Representation of certain self-similar quasiperiodic tilings with perfect matching rules by discrete point sets

1994 ◽  
Vol 4 (6) ◽  
pp. 893-904 ◽  
Author(s):  
Richard Klitzing ◽  
Michael Baake
Keyword(s):  
Perfect Matching ◽  
Discrete Point ◽  
Point Sets ◽  
Self Similar ◽  
Matching Rules ◽  
Quasiperiodic Tilings

PERFECT MATCHING RULES FOR UNDECORATED TRIANGULAR TILINGS WITH 10-, 12-, AND 8-FOLD SYMMETRY

1993 ◽  
Vol 07 (06n07) ◽  
pp. 1455-1473 ◽  
Author(s):  
RICHARD KLITZING ◽  
MARTIN SCHLOTTMANN ◽  
MICHAEL BAAKE
Keyword(s):  
Perfect Matching ◽  
Decomposition Approach ◽  
Matching Rules

Perfect matching rules are derived for quasiperiodic triangular tilings with 10-, 12-, and 8-fold symmetry. We use the composition/decomposition approach via local inflation/deflation symmetry and emphasize the locality of our procedure: The matching rules given here are formulated using certain decorations of the tilings. These decorations turn out to be redundant, i.e., they are locally derivable from the undecorated tilings. Hence, the latter are determined by perfect matching rules as well.

Selfishness of convex bodies and discrete point sets

2019 ◽  
Vol 80 ◽  
pp. 416-431
Author(s):  
Liping Yuan ◽  
Tudor Zamfirescu ◽  
Yue Zhang
Keyword(s):  
Convex Bodies ◽  
Discrete Point ◽  
Point Sets

BASIC IDEAS OF AMMANN BAR GRIDS

1993 ◽  
Vol 07 (06n07) ◽  
pp. 1437-1453 ◽  
Author(s):  
REINHARD LÜCK
Keyword(s):  
Strong Relationship ◽  
Matrix Formalism ◽  
Periodic Patterns ◽  
Characteristic Quantity ◽  
The Matrix ◽  
Related Series ◽  
Inflation Factor ◽  
Self Similar ◽  
Basic Ideas ◽  
Matching Rules

The formalism of the Ammann bar grid with two spacings is systematically analyzed. Simple two-by-two matrices describe the self-similar deflation. The deflation and the inflation factors are the two eigenvalues of the matrix. The ratio of the two spacings is derived as another characteristic quantity of the matrices. The maximum number of spacing ratios and the ratios themselves have been derived for a given inflation factor by the matrix formalism. Applications to non-periodic patterns with eightfold, tenfold and twelvefold symmetry are demonstrated. The strong relationship to Fibonacci and Fibonacci related series is explained. The significance of the Ammann bars for matching rules, inflation/deflation procedures and phason movement is outlined.

Fractal atomic surfaces of self-similar quasiperiodic tilings of the plane

1993 ◽  
Vol 3 (9) ◽  
pp. 1921-1939 ◽  
Author(s):  
C. Godrèche ◽  
J. M. Luck ◽  
A. Janner ◽  
T. Janssen
Keyword(s):  
Self Similar ◽  
Quasiperiodic Tilings
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