Non-Periodic Rhomb Substitution Tilings that Admit Order n Rotational Symmetry

For each n ∈ ℕ we construct a substitution rule using the set of rhombs with angles (πk)/n. These substitution rules generate a local isomorphism class of tilings closed under rotation of order 2n, and also admit singular tilings fixed under a rotation of order n. The scaling factors for this set of substitution rules includes algebraic numbers of every rank.

[1]  V. Berthé,et al.  Purely periodic beta-expansions in the Pisot non-unit case , 2004, math/0407282.

[2]  J. Socolar,et al.  Simple octagonal and dodecagonal quasicrystals. , 1989, Physical review. B, Condensed matter.

[3]  Michael Baake,et al.  Directions in Mathematical Quasicrystals , 2000 .

[4]  F P M Beenker,et al.  Algebraic theory of non-periodic tilings of the plane by two simple building blocks : a square and a rhombus , 1982 .

[5]  A. Mishchenko.  C*-Algebras and K-theory , 1979 .

[6]  Branko Grünbaum,et al.  Aperiodic tiles , 1992, Discret. Comput. Geom..

[7]  The Penrose, Ammann and DA tiling spaces are Cantor set fiber bundles , 2001, Ergodic Theory and Dynamical Systems.

[8]  A. Katz.  Matching Rules and Quasiperiodicity: the Octagonal Tilings , 1995 .

[9]  Richard Kenyon.  The construction of self-similar tilings , 1995 .

[10]  Johannes Kellendonk,et al.  Tilings, C∗-algebras and K-theory , 2000 .

[11]  M. Lothaire.  Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications) , 2005 .

[12]  C. Goodman-Strauss.  MATCHING RULES AND SUBSTITUTION TILINGS , 1998 .

[13]  E. J. W. Whittaker,et al.  Some generalized Penrose patterns from projections of n-dimensional lattices , 1988 .

[14]  N. Priebe,et al.  Characterization of Planar Pseudo-Self-Similar Tilings , 2001, Discret. Comput. Geom..

[15]  J. Karhumäki,et al.  ALGEBRAIC COMBINATORICS ON WORDS (Encyclopedia of Mathematics and its Applications 90) By M. LOTHAIRE: 504 pp., 60.00, ISBN 0 521 81220 8 (Cambridge University Press, 2002) , 2003 .

[16]  Michael Baake,et al.  LIMIT-(QUASI)PERIODIC POINT SETS AS QUASICRYSTALS WITH P-ADIC INTERNAL SPACES , 1998, math-ph/9901008.

[17]  M. Baake,et al.  Self-Similar Measures for Quasicrystals , 2000, math/0008063.

[18]  P. Arnoux,et al.  Pisot substitutions and Rauzy fractals , 2001 .

[19]  Michael Baake,et al.  Designer quasicrystals: Cut-and-project sets with pre-assigned properties , 2000 .

[20]  K.-P Nischke,et al.  A construction of inflation rules based onn-fold symmetry , 1996, Discret. Comput. Geom..

[21]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .