A Guide to Penrose Tilings

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Springer Nature, 19 d’ag. 2023 - 199 pàgines

This book provides an elementary introduction, complete with detailed proofs, to the celebrated tilings of the plane discovered by Sir Roger Penrose in the '70s. Quasi-periodic tilings of the plane, of which Penrose tilings are the most famous example, started as recreational mathematics and soon attracted the interest of scientists for their possible application in the description of quasi-crystals. The purpose of this survey, illustrated with more than 200 figures, is to introduce the curious reader to this beautiful topic and be a reference for some proofs that are not easy to find in the literature. The volume covers many aspects of Penrose tilings, including the study, from the point of view of Connes' Noncommutative Geometry, of the space parameterizing these tilings.


 

Pàgines seleccionades

Continguts

1 Introduction
1
2 Tilings and Puzzles
6
3 Robinson Triangles
33
Chapter4PenroseTilings
85
InthepreviouspictureitisalsoillustratedhowtopassfromPenroserhombitotheprotoset
87
Fig41Thinrhombus
88
Fig43Penrosesnowflake
89
41AperiodicityClassificationLocalPropertiesSymmetries
90
47
104
bwwbwbwwwbwwbwbwwb
105
Definition41Adoublyinfinitesequenceof
106
Givenaprotosetoftwoclosedsegmentswhoselengthratiois
107
Thevertexneighborhoodsofthevertices
111
Ineverycaseweseethatthepairoftiles
112
ProofLet
114
v1v2v1v2
116

whenweremovetheedgeswithasinglearrowOneimmediatelyseesthatthewheelbecomestheSunthestarremainsaStarthediamondiscontainedinanAceLemma
91
HoweverwhenpassingfromRobinsontrianglestorhombisomeverticesmaydisappearwehavetoapplyComposition1andthenremovethebasesofallthetriangle...
92
47
94
35
95
intooneortwotrianglesintheprotoset
96
43ConwayWorms
98
Fig48Cartwheelwithrhombi
101
whereeveryblackkeyisalwaysfollowedbyawhiteoneandthereareatmosttwoconsecutivewhitekeys
102
v5
117
5 De Bruijns Pentagrids
121
6 The Noncommutative Space of Penrose Tilings
156
Appendix A Some Useful Formulas
191
A2 Fibonacci Numbers
193
A3 Cut and Practice with Robinson Triangles
195
Appendix References
196
Copyright

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A Guide to Penrose Tilings
Francesco D'Andrea
Previsualització no disponible - 2023
A Guide to Penrose Tilings
Francesco D'Andrea
Previsualització no disponible - 2023

Frases i termes més freqüents

5-fold symmetry algebra angle apply Composition big kite bijection binary sequences bottom vertex Bratteli diagram C*-algebra called Cantor set Cartwheel compact compute contained convex convex polygon Conway worm Decomposition define denote direct isometry disk double arrow edge-to-edge tiling equivalence classes equivalence relation example Fibonacci Fibonacci tiling finite follows golden gnomon grid Hausdorff Hausdorff distance homeomorphism index sequence induction infinite integer intersection kites and darts lattice Lemma matching rule matrix monohedral tilings musical sequence non-empty non-periodic P₁ partial tiling patch Penrose rhombi Penrose tilings pentagons picture plane with protoset polygon Proof Let Proposition prototiles prove rhombus tiling ribbon Robinson tiling Robinson triangles rotation Sect segments single arrow subset subspace topology tail equivalence Theorem tiling by kites tiling by Penrose tiling obtained tiling with protoset topological space topology transforms translation triangle attached vertex neighborhood vertices Wang tiles whole plane

Sobre l'autor (2023)

Francesco D'Andrea has a master in Theoretical Physics (Univ. Sapienza of Rome) and a Ph.D. in Mathematics (SISSA, Trieste). He is currently an associate professor in Geometry at the University of Naples Federico II. In the past, he has been a junior research fellow at the Erwin Schroedinger Institute of Vienna, a postdoctoral researcher at the Catholic University of Louvain-La-Neuve, Belgium, a visiting professor at IMPAN, Warsaw (Simons Professorship), and at Penn State University, USA (Shapiro Visitor Program).

His main interests are in Connes' noncommutative geometry, C*-algebras, and differential geometry.


Informació bibliogràfica

TítolA Guide to Penrose Tilings
AutorFrancesco D'Andrea
EditorSpringer Nature, 2023
ISBN3031284283, 9783031284281
Nre. de pàgines199 pàgines
  
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