{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T10:15:03Z","timestamp":1760955303969,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2011,7,7]],"date-time":"2011-07-07T00:00:00Z","timestamp":1309996800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein\u2019s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein\u2019s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version.<\/jats:p>","DOI":"10.3390\/sym3030402","type":"journal-article","created":{"date-parts":[[2011,7,7]],"date-time":"2011-07-07T10:36:02Z","timestamp":1310034962000},"page":"402-442","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content"],"prefix":"10.3390","volume":"3","author":[{"given":"Mihai","family":"Prunescu","sequence":"first","affiliation":[{"name":"Brain Products, Freiburg, Germany, and Simion Stoilow Institute of Mathematics of the Romanian Academy Research Unit 5, P.O. Box 1-764, RO-014700 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2011,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1142\/S0218348X10004701","article-title":"Recurrent double sequences that can be generated by context-free substitutions","volume":"18","author":"Prunescu","year":"2010","journal-title":"Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1215\/00294527-2008-004","article-title":"Undecidable properties of recurrent double sequences","volume":"49","author":"Prunescu","year":"2008","journal-title":"Notre Dame J. Formal Logic"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"866","DOI":"10.1016\/j.ejc.2008.08.002","article-title":"Self-similar carpets over finite fields","volume":"30","author":"Prunescu","year":"2009","journal-title":"Eur. J. 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Automatic Sequences-Theory, Applications, Generalizations, Cambridge University Press.","DOI":"10.1017\/CBO9780511546563"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Allouche, J.-P., and Shallit, J. (1999). The Ubiquitous Prouhet-Thue-Morse Sequence, Springer. Sequences and their applications (Singapore 1998), springer series discrete mathematics in theoretical computer science.","DOI":"10.1007\/978-1-4471-0551-0_1"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1070\/RM2009v064n05ABEH004641","article-title":"Sequences close to periodic","volume":"64","author":"Muchnik","year":"2009","journal-title":"Russ. Math. Surv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/0166-218X(84)90082-9","article-title":"Cellular automata can generate fractals","volume":"8","author":"Willson","year":"1984","journal-title":"Discrete Appl. Math."},{"key":"ref_11","unstructured":"Mandelbrot, B.B. (1982). The Fractal Geometry of Nature, W.H. Freeman and Company."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.tcs.2004.11.021","article-title":"Theory of cellular automata: A survey","volume":"334","author":"Kari","year":"2005","journal-title":"Theor. Comput. Sci."},{"key":"ref_13","unstructured":"Wolfram, S. (2002). A New Kind of Science, Wolfram Media."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/BF03024384","article-title":"Pentaplexity","volume":"2","author":"Penrose","year":"1979","journal-title":"Math. Intell."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1145\/367177.367224","article-title":"Proving theorems by pattern recognition","volume":"3","author":"Wang","year":"1960","journal-title":"Commun. ACM"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/BF01418780","article-title":"Undecidability and nonperiodicity for tilings of the plane","volume":"12","author":"Robinson","year":"1971","journal-title":"Invent. Math."},{"key":"ref_17","unstructured":"Grunbaum, B., and Shephard, G.C. (2011). Tilings and Patterns, Dover Publications."},{"key":"ref_18","unstructured":"Lindenmayer, A., and Prusinkiewicz, P. (1996). The Algorithmic Beauty of Plants, Springer Verlag."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Rozenberg, G., and Salomaa, A. (1992). Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics, and Developmental Biology, Springer Verlag.","DOI":"10.1007\/978-3-642-58117-5"},{"key":"ref_20","unstructured":"Baake, M., and Moody R., V. (2000). Directions in Mathematical Quasicrystals, AMS."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Moody, R.V. (1997). The Mathematics of Aperiodic Order. Proceedings of the NATO Advanced Study Institute on Long Range Aperiodic Order, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-8784-6"},{"key":"ref_22","first-page":"619","article-title":"Duality of model sets generated by substitution","volume":"50","year":"2005","journal-title":"Rom. J. Pure Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1007\/BF01449896","article-title":"Zur Theorie der matrices","volume":"64","author":"Perron","year":"1907","journal-title":"Math. 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