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## QUASIPERIODIC TILINGS and CUBIC IRRATIONALITIES

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**QUASIPERIODIC TILINGS and CUBIC IRRATIONALITIES**Shutov A.V., Maleev A.V., Zhuravlev V.G. Vladimir, Russia**Consider a cubic equation**then it has a real root and two complex roots with Cubic Irrationalities with the coefficients under the conditions: We split the set Q of cubic equations with the above conditions on three sets Q3 (triangles), Q4 (squares), and Q5 (pentagons).**By using a greedy algorithm, any**from the real integer ring Greedy Algorithms canbe decomposed in a finite series under the condition for anym>0.**Letζbe the real root of the equation**In the case p = q = 1 (so called Rauzy case) the above order isequivalent to theconditions The Lexicographic Order of the Q3-type. Then thedigits aisatisfy a condition where means a lexicographic order. i.e. the word 111 is absent in the corresponding greedy algorithm. Similarly, in a general case, every Qk-type (k = 3; 4; 5) defines its own condition on the digits ai.**with thecoeficients**under the corresponding Qk-restriction. The Nuclear Let Nucl = Nucl(p; q) be a set of complex numbers Then Nucl called a nuclear is a compact fractal tile.**Q3 equation**The Nuclear**Q3 equation**The Nuclear**Q4 equation**The Nuclear**Q5 equation**The Nuclear**Partitions of a Nuclear**Each nuclear Nucl = Nucl(p; q) can be divided into small tiles respect to the first (k-1)elements a1, a1, ... , ak-1, where k is the type of an equation Qk.**- Green Tile**- Blue Tile Partitions of a Nuclear In case - Red Tile**- Red Tile**- Green Tile - Blue Tile Partitions of a Nuclear In case**Partitions of a Nuclear**In case - Red Tile - Green Tile - Blue Tile - Aqua Tile**Partitions of a Nuclear**In case - Red Tile - Red Tile - Green Tile - Green Tile - Blue Tile - Blue Tile - Aqua Tile - Aqua Tile - Yellow Tile**Inflations with theβ-renormalizations generate**quasiperiodictilings of levels l = 0,1,2…∞**Inflations with theβ-renormalizations generate**quasiperiodictilings of levels l = 0,1,2…∞ Level 1 2 3 4 5 6 7 8**The Rauzy Points**The Rauzy Points Every tile includes an inner point (the Rauzy Point) which is image of the zeropointof the nuclear Nucl under some similarity. 17**Let R(p; q) be a set of all Rauzy points, and Rm(p, q) a set**of Rauzy pointsof type m tiles. Then I(p,q) = R(p,q)'and Im(p, q) = Rm(p; q)'are correspondingparameter sets, where the dash ' is a real conjugation in the cubic field The set Im(p,q) is an intersection of the ring with some right-open interval.Moreover, the closure Im(p,q)cis the same segment, and the closure I(p,q)cis a unionof a finite number of segments. Weak Parameterization**The case Q3: x3+x2+x=1**The case Q3: x3+x=1 The case Q4: x3 - x2+2x=1 The case Q5: x3+x2=1**Strong parameterization for the tiling**We call the strong parameterization the partition of the parameter set into intervalswhich define the 1-crown of each tile having a parameter in the fixed interval.For this purpose we use the nuclear Nucl and the Complexity theorem.**The Nuclear Nucl consists of all tiles which parameters are**the left ends of intervalsin the weak parameterization. The case Q3: x3+x2+x=1 The case Q3: x3+x=1 The case Q4: x3 - x2+2x=1 The case Q5: x3+x2=1**The Complexity Theorem**Every n-crown Crnn(T) of tiles T in Til one-to-onecorresponds to tiles Tin the n-crown Crnn(Nucl) of the nuclear Nucl.Moreover, n-crowns Crnn(Ti) of all tiles Ti in Crnn(Nucl) give all types of n-crowns in the tilings.**The Strong Parameterization and the Partition of the**Parameter Set**The layerwise growth of the tilings**N=1 N=2 N=3 N=4 N=5 N=6 N=7 N=8**Growth Form Conjecture**From this follows that the complexity function is asymptotically equivalent to Here is area of growth polygon. Tilings have polygonal growth forms, i.e. Moreover, with some constant c.**Publications**• Shutov, A. V.; Maleev, A. V.; Zhuravlev, V. G. Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetry. // ActaCrystallographica Section A, 2010, 66, 427-437. • Shutov, A. V.; Maleev, A. V. Quasiperiodic plane tilings based on stepped surfaces. // ActaCrystallographica Section A, 2008, 64, 376–382. • Zhuravlev, V. G.; Maleev, A. V. Layer-By-Layer Growth of Quasi-Periodic Rauzy Tiling. // Crystallography Reports, 2007, 52, 180–186. • Zhuravlev, V. G.; Maleev, A. V. Complexity Function and Forcing in the 2D Quasi-Periodic Rauzy Tiling. // Crystallography Reports, 2007, 52, 582–588. • Zhuravlev, V. G.; Maleev, A. V. Quasi-Periods of Layer-by-Layer Growth of Rauzy Tiling. // Crystallography Reports, 2008, 53, 1–8. • Zhuravlev, V. G.; Maleev, A. V. Diffraction on the 2D Quasi-Periodic Rauzy Tiling. // Crystallography Reports, 2008, 53, 921–929. • Zhuravlev, V. G.; Maleev, A. V. Construction of 2D Quasi-Periodic Rauzy Tiling by Similarity Transformation. // Crystallography Reports, 2009, 54, 359–369. • Zhuravlev, V. G.; Maleev, A. V. Similarity Symmetry of a 2D Quasi-Periodic Rauzy Tiling. // Crystallography Reports, 2009, 54, 370–378. • Maleev, A. V.; Shutov, A. V.; Zhuravlev, V. G. 3D QuasiperiodicRauzy Tilling as a Section of 3D periodic tilling . // Crystallography Reports, 2010, 55, 427-437.