Algebra and Tiling: Homomorphisms in the Service of Geometry

222 Pages·1994·6.48 MB·English

by  Sherman Stein

Algebra and Tiling
Homomorphisms in the Service of Geometry

The Cams Mathematical Monographs
Number Twenty-five
Algebra and Tiling
Homomorphisms in the Service of Geometry
Sherman K. Stein
University of California, Davis
and
Sändor Szabo
University of Bahrain
Published and Distributed by
THE MATHEMATICAL ASSOCIATION OF AMERICA
© 1994 by
The Mathematical Association of America (Incorporated)
Library of Congress Catalog Card Number 2006925584
Hardcover (out of print) ISBN 978-0-88385-028-2
Paperback ISBN 978-0-88385-041-1
eISBN 978-1-61444-024-6
Printed in the United States of America
Current Printing (last digit):
10 9 8 7 6 5 4 3
THE
CARUS MATHEMATICAL MONOGRAPHS
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Preface
If n-dimensional space is tiled by a lattice of parallel unit cubes,
must some pair of them share a complete (η -1)-dimensional face?
Is it possible to tile a square with an odd number of triangles,
all of which have the same area?
Is it possible to tile a square with 30°-60°-90° triangles?
For positive integers k and η a (k, n)-semicross consists of
kn + l parallel η-dimensional unit cubes arranged as a corner cube
with η arms of length k glued on to η non-opposite faces of the
corner cube. (If η is 2, it resembles the letter L, and, if η is 3, a
tripod.) For which values of k and η does the (k, n)-semicross tile
space by translates?
The resolution of each of these questions quickly takes us away
from geometry and places us in the world of algebra.
The first one, which grew out of Minkowski's work on dio-
phantine approximation, ends up as a question about finite abelian
groups, which is settled with the aid of the group ring, characters of
abelian groups, factor groups, and cyclotomic fields.
Tiling by triangles of equal areas leads us to call on valuation
theory and Sperner's lemma, while tiling by similar triangles turns
out to involve isomorphisms of subfields of the complex numbers.
The semicross forces us to look at homomorphisms, cosets,
factor groups, number theory, and combinatorics.
Of course there is a long tradition of geometric questions re-
quiring algebra for their answers. The oldest go back to the Greeks:
vii
vüi ALGEBRA AND TILING
"Can we trisect every angle with straightedge and compass?" "Can
we construct a cube with twice the volume of a given cube?" "Can
we construct a square with the same area as that of a given disk?"
These were not resolved until we had the notion of the dimension
of a field extension and also knew that π is transcendental.
We consider only the algebra that has been used to solve tiling
and related problems. Even so, we do not cover all such problems.
For instance, we do not describe Conway's application of finitely
presented groups to tiling by copies of a given figure. See [19] in
the Bibliography on pp. 200- 201. Nor do we treat Barnes' use of al-
gebraic geometry [1]. Thurston [22] has written a nice exposition of
Conway's work, and providing the algebraic background for Barnes'
work would take too many pages. The group with generators a and b
and relations a2 = e = b3 plays a key role in obtaining the Banach-
Tarski paradox, which asserts that a pea can be divided into a finite
number of pieces that can be reassembled to form the sun. A clear
exposition of the argument was given by Meschkowski [16].
We had two types of readers in mind as we wrote, the under-
graduate or graduate student who has had at least a semester of
algebra, and the experienced mathematician. To make the exposi-
tion accessible to the beginner we have added a few appendices that
cover some special topics not usually found in a typical introduc-
tory algebra course and also included exercises to serve as a study
guide. For both the beginner and the expert we include questions
that have not yet been answered, which we call "Problems," to dis-
tinguish them from the exercises.
Now a word about the organization of this book and the order
in which the chapters may be read.
Chapter 1 describes the history leading up to Minkowski's con-
jecture on tiling by cubes. We give the solution of that problem in
the form of Redei's broad generalization of Hajos's original solu-
tion. Its proof, which is much longer than the proofs in the other
chapters, is delayed until Chapter 7. (However, the proof uses only
such basic notions as finite abelian groups, factor groups, homo-
morphisms from abelian groups into the complex numbers, finite
fields, and polynomials over those fields.)
Preface ix
The beginner might start with Chapter 1, go to de Bruijn's har-
monic bricks in Chapter 2, and then move on to Chapters 3 and 4,
which concern the semicross and its centrally symmetric compan-
ion, the cross. After that, Chapters 5 and 6, which concern tiling by
triangles, can be taken in either order. With the experience of study-
ing these chapters, the beginner would then be ready for the rest of
Chapter 2 and the proof of Redei's theorem. The advanced reader
may examine the chapters in any order, since they are essentially
independent.
We hope that instructors will draw on these chapters in their
algebra courses, in order to bring abstract algebra ideas down to
earth by applying them in geometric settings.
The little that we include in the seven chapters is only the tip
of the iceberg. The references at the end of each chapter and the
bibliography at the end of the book will enable the reader to pursue
the topics much further.
Several of these describe quite recent work. In [14] Laczkovich
and Szekeres obtain the following result. Let r be a positive real
number. Then a square can be tiled by rectangles whose width and
length are in the ratio r if and only if r is algebraic and the real
parts of its conjugates are positive. This was done in 1990. Indepen-
dently, Freiling and Rinne obtained the same result in 1994 by simi-
lar means. Kenyon [12] considers the question: Which polygons can
be tiled by a finite number of squares? Gale [8] obtains a short proof
using matrices of Dehn's theorem, which asserts that in a tiling of
a unit square by a finite number of squares all the squares have ra-
tional sides.
We wish to acknowledge the contributions of several people
to this book. Victor Klee, Miklos Laczkovich, Lajos Posa, Fred
Richman, and John Thomas, in response to our requests, described
the background of their work. Mark Chrisman and Dean
Hickerson graciously permitted us to include some of their unpub-
lished proofs. Aaron Abrams, while a senior at the University of
California at Davis, read two earlier versions of the manuscript and
made many suggestions that should help the book serve a broader