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A Fibonacci tiling of the plane

Charles W. Huegy, Douglas B. West
2002 Discrete Mathematics  
We describe a tiling of the plane, motivated by architectural constructions of domes, in which the Fibonacci series appears in many ways.  ...  Here we describe a way of designing such a framework that gives rise naturally to the Fibonacci recurrence in various aspects of a tiling of the plane.  ...  Viewed from above, the framework is a tiling of a disc. This extends to a tiling of the plane.  ... 
doi:10.1016/s0012-365x(01)00239-4 fatcat:ser4lr7tizeyro6vnx4633mdv4

A molecular overlayer with the Fibonacci square grid structure

Sam Coates, Joseph A. Smerdon, Ronan McGrath, Hem Raj Sharma
2018 Nature Communications  
Here, we show an experimental realisation of the Fibonacci square grid structure in a molecular overlayer.  ...  Previous theoretical work explored the structure and properties of a hypothetical four-fold symmetric quasicrystal-the so-called Fibonacci square grid.  ...  Shimoda from the National Institute for Materials Science, Tsukuba, Japan for supplying the programme to analyse the surface atomic structure of the Al-Pd-Mn quasicrystal.  ... 
doi:10.1038/s41467-018-05950-7 pmid:30143631 pmcid:PMC6109137 fatcat:vh6errsg6farherx4idw7usiby

About the domino problem in the hyperbolic plane from an algorithmic point of view

Maurice Margenstern
2008 RAIRO - Theoretical Informatics and Applications  
In this paper, we prove that the general problem of tiling the hyperbolic plane with \'a la Wang tiles is undecidable.  ...  In the same line, I have to thank very much Tero Harju.  ...  tile 4 is again a standard Fibonacci tree rooted at the tile 12.  ... 
doi:10.1051/ita:2007045 fatcat:clxzsba47vfzjkepzngc6uhvom


2008 International Journal of Foundations of Computer Science  
In this paper, we consider the finite tiling problem which was proved undecidable in the Euclidean plane by Jarkko Kari in 1994.  ...  Here, we prove that the same problem for the hyperbolic plane is also undecidable.  ...  a set of tiles spanned by a Fibonacci tree.  ... 
doi:10.1142/s0129054108006078 fatcat:td7cinhouvbzne6xbuequm73ku

Fibonacci words, hyperbolic tilings and grossone

Maurice Margenstern
2015 Communications in nonlinear science & numerical simulation  
In this paper, we study the contribution of the theory of grossone to the study of infinite Fibonacci words, combining this tool with the help of a particular tiling of the hyperbolic plane : the tiling  ...  With the help of the numeral system based on grossone, we obtain a richer family of infinite Fibonacci words compared with the traditional approach.  ...  Acknowledgment The author is very much in debt to Yaroslav Sergeyev for his interest to this work.  ... 
doi:10.1016/j.cnsns.2014.07.032 fatcat:7yaplbxu4ra2lonmwxi6zva32a

Surface structure ofi−Al68Pd23Mn9:An analysis based on theT*(2F)tiling decorated by Bergman polytopes

G. Kasner, Z. Papadopolos, P. Kramer, D. E. Bürgler
1999 Physical Review B (Condensed Matter)  
A Fibonacci-like terrace structure along a 5fold axis of i-Al(68)Pd(23)Mn(9) monograins has been observed by T.M. Schaub et al. with scanning tunnelling microscopy (STM).  ...  We derive a picture of "geared" layers of Bergman polytopes from the projection techniques as well as from a huge patch.  ...  We would also like to thank the Geometry-Center at the University of Minnesota for making Geomview freely available, which proved to be a valuable tool throughout our work.  ... 
doi:10.1103/physrevb.60.3899 fatcat:ebvfovne6vb2vova465d5nwqrq

Writing sequences on the plane

E. Soijanin
2002 IEEE Transactions on Information Theory  
We look into three unusual ways to write a sequence in the plane: by Penrose tilings, by space-filling curves, and by cylindrical and spiral lattices.  ...  A good arrangement should enable the one-dimensional sequences to be modeled as Markov chains or shifts of finite type.  ...  ACKNOWLEDGMENT The author is grateful to the late A. D. Wyner who taught her what a great fortune it is to be able to do for a living what you would do for fun.  ... 
doi:10.1109/tit.2002.1003825 fatcat:p6esifk37ra5dkayzoknafykte

Penrose tiling approximants

O. Entin-Wohlman, M. Kléman, A. Pavlovitch
1988 Journal de Physique  
Abstract. 2014 The modification of the Penrose tiling into a periodic structure is considered.  ...  Detailed analysis of the strip method and the dual transformation which yield approximants of the perfect tiling is presented in such a way that a complete classification of approximants is provided.  ...  This research has been supported in part by the Fund for Basic Research administered by the Israel Academy of Sciences and Humanities and by the CNRS.  ... 
doi:10.1051/jphys:01988004904058700 fatcat:kqfcnfisongclfppgkoq5x3hne

