I see one description of the phenomenon as being of a surface (2D) and the "other" two dimensions coming from the exitation "phason." >Common to all these is t

I see one description of the phenomenon as being of a surface (2D) and the "other" two dimensions coming from the exitation "phason." >Common to all these is the existence of an intrinsic degree of freedom, the phason, which in many instances is experimentally accessible and fully controllable. The phason space augments the physical space and supplies additional virtual dimensions, hence enabling physical phenomena beyond what can be ordinarily observed in our physical space. In particular, the phason can be used as an adiabatic knob to engineer topological pumping, which is one of the main applications of the formalism, as evidence by the literature cited above. https://www.nature.com/articles/s42005-021-00630-3 https://www.nature.com/articles/s42005-021-00630-3 >Here, we formulate the theoretical principles and manufacture acoustic crystals composed of arrays of acoustic cavities strongly coupled through modulated channels to evidence one-dimensional (1D) and two-dimensional (2D) dynamic topological pumpings. In particular, the higher-order topological edge-bulk-edge and corner-bulk-corner transport are physically illustrated in finite-sized acoustic structures. We delineate the generated 2D and four-dimensional (4D) quantum Hall effects https://www.nature.com/articles/s41467-021-25305-z https://www.nature.com/articles/s41467-021-25305-z The other notion I see is to view quasicrystalline structures as lower dimension slices of higher dimension periodic arrangements: >The structure of quasicrystals is best described in the framework of the superspace formalism, where the quasicrystal is embedded in a higher dimension (see Janot, 1992 ▸; Janssen et al., 2007 ▸, for an introduction). In the case of icosahedral (i) QCs, the superspace periodic lattice is defined in six-dimensions (six-dimensional) and it is decomposed into two three-dimensional subspaces: the parallel (or physical or external) space, An external file that holds a picture, illustration, etc. Object name is m-03-00247-efi1.jpg, and the perpendicular (or complementary or internal) space, An external file that holds a picture, illustration, etc. Object name is m-03-00247-efi2.jpg. The quasiperiodic atomic arrangement in An external file that holds a picture, illustration, etc. Object name is m-03-00247-efi1.jpg is obtained by an irrational cut through the six-dimensional periodic lattice decorated with three-dimensional objects, lying in An external file that holds a picture, illustration, etc. Object name is m-03-00247-efi2.jpg, called occupation domains (ODs) or atomic surfaces. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4937780/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4937780/