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A Fibonacci tiling of the plane

2002
*
Discrete Mathematics
*

We describe

doi:10.1016/s0012-365x(01)00239-4
fatcat:ser4lr7tizeyro6vnx4633mdv4
*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*. ...##
###
A molecular overlayer with the Fibonacci square grid structure

2018
*
Nature Communications
*

Here, we show an experimental realisation

doi:10.1038/s41467-018-05950-7
pmid:30143631
pmcid:PMC6109137
fatcat:vh6errsg6farherx4idw7usiby
*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. ...##
###
About the domino problem in the hyperbolic plane from an algorithmic point of view

2008
*
RAIRO - Theoretical Informatics and Applications
*

In this paper, we prove that

doi:10.1051/ita:2007045
fatcat:clxzsba47vfzjkepzngc6uhvom
*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. ...##
###
THE FINITE TILING PROBLEM IS UNDECIDABLE IN THE HYPERBOLIC PLANE

2008
*
International Journal of Foundations of Computer Science
*

In this paper, we consider

doi:10.1142/s0129054108006078
fatcat:td7cinhouvbzne6xbuequm73ku
*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. ...##
###
Fibonacci words, hyperbolic tilings and grossone

2015
*
Communications in nonlinear science & numerical simulation
*

In this paper, we study

doi:10.1016/j.cnsns.2014.07.032
fatcat:7yaplbxu4ra2lonmwxi6zva32a
*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. ...##
###
Surface structure ofi−Al68Pd23Mn9:An analysis based on theT*(2F)tiling decorated by Bergman polytopes

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. ...

##
###
Writing sequences on the plane

2002
*
IEEE Transactions on Information Theory
*

We look into three unusual ways to write

doi:10.1109/tit.2002.1003825
fatcat:p6esifk37ra5dkayzoknafykte
*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. ...##
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Penrose tiling approximants

1988
*
Journal de Physique
*

Abstract. 2014

doi:10.1051/jphys:01988004904058700
fatcat:kqfcnfisongclfppgkoq5x3hne
*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. ...##
###
Tiling Rectangles and the Plane using Squares of Integral Sides
[article]

2021
*
arXiv
*
pre-print

We study

arXiv:2110.00839v1
fatcat:oyopcu7zmjbzpkljkkvua7rh2a
*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*. ...##
###
About Fibonacci trees. I
[article]

2019
*
arXiv
*
pre-print

In this first paper, we look at

arXiv:1904.12135v1
fatcat:3fkisr3r5zbtxmody3b4o6hy2m
*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*...##
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Tiling a Rectangular Area Using a Set of Unique Squares

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. ...

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

2014
*
arXiv
*
pre-print

In this paper we study

arXiv:1402.4337v1
fatcat:kjwuretx35dc7cufuermoqayme
*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. ...##
###
The periodic domino problem is undecidable in the hyperbolic plane
[article]

2007
*
arXiv
*
pre-print

In this paper, we consider

arXiv:cs/0703153v1
fatcat:l6gwmp4to5dvvf2unuzzglzmqu
*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. ...##
###
An application of Grossone to the study of a family of tilings of the hyperbolic plane

2012
*
Applied Mathematics and Computation
*

In this paper, we look at

doi:10.1016/j.amc.2011.04.014
fatcat:xej2dptqxnb7vahtsloewjgjjy
*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 ...##
###
WHAT IS... a Rauzy Fractal?

2014
*
Notices of the American Mathematical Society
*

Elementary algebra shows that

doi:10.1090/noti1144
fatcat:4i7dn2vetre6raajhprlemywbe
*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*. ...
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