Fig. 1

Metallic-mean aperiodic approximants. a Schematic showing the role of aperiodic approximants as a link between QCs and periodic crystals illustrated by bright-field transmission electron micrographs of dodecagonal QCs and (3².4.3.4) Archimedean tiling observed in star terpolymer-homopolymer blend used in ref. ¹⁷; the latter consists exclusively of the rugby-ball local motifs (outline) whereas the former features various local motifs including the dodecagonal wheel. b First-generation bronze-mean k = 3 (n = 1, m = 1) tiling. c–f First-generation type IA tilings with k = 6 (n = 3, m = 4), k = 9 (n = 5, m = 7), k = 12 (n = 7, m = 10), and k → ∞, respectively. g–j First-generation type IB tilings with \(k = 6\, (\tilde n = 2,\tilde m = 1)\), \(k = 9\, (\tilde n = 3,\tilde m = 2)\), \(k = 12\, (\tilde n = 4,\tilde m = 3)\), and k → ∞, respectively. In f, j the k → ∞ patterns are drawn schematically, showing only a patch of majority tiles. k, l Subdivision patterns for the rectangle (R), small triangle (ST), and large triangle (LT) tiles in k = 6 type IA and IB tilings, respectively, which also illustrate the meaning of \(n,m,\tilde n\), and \(\tilde m\). m Second-generation k = 6 type IA tiling, with dark blue outlines showing the superimposed first-generation tiling magnified by a factor of \(\beta _6 = 3 + \sqrt {10}\) so as to emphasize self-similarity