Tiling Rectangles and the Plane using Squares of Integral Sides [article]

Bahram Sadeghi Bigham, Mansoor Davoodi, Samaneh Mazaheri, Jalal Kheyrabadi
2021 arXiv   pre-print
We study the problem of perfect tiling in the plane and exploring the possibility of tiling a rectangle using integral distinct squares.  ...  Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given, and one has to decide whether it can tile the plane or a rectangle or not.  ...  The growth rate of the Fibonacci numbers is the golden ratio  = 1+√5 2 (i. e. +1 = 1+√5 2 ). It has been shown that a sequence with a higher growth rate than  cannot tile the plane.  ... 
arXiv:2110.00839v1 fatcat:oyopcu7zmjbzpkljkkvua7rh2a

About Fibonacci trees. I [article]

Maurice Margenstern
2019 arXiv   pre-print
In this first paper, we look at the following question: are the properties of the Fibonacci tree still true if we consider a finitely generated tree by the same rules but rooted at a black node?  ...  The direct answer is no, but new properties arise, a bit more complex than in the case of a tree rooted at a white node, but still of interest.  ...  In the case of the pentagrid, such a sector is a quarter of the plane: it is delimited by two perpendicular half-lines stemming from the same vertex V of a tile τ and passing through the other ends of  ... 
arXiv:1904.12135v1 fatcat:3fkisr3r5zbtxmody3b4o6hy2m

Tiling a Rectangular Area Using a Set of Unique Squares

Bahram Sadeghi Bigham Jalal Khairabadi
2013 Zenodo  
A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of side length n for every n in the set. From [2] we know that N itself tiles the plane.  ...  From that and [3] we know that the set of even numbers tiles the plane while the set of odd numbers does not. According to [1] it is possible to tile the plane using only an odd square.  ...  Abstract: -A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of side length n for every n in the set.  ... 
doi:10.5281/zenodo.3783840 fatcat:v3fkjsimsngu5pzg3qfwpr4khu

A note on groups of a family of hyperbolic tessellations [article]

Anthony Gasperin, Maurice Margenstern
2014 arXiv   pre-print
In this paper we study the word problem of groups corresponding to tessellations of the hyperbolic plane.  ...  In particular using the Fibonacci technology developed by the second author we show that groups corresponding to the pentagrid or the heptagrid are not automatic.  ...  The proof given in [11] constructs a bijection between the tiling of the southwestern quarter of the hyperbolic plane, say Q, and a special infinite tree: the Fibonacci tree.  ... 
arXiv:1402.4337v1 fatcat:kjwuretx35dc7cufuermoqayme

The periodic domino problem is undecidable in the hyperbolic plane [article]

Maurice Margenstern
2007 arXiv   pre-print
In this paper, we consider the periodic tiling problem which was proved undecidable in the Euclidean plane by Yu. Gurevich and I. Koriakov in 1972.  ...  Here, we prove that the same problem for the hyperbolic plane is also undecidable.  ...  a set of tiles spanned by a Fibonacci tree.  ... 
arXiv:cs/0703153v1 fatcat:l6gwmp4to5dvvf2unuzzglzmqu

An application of Grossone to the study of a family of tilings of the hyperbolic plane

Maurice Margenstern
2012 Applied Mathematics and Computation  
In this paper, we look at the improvement of our knowledge on a family of tilings of the hyperbolic plane which is brought in by the use of Sergeyev's numeral system based on grossone.  ...  It appears that the information we can get by using this new numeral system depends on the way we look at the tilings.  ...  Theorem 1 (Poincaré) − A triangle of the hyperbolic plane whose angles are of the form 2π p , 2π q and 2π r , where p, q and r are positive integers, generates a tiling of the hyperbolic plane by tessellation  ... 
doi:10.1016/j.amc.2011.04.014 fatcat:xej2dptqxnb7vahtsloewjgjjy

WHAT IS... a Rauzy Fractal?

Pierre Arnoux, Edmund Harriss
2014 Notices of the American Mathematical Society  
Elementary algebra shows that the lengths of the words a, ab, aba, abaab . . . are the Fibonacci numbers, the ratio of the frequencies of a and b tends to the golden number φ = 1+ √ 5 2 , and each word  ...  We can understand this word better by giving it a geometric representation as a broken line in the plane. Every a (resp. b) gives a step to the right (resp. a step up).  ...  Secondly, these three pieces admit both a periodic tiling and a nonperiodic self-similar tiling of the contracting plane.  ... 
doi:10.1090/noti1144 fatcat:4i7dn2vetre6raajhprlemywbe
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