Symmetry: A Mathematical Exploration: Tapp

225 Pages·2012·13.139 MB·English

by  Tapp, Kristopher

Symmetry

Kristopher Tapp
Symmetry
A Mathematical Exploration
Kristopher Tapp
Department of Mathematics
Saint Joseph’s University
5600 City Ave.
Philadelphia, PA 19131
USA
[email protected]
ISBN 978-1-4614-0298-5 e-I SBN978-1-4614-02 99-2
DOI 10.1007/ 978-1-4614-0299-2
Springer New York Dordrecht Heidelberg London

LibraryofCongressControlNumber:2011940690
M athematics Subject Classification (2010): 00-01, 00A06, 00A09, 00A66, 00A99, 20-01, 51-01, 97G40,
97U20

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Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)
(cid:19)(cid:148)(cid:135)(cid:136)(cid:131)(cid:133)(cid:135)(cid:3)
Symmetry is a ubiquitous
concept in mathematics and
science. Certain shapes and
images seem more symmetric than
others, yet it is not immediately
obvious how to best measure and
understand an object’s symmetry.
In fact, the quest to more precisely
understand symmetry has been a
driving force in science and mathe-
Vitruvian Manby Leonardo Da Vinci
matics, and will form the central
theme of this book. You will learn the ways in which mathe-
maticians study the topic of symmetry.
If you are curious about the
mathematical patterns underlying the
symmetry that you observe in the phy-
sical world, then this book is for you.
Why are honeycombs hexagonal?
Why are bubbles spherical? Why did
the HIV virus evolve its icosahedral
shape? What is the shape of the
universe, and how might this shape be
An icosahedral HIV virus
related to the shape of a virus? How
can one understand the symmetry of molecules or crystal
formations? How might the symmetry in a painting enhance its
artistic appeal? Parts of these answers are found in other
disciplines – biology, chemistry, physics, and art – but the
common thread is mathematics. Mathematics provides the tools
to understand and classify the possible types of symmetry that
v
vi Preface
objects may possess, which is a crucial prerequisite for
addressing questions like those above.
No background beyond high school level algebra is
required to read this book. The mathematical topics are drawn
from diverse fields including graph theory, abstract algebra, linear
algebra and topology, all of which are essential to rigorously
study symmetry. Although some of these topics are advanced,
the presentation in this book is intended to be precise and
rigorous, yet accessible to a general audience. The only real pre-
requisite is that you discard any preconceived notions of what
math is and is not, and begin this mathematical journey with an
open mind and a willingness to begin actively doing what
mathematicians do: dis-
covering patterns, invent-
ing precise language for
discussing the mathe-
matical principles under-
lying those patterns,
forming conjectures, and
eventually proving beauti-
ful theorems.
Honeycomb photo by Ken Tapp
Intended Audience
This book is primarily intended as a textbook for a one-
semester math course for math or non-math majors, including
humanities majors, with the goal of encouraging effective ana-
lytical thinking and exposing students to elegant mathematical
ideas. It includes some of the topics which are commonly found
in sampler textbooks, such as Platonic solids, Euler’s formula,
irrational numbers, countable sets, permutations and a proof of
the Pythagorean Theorem. All of these topics serve a single
compelling goal: to understand the mathematical patterns under-
lying the symmetry that we observe in the physical world. I hope
Preface vii
that students from all majors will enjoy the many beautiful
mathematical topics herein, and will come to better appreciate
the powerful cumulative nature of mathematics as these topics
are woven together into a single story about symmetry.
Instructor resources, including PowerPoint lectures and
access to all images in the book, can be found at http://www.sju.edu/
~ktapp/Symmetry
Acknowledgments
I am delighted to thank Paul Klingsberg and Susan Ramee
for suggesting many improvements to the exposition. I am also
grateful to the following artists who generously permitted me
to include their images in my book: Vladimir Bulatov, Robert
Fathauer, Brian Sanderson, Paul Söderholm and Ken Tapp.

Taablle of Nootaatioonn
D = thee nnth dihheddrall grouup = tthee syymmeetryy grrouup of a rreggulaar
nn
n-ggonn. Itt haas 22n tottal mmeembberrs (nn rootaations and nn flips)).

C = thee nnth cycclicc grouup = tthee syymmmeetryy grouup of ann orriennteed
nn
reggulaar n-ggonn. It haas n tottal memmbeer (alll rootaations)). AA
shoorthhannd nottation forr thhese nn rootattionns iis {0, 1, 22, ……, n––1}.
P = thee nnth peermmutaatioon grooupp == tthe coolleection off all pperrmuu-
nn
tationns oof nn orrdeeredd thhinggs. It hhass n! tootal meemmbeers.

A = thee nnth alternaatinng grooupp == thee ccollecttionn oof alll eeveen
nn
permutaatioons of n oordereed tthinngss. Itt haas hhalf thhe ssizee oof PP .
n

Z = {…… –33, ––2, –1, 0, 1, 2,, 3, ……} == thhe sset of all inttegeerss.

Q = thee seet oof aall rratioonaal nnummbeers (frracttionns llikee 8//5 aandd –33/77).

R = thee seet oof aall rreal nuumbers ((rattionnal annd irrationnal nuumbberrs).

Rnn = nn-dimeenssionnal Euucliddeaan spacee. FForr exxammple, R22 is caalleed
thee pllane aandd R33 iss caalleed sspaace.

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