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Osmo Pekonen, Editor
Group Theory in the
Bedroom, and Other
Mathematical
Diversionsby Brian Hayes
NEW YORK: HILL AND WANG, 2008, 269 PP.
US $25.00, ISBN-10: 0-8090-5219-9, ISBN-13:
978-0-8090-5219-6
REVIEWED BY JEANINE DAEMS AND IONICA
SMEETS1
WWhen we first heard aboutGroup Theory in the Bed-
room, we became very
enthusiastic. We were reminded ofMathematics and Sex by Clio
Cres-swell. We enjoyed that book a fewyears ago, but it contained
too littlemathematics to our taste (and surpris-ingly little about
sex, for that matter).The title of Hayess book sounded
verypromising: More serious mathematicsin the bedroom! However,
this bed-room does not appear until the finalchapter of the book,
which deals withmattress flipping. And in that chapter,the author
soon turns to group theory
in the garage. We understand why theauthor chose this
eye-catching title,even though it does not really coverthe
contents.
In the early 1980s, Brian Hayes wasworking as an editor at
ScientificAmerican magazine. The magazinewanted to launch a new
monthly col-umn called Computer Recreations, andalthough he had no
real knowledge ofcomputers at the time, Hayes volun-teered to write
it. But he managed, andas he writes in the preface, I discov-
ered that the computer is not like theviolin: It doesnt take
inborn genius ora lifetime of practice to get sweet musicout of it.
Hayes wrote the column forjust a few months, but his newlydeveloped
interest stayed, and later hewrote some pieces for Computer
Lan-guageand The Sciences. Since 1993, he
has written another column for Amer-ican Scientist. The essays
in GroupTheory in the Bedroom are all reprints,with one appearing
in The Sciencesand the rest appearing in AmericanScientist. Some of
the essays are over10 years old and show their age. Hayeshas added
a section called After-thoughts to each essay, in which he
describes more recent developmentsand discusses some of the
reactions hegot from readers after first publication.
The topics of the essays are quitediverse, although all of them
have analgorithmic and/or mathematical fla-vor. Subjects include:
The astronomicalclock of Strasbourg Cathedral; ran-domness;
statistics of wars; the historyof gears and their relation to
comput-ing; the ternary system; and, obviously,group theory. Hayes
is no more amathematician than a computer scien-
tist, but his fascination for the subjectshows and he knows his
audiencewell. Im not a mathematician, but Ivebeen hanging around
with some ofthem long enough to know how thegame is played. Once
youve solved aproblem, the next step is to generalizeit beyond all
recognition.
Your two reviewers could not agreeon one favorite chapter.
Jeanine reallyenjoyed The Clock of Ages. This essaywas written at
the end of 1999, whenpeople were preparing themselves for
the upcoming New Year. As the worldspirals on toward 01-01-00,
survivalistsare hoarding cash, canned goods, andshotgun shells. Its
not the Rapture orthe Revolution they await, but a tech-nological
apocalypse. Y2K! Of course,the apocalypse did not happen onJanuary
1, 2000. But how could thecomputer programmers of the 1960sand
1970s fail to look beyond 1999?Hayes gives them the benefit of
thedoubt. No programmer back thenwould think his programs would
still
be in use by the year 1999. Theimportant questions of this
chapter arewhether there is any sense in buildingthings to last,
and to what extent?
A very remarkable example ofsomething that was definitely built
tolast for a very long time is the astro-nomical clock of
Strasbourg Cathedral
(see On Picturing the Past: Arithmeticand Geometry as Wings of
the Mindby Volker Remmert, this magazine,Vol. 31, No. 3). In its
present form, itwas started up in 1574. It is more anastronomical
and calendar computerthan a clock. It keeps track of a host
ofobjects and events: The positions of5,000 stars, the six inner
planets, the
current phase of the moon, siderealtime, local solar time, local
lunar time,mean solar time, the present year, theday of the year
(including February29th in leap years), and the movablefeasts of
the ecclesiastical calendaramong others. All this is done
withgears, with the clocks error being lessthan a second per
century. Schwilgue,the maker of the clock, was thinkinglong-term.
He included parts in theleap-year mechanism that engage onlyevery
400 years and that were first
tested in the year 2000. The clock canrepresent the years until
9999, butSchwilgue suggested that after thatthere just should be a
1 painted onthe left of the thousands digit. Theexistence of a
clock like this raises thequestion: Is the building of
multimil-lennial machines a good idea? Hayes isdoubtful.
