  Aperiodic crystals constitute a fascinating class of materials that includes
incommensurate (IC) modulated structures and quasicrystals (QCs). Although
these two categories share a common foundation in the concept of superspace,
the relationship between them has remained enigmatic and largely unexplored.
Here, we show "any metallic-mean" QCs, surpassing the confines of Penrose-like
structures, and explore their connection with IC modulated structures. In
contrast to periodic approximants of QCs, our work introduces the pivotal role
of "aperiodic approximants", articulated through a series of $k$-th
metallic-mean tilings serving as aperiodic approximants for the honeycomb
crystal, while simultaneously redefining this tiling as a metallic-mean IC
modulated structure, highlighting the intricate interplay between these
crystallographic phenomena. We extend our findings to real-world applications,
discovering these unique tiles in a terpolymer/homopolymer blend and applying
our QC theory to a colloidal simulation displaying planar IC structures. In
these structures, domain walls are viewed as essential components of a
quasicrystal, introducing additional dimensions in superspace. Our research
provides a fresh perspective on the intricate world of aperiodic crystals,
shedding light on their broader implications for domain wall structures across
various fields.
