Preface
This boo k i s a guid e t o discoverin g mathematics .
Every mathematic s textboo k i s fille d wit h result s an d technique s whic h
once were unknown. Th e result s wer e discovered b y mathematicians wh o exper -
iment ed, conjectured , discusse d thei r wor k wit h others , an d the n experimente d
some more. Man y promisin g idea s turned ou t t o b e dead-ends, an d lot s o f har d
work resulte d i n littl e output . Ofte n th e first progres s was the understandin g o f
some special cases. Continue d wor k led to greater understanding , an d sometime s
a comple x picture bega n to be seen as simple and familiar . B y the time the wor k
reaches a textbook , i t bear s n o resemblanc e t o it s earl y form , an d th e detail s of
its birt h an d adolescenc e hav e been lost . Th e precis e and methodica l expositio n
of a typica l textboo k ofte n lead s peopl e t o mistakenl y thin k tha t mathematic s
is a dry , rigid , an d unchangin g subject .
The mos t excitin g par t o f mathematics i s the proces s o f inventio n an d dis -
covery. Th e ai m o f this boo k i s to introduc é tha t proces s t o you . B y mean s o f a
wide variety o f tasks, this boo k will lead yo u to discover some real mathematics .
There ar e no formula s t o memorize. Ther e ar e no procedures t o follow. B y look -
ing a t examples , searchin g fo r pattern s i n thos e examples , an d the n searchin g
for th e reason s behin d thos e patterns , yo u wil l develo p you r ow n mathematica l
ideas. Th e boo k i s only a guide; its job i s to star t yo u in the righ t direction , an d
to brin g yo u bac k i f you stra y to o far . Th e discover y i s left t o you .
This boo k i s suitable fo r a one semester cours e at th e beginnin g undergrad -
uate level . Ther e ar e n o prerequisites . An y colleg e student intereste d i n discov -
ering th e beaut y o f mathematic s ca n enjo y a cours e taugh t fro m thi s book . A n
interested hig h schoo l studen t wil l find thi s boo k t o b e a pleasan t introductio n
to som e moder n area s o f mathematics .
I than k Dav e Baye r fo r showin g m e hi s metho d o f drawin g th e Cayle y
diagrams o f wallpatter n groups . Whil e preparin g thi s boo k I wa s fortunat e t o
have acces s t o excellen t note s take n b y Hui-Chu n Le e and b y Eli e Levine . I t i s
a pleasur e t o than k Benj i Fisher , Klau s Peters , Sand y Rhoades , Te d Stanford ,
John Sullivan , an d Gretche n Wrigh t fo r helpfu l comment s o n earlie r version s of
this book .
David W. Farmer
September, 1995
Vil
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