Ionica found the story about theclocks rather dull, but she
lovedInventing the Genetic Code. Hayesdescribes how physicists and
mathe-
maticians in the early 1950s tried tobreak the code of DNA right
after biol-ogists discovered that the language ofthe double helix
consisted of just fourletters: A, T, G and C. The big questionwas
how these four letters coded 20different amino acids. Many
beautifulcoding schemes were invented, veryefficient codes with
nice symmetriesand error-correcting possibilities. Withthe
discovery of the actual encodingscheme, Hayes admits that he
wasrather disappointed by natures real
solution. He jokes that we might havebeen better off with the
genetic codedevised by one of the mathematicians:Life would be a
lot more reliable ifSolomon Golomb were in charge.
We both liked the final chapter ofthe book, the one that gave
the bookits title. It presents a nice way to
1Jeanine Daems and Ionica Smeetsoften referred to as The Math
Girls (wiskundemeisjesin Dutch)are two female Dutch Ph.D students
of mathematics who started,
in March 2006, a hugely successful website devoted to cultural
aspects of mathematics: http://www.wiskundemeisjes.nl. They have
also appeared as celebrities on TV
shows and in other media in The Netherlands.O. P.
2009 The Author(s). This article is published with open access
at Springerlink.com
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explain group theory. Group theoryhelps when the author cant
sleep atnight: Having run out of sheep theother night, I found
myself countingthe ways to flip a mattress. In the longintervals
between seasonal flips, healways forgets which way he flippedthe
mattress the last time, so he asksif there is a so-called golden
rule
for mattress flipping, i.e., a consistentrule telling you what
to do thatresults in the mattress cycling throughall its possible
configurations. Usingthe transformations one could performwhile
flipping a mattress into anotherproper position and the structure
ofthe mattresss symmetry group, heanswers the question.
Then the author turns to grouptheory in the garage, asking for a
sim-ilar golden rule for rotating the tires ofa car. The structures
of the Klein 4-
group and the cyclic 4-group areexplained in the process. Hayes
evendiscusses ways to flip a hypotheticalcubical mattress, daring
the readers todo the same with a 4- or even more-dimensional cube.
However, in theAfterthoughts, it becomes clear eventhis is not
enough generalization formathematicians. Readers have writtenhim
about polygon-shaped mattresses,circular mattresses and even a
Mobiusmattress.
It is not very clear what kind of
audience this book aims at; we think apotential reader should be
familiarwith algorithmic thinking and somemathematics. Hayes is not
wholly
consistent in the prior knowledge heassumes his readers have.
Most expla-nations of the mathematics and algo-rithms are very
clear, but some maybe too brief for nonmathematicians.Luckily, his
examples are very wellchosen and most of them are appeal-ing. For
example, To appreciate thevalue of randomness, imagine a world
without it. What would replace thereferees coin flip at the
start of afootball game?
Our main criticism is that 12 of theseessays is too much of the
same. Hayeslikes computer simulations a bit morethan we do, and
after a few chapterswe started groaning when he proudlypresented
another homemade dia-gram. At one point we were alsoslightly
annoyed with his lack ofunderstanding of the mathematics. Inthe
(highly enjoyable) chapter The
Easiest Hard Problem, Hayes countsthe number of perfect
partitions of aset of n integers. He nicely illustratesthis problem
with picking teams for aball game where you want the teams tobe as
equal as possible. In mathemat-ical terms: Given a set of integers,
youwant to divide it into two subsets thathave the same sum of
values. If thesum of all values in the original set isodd, then
this is impossible. In thiscase, a perfect partition is given by
twosubsets whose sums differ by exactly
one. Of course, Hayes could not resistdoing a bunch of computer
experi-ments, and he was surprised that setswhose sum is an odd
number have
about twice as many partitions, onaverage, as similar sets whose
sum isan even number. He emailed sometop-range mathematicians about
thisstrange phenomenon and they kindlyhelped him solve this
mystery.
This example is the exception,though. Hayes took a lot of effort
todelve into the subjects, and his view of
mathematics from the outside is refresh-ing. In many cases, it
is charming to seehis personal struggle with the material.
We recommend this book to peoplewho are already know a bit about
math-ematics and algorithms and who loveto read about problems and
questionsin the real world and its generaliza-tions. We think that,
for those readers,the book is extremely suitable forreading in the
bedroom. So, at least inthis sense, the title fits!
OPEN ACCESS
This article is distributed under the terms
of the Creative Commons Attribution
Noncommercial License which permits
any noncommercial use, distribution,
and reproduction in any medium, pro-
vided the original author(s) and source
are credited.
Mathematical Institute
University of Leiden
P.O. Box 9512, 2300 RA Leiden
The Netherlands
e-mail: [email protected]
e-mail: [email protected]
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