Tilings of the Plane: From Escher via Möbius to Penrose

284 Pages·2022·27.991 MB·English

by  Ehrhard Behrends

Ehrhard Behrends
Tilings of
the Plane
From Escher via Möbius to Penrose
Mathematics Study Resources
Band 2
Series Editors
Kolja Knauer, Departament de Matemàtiques Informàtic,
Universitat de Barcelona, Barcelona, Barcelona, Spain
Elijah Liflyand, Department of Mathematics, Bar-Ilan University,
Ramat-Gan, Israel
This series comprises direct translations of successful foreign language titles, espe-
cially from the German language.
Powered by advances in automated translation, these books draw on global teach-
ing excellence to provide students and lecturers with diverse materials for teaching
and study.
Ehrhard Behrends
Tilings of the Plane
From Escher via Möbius to Penrose
Ehrhard Behrends
Fachbereich Mathematik und Informatik
Freie Universität Berlin
Berlin, Germany
ISSN 2731-3824 ISSN 2731-3832 (electronic)
Mathematics Study Resources
ISBN 978-3-658-38809-6 ISBN 978-3-658-38810-2 (eBook)
https://doi.org/10.1007/978-3-658-38810-2
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien
Wiesbaden GmbH, part of Springer Nature 2022
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether
the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse
of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
transmission or information storage and retrieval, electronic adaptation, computer software, or by similar
or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, expressed or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
Responsible Editor: Iris Ruhmann
This Springer imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH,
part of Springer Nature.
The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany
Preface
The present book deals with three special aspects of the topic “tilings of the
plane”, which have an interesting mathematical background. By a tiling is meant
a seamless and non-overlapping coverage of the plane, where the individual build-
ing blocks arise from a “simple” law of formation - for example by the action of a
group of motions.
In the first part we consider motions of the plane, which preserve distances
and then study objects which are invariant under certain motions: This leads to the
concept of symmetry. As a very simple example for illustration one could consider
the letter “M”: If one reflects it at the median line, it merges into itself. Much more
interesting are of course examples from architecture (rotational and mirror sym-
metry) or art: The pictures of the Dutch graphic artist Maurits Cornelis Escher,
who was inspired by the patterns in the Alhambra in Granada, are well known and
then realized various aspects of symmetry in his graphics masterfully. We will
look “over the shoulder” of Escher and derive those mathematical results and con-
struction methods which he was able to find by artistic intuition, without ever hav-
ing had a mathematical education.

A rosette from the Museum of Applied Arts in Vienna
V
VI Preface
In the second part the topic of symmetry is interpreted from the perspective of
function theory. The natural “movements” of the complex number sphere are those
which are firstly holomorphic and secondly bijective. They are easy to describe,
they have the form z →(az+b)/(cz+d), where a,b,c,d are complex numbers
with ad−bc�=0. Today they are called Möbius transformations. We will classify
Möbius transformations and see how they and the groups they generate give rise to
interesting tessellations of the plane.

A visualization of a special Möbius transformation
The third part is finally dedicated to Penrose tessellations. Starting from two sim-
ple triangles, where the ratio of their sides is the golden section. If you have an
unlimited number of these triangles available and postulate some laying rules, it
turns out: You can tessellate the plane in principle in infinitely many different ways
with these triangles, but none of these tessellations is periodic, that is, it cannot be
brought into itself by a non-trivial translation. This solves the long-standing open
problem of whether, in the case of a tessellation with certain building blocks, there
must also be a periodic tessellation with these building blocks.

