Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity that is mathematically described by fractal geometry. Here, we extend the reach of fractal concepts in photonics by experimentally demonstrating multifractality of light in arrays of dielectric nanoparticles that are based on fundamental structures of algebraic number theory. Specifically, we engineered novel deterministic photonic platforms based on the aperiodic distributions of primes and irreducible elements in complex quadratic and quaternions rings. Our findings stimulate fundamental questions on the nature of transport and localization of wave excitations in deterministic media with multi-scale fluctuations beyond what is possible in traditional fractal systems. Moreover, our approach establishes structure–property relationships that can readily be transferred to planar semiconductor electronics and to artificial atomic lattices, enabling the exploration of novel quantum phases and many-body effects.

许多自然形态和形状（例如蜿蜒的海岸线，云层或湍流）表现出特征复杂性，这在数学上可以通过分形几何来描述。在这里，我们通过实验证明基于代数数论基本结构的电介质纳米粒子阵列中的光的多重分形性，扩展了分形概念在光子学中的范围。具体来说，我们设计了新颖的确定性光子平台，该平台基于复杂的二次和四元数环中质数和不可约元素的非周期性分布。我们的发现激起了关于波激发在确定性介质中的传输和局域化性质的基本问题，这些介质具有超过传统分形系统可能发生的多尺度波动。此外，