoretical studies also explore the stability limits of quasicrystals under varying thermodynamic conditions. Perturbation analyses indicate that quasicrystals occupy narrow regions in phase space, bordered by periodic crystals and disordered phases. High-throughput computational screening identifies interaction potentials most likely to yield quasicrystals, with results suggesting that two-length-scale potentials and anisotropic interactions are common prerequisites. These findings guide the design of novel aperiodic nanostructures through inverse statistical mechanics approaches.

In summary, the self-assembly of quasicrystals and aperiodic nanostructures is deciphered through an interplay of matching rules, phason strain, and entropy stabilization. Computational models and mathematical formalisms continue to uncover the subtle balances between energy, entropy, and kinetics that govern these exotic states of matter. As theoretical frameworks mature, they enable the predictive design of aperiodic nanomaterials with tailored properties, advancing applications from photonics to catalysis. The absence of periodicity no longer precludes understanding; instead, it invites richer descriptions of order in the nanoworld.