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Such lattices acquire periodicity only for rotation angles cos\u2009\u03b8\u2009=\u2009<jats:italic>a<\/jats:italic>\/<jats:italic>c<\/jats:italic>, sin\u2009\u03b8\u2009=\u2009<jats:italic>b<\/jats:italic>\/<jats:italic>c<\/jats:italic>, set by Pythagorean triples of natural numbers (<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>, <jats:italic>c<\/jats:italic>). While linear eigenmodes supported by lattices associated with Pythagorean triples are always extended, composite patterns generated for intermediate rotation angles allow observation of the localization-delocalization transition of eigenmodes upon modification of the relative strength of two sublattices forming the composite pattern. Sharp delocalization of supported modes for certain \u03b8 values can be used for visualization of Pythagorean triples. The effects predicted here are general and also take place in composite structures generated by two rotated hexagonal lattices.<\/jats:p>","DOI":"10.1038\/srep32546","type":"journal-article","created":{"date-parts":[[2016,9,2]],"date-time":"2016-09-02T08:56:37Z","timestamp":1472806597000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":83,"title":["Localization-delocalization wavepacket transition in Pythagorean aperiodic potentials"],"prefix":"10.1038","volume":"6","author":[{"given":"Changming","family":"Huang","sequence":"first","affiliation":[]},{"given":"Fangwei","family":"Ye","sequence":"additional","affiliation":[]},{"given":"Xianfeng","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Yaroslav V.","family":"Kartashov","sequence":"additional","affiliation":[]},{"given":"Vladimir V.","family":"Konotop","sequence":"additional","affiliation":[]},{"given":"Lluis","family":"Torner","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,9,2]]},"reference":[{"key":"BFsrep32546_CR1","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1103\/PhysRev.28.1049","volume":"28","author":"E Schr\u00f6dinger","year":"1926","unstructured":"Schr\u00f6dinger, E. 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Phys 52, 555\u2013600 (1929).","journal-title":"Z. Phys"},{"key":"BFsrep32546_CR8","doi-asserted-by":"crossref","first-page":"1492","DOI":"10.1103\/PhysRev.109.1492","volume":"109","author":"PW Anderson","year":"1958","unstructured":"Anderson, P. W. Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958).","journal-title":"Phys. Rev."},{"key":"BFsrep32546_CR9","unstructured":"Mandelbrot, B. B. The fractal geometry of nature. Vol. 173 (Macmillan, 1983)."},{"key":"BFsrep32546_CR10","doi-asserted-by":"crossref","first-page":"2477","DOI":"10.1103\/PhysRevLett.53.2477","volume":"53","author":"D Levine","year":"1984","unstructured":"Levine, D. & Steinhardt, P. Quasicrystals: A new class of ordered structures. Phys. Rev. Lett. 53, 2477 (1984).","journal-title":"Phys. Rev. Lett."},{"key":"BFsrep32546_CR11","doi-asserted-by":"crossref","unstructured":"Janot, C. Quasicrystals: A Primer. 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Localization of light in a parity-time-symmetric quasi-periodic lattice. Opt. Lett. 40, 2758\u20132761 (2015).","journal-title":"Opt. Lett."},{"key":"BFsrep32546_CR31","doi-asserted-by":"crossref","first-page":"3566","DOI":"10.1103\/PhysRevB.33.3566","volume":"33","author":"C Allain","year":"1986","unstructured":"Allain, C. & Cloitre, M. Optical diffraction on fractals. Phys. Rev. B 33, 3566 (1986).","journal-title":"Phys. Rev. B"},{"key":"BFsrep32546_CR32","doi-asserted-by":"crossref","first-page":"5751","DOI":"10.1103\/PhysRevA.36.5751","volume":"36","author":"C Allain","year":"1987","unstructured":"Allain, C. & Cloitre, M. Spatial spectrum of a general family of self-similar arrays. Phys. Rev. A 36, 5751 (1987).","journal-title":"Phys. Rev. A"},{"key":"BFsrep32546_CR33","doi-asserted-by":"crossref","first-page":"10021","DOI":"10.1364\/OE.14.010021","volume":"14","author":"A Della Villa","year":"2006","unstructured":"Della Villa, A. et al. 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Design of photonic band gap structures through a dual-beam multiple exposure technique. Opt. Laser Technol. 36, 625\u2013633 (2004).","journal-title":"Opt. Laser Technol."},{"key":"BFsrep32546_CR46","doi-asserted-by":"crossref","first-page":"013832","DOI":"10.1103\/PhysRevA.84.013832","volume":"84","author":"M Boguslawski","year":"2011","unstructured":"Boguslawski, M., Rose, P. & Denz, C. Increasing the structural variety of discrete nondiffracting wave fields. Phys. Rev. A 84, 013832 (2011).","journal-title":"Phys. Rev. A"},{"key":"BFsrep32546_CR47","doi-asserted-by":"crossref","first-page":"141104","DOI":"10.1063\/1.4754136","volume":"101","author":"W Jin","year":"2012","unstructured":"Jin, W. & Gao, Y. Optically induced two-dimensional photonic quasicrystal lattices in iron-doped lithium niobate crystal with an amplitude mask. Appl. Phys. Lett. 101, 141104 (2012).","journal-title":"Appl. Phys. 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