,

QUASIPERIODIC TILINGS and CUBIC IRRATIONALITIES Shutov A.V., Maleev A.V., Zhuravlev V.G.Vladimir, Russia

2Cubic Irrationalities Consider a cubic equation

with the coefficients under the conditions:and two complex roots

then it has a real root

We split the set Q of cubic equations with the above conditions on three sets Q3(triangles), Q4 (squares), and Q5 (pentagons).

with

3Greedy Algorithmscan be decomposed in a finite seriesfor any m>0.

from the real integer ringBy using a greedy algorithm, any

under the condition

4The Lexicographic Orderof the Q3-type. Then the digits ai satisfy a conditionwhereLet be the real root of the equation

means a lexicographic order.In the case p = q = 1 (so called Rauzy case) the above order is equivalent to the conditions

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.5The Nuclear

Let Nucl = Nucl(p; q) be a set of complex numbers

with the coeficientsunder the corresponding Qk-restriction.Then Nucl called a nuclear is a compact fractal tile. The NuclearQ3 equation

The NuclearQ3 equation

The NuclearQ4 equation

The NuclearQ5 equation

10Partitions of a NuclearEach 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.

Partitions of a Nuclear In case

- Red Tile

- Green Tile

- Blue Tile

12In case

- Red Tile

- Green Tile

- Blue TilePartitions of a Nuclear

13

In case- Red Tile- Green Tile- Blue Tile- Aqua TilePartitions of a Nuclear

14

In case- Red Tile- Green Tile- Blue Tile- Aqua Tile- Yellow TilePartitions of a Nuclear

- Red Tile- Green Tile- Blue Tile- Aqua Tile

Inflations with the -renormalizations generate quasiperiodic tilings of levels l = 0, 1, 2

16Inflations with the -renormalizations generate quasiperiodic tilings of levels l = 0, 1, 2

Level 12345678The Rauzy PointsEvery tile includes an inner point (the Rauzy Point) which is image of the zeropoint of the nuclear Nucl under some similarity.17

The Rauzy Points18Weak Parameterization

Let R(p; q) be a set of all Rauzy points, and Rm(p, q) a set of Rauzy points of type m tiles. Then I(p,q) = R(p,q)' and Im(p, q) = Rm(p; q)' are corresponding parameter sets, where the dash ' is a real conjugation in the cubic fieldThe set Im(p,q) is an intersection of the ring with some right-open interval. Moreover, the closure Im(p,q)c is the same segment, and the closure I(p,q)c is a union of a finite number of segments.Weak Parameterization

20

The case Q3: x3+x2+x=1The case Q3: x3+x=1The case Q4: x3 - x2+2x=1The case Q5: x3+x2=121Strong parameterization for the tilingWe call the strong parameterization the partition of the parameter set into intervals which 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.22The Nuclear Nucl consists of all tiles which parameters are the left ends of intervals in the weak parameterization.The case Q3: x3+x2+x=1The case Q3: x3+x=1The case Q4: x3 - x2+2x=1The case Q5: x3+x2=1The Complexity TheoremEvery n-crown Crnn(T) of tiles T in Til one-to-one corresponds to tiles T in 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.2425

The Strong Parameterization and the Partition of the Parameter Set26N=1N=2N=3N=4N=5N=6N=7N=8The layerwise growth of the tilingsGrowth Form Conjecture27Tilings have polygonal growth forms, i.e.

Moreover,

with some constant c. From this follows that the complexity function is asymptotically equivalent to Here is area of growth polygon.

SimilarityTransformations of the tilings28

Generators of the symmetry semigroup29

PublicationsShutov, A. V.; Maleev, A. V.; Zhuravlev, V. G. Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetry. // Acta Crystallographica Section A, 2010, 66, 427-437.Shutov, A. V.; Maleev, A. V. Quasiperiodic plane tilings based on stepped surfaces. // Acta Crystallographica Section A, 2008, 64, 376382.Zhuravlev, V. G.; Maleev, A. V. Layer-By-Layer Growth of Quasi-Periodic Rauzy Tiling. // Crystallography Reports, 2007, 52, 180186. Zhuravlev, V. G.; Maleev, A. V. Complexity Function and Forcing in the 2D Quasi-Periodic Rauzy Tiling. // Crystallography Reports, 2007, 52, 582588. Zhuravlev, V. G.; Maleev, A. V. Quasi-Periods of Layer-by-Layer Growth of Rauzy Tiling. // Crystallography Reports, 2008, 53, 18. Zhuravlev, V. G.; Maleev, A. V. Diffraction on the 2D Quasi-Periodic Rauzy Tiling. // Crystallography Reports, 2008, 53, 921929.Zhuravlev, V. G.; Maleev, A. V. Construction of 2D Quasi-Periodic Rauzy Tiling by Similarity Transformation. // Crystallography Reports, 2009, 54, 359369.Zhuravlev, V. G.; Maleev, A. V. Similarity Symmetry of a 2D Quasi-Periodic Rauzy Tiling. // Crystallography Reports, 2009, 54, 370378.Maleev, A. V.; Shutov, A. V.; Zhuravlev, V. G. 3D Quasiperiodic Rauzy Tilling as a Section of 3D periodic tilling . // Crystallography Reports, 2010, 55, 427-437.30Thank You!

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