A Penrose tessellation (detail)
Preface VII
The book is very detailed and therefore not only suitable as a template for a lec-
ture or seminar, but also for self-study. Through numerous images, the mathemati-
cal facts are visualized, and the readers may be inspired to create a picture à la
Escher themselves, to create an attractive image using a group of Möbius transfor-
mations, or to try a Penrose tiling.
In the context of proseminars and seminars, I have repeatedly addressed the
various aspects of the topic “tilings”, and in the winter semester of 2017/2018
there was a lecture announced under the title of this book. In it I received many
suggestions from the participants, to whom I would like to express my sincere
thanks at this point.
Berlin, Germany Ehrhard Behrends
2018
Contents
1 Introduction ............................................... 1
Part I Looking over Escher’s Shoulder
2 Symmetries and Fundamental Domains ........................ 9
2.1 What is Symmetry? .................................... 9
2.2 What Movements are There? ............................. 11
2.3 Groups of Movements .................................. 16
2.4 Discontinuous Groups and Fundamental Domains ............ 21
3 The Discontinuous Symmetry Groups of the Plane ............... 25
3.1 How Many Different Groups of Movements are There? ........ 25
3.2 Finite Groups of Movements ............................. 31
3.3 The Subgroup of Translations ............................ 37
3.4 The 7 Frieze Groups .................................... 40
3.4.1 F1: Only Translations (nnnn) ...................... 43
3.4.2 F1: Only Reflections of Type 1 (jnnn). . . . . . . . . . . . . . . 44
1
3.4.3 F2: Only Reflections of Type 2 (njnn) ............... 45
1
3.4.4 F3: Proper Glide Reflections (nnnj) ................ 46
1
3.4.5 F2: Only Rotations (nnjn) ........................ 47
3.4.6 F1: Rotations, Type-1 and Type-2 Reflections (jjjn) .... 48
2
3.4.7 F2: Proper Glide Reflections, Type-2 Reflections, and
2
Rotations (njjj) ................................. 50
3.4.8 Summary ..................................... 51
3.4.9 Classification: A Test ............................ 51
3.4.10 Hints for Artists ................................ 53
3.5 The 17 Plane Crystal Groups ............................. 53
3.5.1 The Crystallographic Restriction ................... 54
3.5.2 Translations, Reflections: 4 Groups ................. 56
3.5.3 Translations, 2-Rotations, Reflections: 5 Groups ...... 68
3.5.4 Translations, 3-Rotations, (Glide) Reflections:
3 Groups ...................................... 76
IX
X Contents
3.5.5 Translations, 4-Rotations, Reflections: 3 Groups ...... 84
3.5.6 Translations, 6-Rotations, Reflections: 2 Groups ...... 88
3.5.7 Classification: A Test ............................ 91
4 The Heesch Constructions ................................... 101
4.1 Lattices and Nets ...................................... 102
4.2 The Heesch Construction: Motivation ...................... 109
4.3 The Heesch Constructions: 28 Methods .................... 112
References for Part I
Part II Möbius Transformations
5 Möbius Transformations .................................... 145
5.1 Complex Numbers: Some Reminders ...................... 145
5.2 Möbius Transformations: Definitions and First Results ........ 147
5.3 Möbius Transformations and Circles ....................... 150
5.4 Fixed Points of Möbius Transformations .................... 156
5.5 Conjugate Möbius Transformations ........................ 158
5.6 Characterization: Fixed Points in {0, } .................... 160
∞
5.7 Characterization: the General Case ........................ 167
5.8 Wish List/Visualization ................................. 173
6 Groups of Möbius Transformations ........................... 177
6.1 First Examples of Groups of Möbius Transformations ......... 177
6.2 Fundamental Domains and Discrete Groups ................. 179
6.3 Special Möbius Transformations .......................... 182
6.4 Digression: Hyperbolic Geometry ......................... 191
6.4.1 Hyperbolic Geometry I: The Upper Half-plane H ...... 192
6.4.2 Hyperbolic Geometry II: The Unit Circle U .......... 198
6.5 The Modular Group .................................... 199
6.6 Groups with Two Generators ............................. 204
6.7 Schottky Groups ....................................... 206
6.8 The Mystery of the Parabolic Commutator .................. 218
6.9 The Structure of Kleinian Groups ......................... 226
6.9.1 The Isometric Circles ............................ 226
6.9.2 The Limit Set .................................. 230
6.9.3 A Fundamental Domain .......................... 233
6.10 Parabolic Commutators: Construction ...................... 238
References for Part II