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,dF .''B- it,, as' ~ ~ ~ ~ ?~ ~ ~ ~ ~ ~ ~ ~ : pheno~, - m the quanmm v~s :~f ammi-c paretic to the umw~e :~I: is ~ man~ion of cha~. Developing organisms change as they ~w. Pop~h~io;~s of i~g :~ - = I {,0 - ~~,0ij; ~3, - ~ [~= 6,~,7 t~ ~> 0( from ~ to ~~. 0O 61 piGii~, 60~7 8~6 `;~10 ;~76 50~= tO 00~q. =.~:~^ =d o~n bandit, ~.~. ~ ~ ~~ =e s~. the cv~e of the seamns, the e~ and 6~w of the tides. ~l~:rs seem ~~re complicated economic recess:~' ou =~' the weather. M) ~~ of ch~s Unguent our Ions :~t is of the go imw~e that we sh - d u~d and :~! t5e chan:~-~ ~~d :n whim we I~e ~ do th~s Docky we mum - ~~e se.~.~e to ~e pat - ~s of ~, incI~ding the discove - ~~ of :hidden =~s in - ~~s that at h=t sight appear pa~. To do this we need to. R~=t changes in ~ =~-~:nsiue Ems, d the fun~:~:~! types of c:~, R~= padicular tVpes of ch=~s when they own, Spay th~ew t=~hniques to the outside world, and ~ ~~! ~ changing u:nive~e ~ our best a~:~- The most emotive medium ~r pedo~ing these tasks is mathematics. With ma~atics ~ build maim unifies and take them apart to see how It tick., ~t Piggy twit )~t Ii Amp: 8~] ~6 deceive andr de\~-p ~~l pnnciples. Sliat3hematics ls t:hg: u.~tim=e ~3

OCR for page 183
184 4~:~Y in "tech;~ol~ tmns:~- p~s pewei~d in ~ single =~mple ~ -~e Pi aCmss the entire sw~m -of sclen~e an~d Ad ~E ~[:E~CS OF CI=E The Id ~~h ~ ~e m~m of cha~ c~ be ~~d up in ::: ~~* m^~ In :~: ~e ch~ mtem Is mo eled all: ~ ~~al equatI~ (~' ~ ~k equation) ~at de~ ~~s the ~a.tion be~e m - ~ ~ cha~ of ~~t w~. As much :~ maxim~ (~h t:h=mi=! and =~) as ~ Deere<] is bro~t ~ - r in, ~ - d to m~ the =u,~- preparing S~ for the sly of =~:~s teas - n ~e central ~~ of ~~l my -me, setting u~p a~ so-~g ~e eQu~.~.s Of calC~s is the li~d -of _. e. A" A: _ _. A'. ... .~ _ _ ' ~ . . ~ ~~s ~~s ~ Phi component of the m~:~S of ch~- Newe: methods Ale as Dante :~cs a~ =~at:on enhance ~~:~r than Irene c~- But m.~ti~ :s itwI-f s^~t to chan~. New p-~ms and new d:iscovenes ~~ the need ~r ~ much -more -~d ran~ -of ment~ eq~:~. Two- :~t t:~s am ~1: -ment:-~:~ng. the use of inc~fs:~y s~hist~1 ale ~0itstion otgt0~=6 =~-et=: ~ ~ ~ b~ b~ ma~ possible by the e=~= income in =~r power. ~ w =~:~-~ti32g Is band on d~31 manipulation, it requ~s a;n ~~ Phi of the -dis-~e as wed aS the continuou~a~ above aL' -it: the I- between the two. The -I twnd is ~ ~j:~r t=~-~h of mathemat:~al imp the use of v!sual ~mW ~ mn~nw ~ ~= quantize of iron ;~o ~ sire mmpreh=~e Dip K ~~r Chip h~ Ed to the tt ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ small n~r of ~~l ~~-~:~c forms. M~ematimans aw ~~t burning ~ unde~d ~~= basic Its blocks of change and to a.~e how theV c~. He melt- ire - 1 has ~ vew c:~t s;~:int tmm t=~:~:~:~:nal -modeling with -di~e;~ti~ ~~:~" it ls mOm like chemistry ~~n -. r~uinng car-~! counterpoint between analysis and Id The gm:~:hical wpresenmlion of Yanker m:~eimatIC~ cons - s ansilng in the smd~ of change haS )~d to the discover of ~ va.~y -of intn-~e shyest -each of which appears in mar I: (Ivnami~] si:~=io-ns send is thus ~ `~-~nive~= o~t :~n the mathematics of 0~hi~ii0~33s 4 hi, ure ~ po~s ~ -~:~r of these I baby :iliustrate well the vast did b-~n t-~s visual methods and the ~~s t~ittonali~

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:~ i, .: . A: : :~ ~ : I::: I : : : :: ~ : ::: ~:~ :: :: :: i:::: :~ i: :::::: ::: : : :: :: : If'-' :::: ::::::: .~ a: A: ^~ o, A. ENCORE ~ ~ N~ Ad. - i~ that ~~ of -~: '~:~s, 0~7 -~6 u~,\~n ~ w111 -~d scle:~ti~s who wach -as readIl~ ~ ~ ~r ~ cOmputer 1;~*min~'' who can d~ c=~e b~`t ]~t~N~yre ~`etches as :~::~y as ~ com:~:~:r =~' and who ~~ in pictures as :~v as in numbers -of formulas. The culture :~t of vowthe m=~1 too! : t;50 ~~:~:~nS sc:~-nti:st w:ll be vew d-~ :~= ~~ ~ bras enrage The pattems <:~f chan~ :-n natu~ a~d :~n mathematics are unco sm:~6 5v 00nNy~:~) (~5 0: tt0~:~- in 0~: to make p~s we mum :~d i:~:tiwN and sensi~;~:~-elv to new types Of pat.~w, Our In, patte=,5 of 1;~o,u,~t =,~t, (~s cha~

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As the twent:~ centu~: ~~ to ~ closet ~ ~~ ~~t ~ = iCs is I ~~leose Id :s vanish. Mathem~i~ ~ ~ ce ~m ~ d - cooping in d.0= my. unctIon OWE: ~~ t~ it, ~~t,00s to sciencep~, I b~ =d s~ ; Mu~ her ~ Is in= - by =~r Or ~~:~ - eX'nmen~ :~r by ~ ~~s Of n=~] phenomena~ ~~- He ~~as ~~r Weir A ~ 6~ 3~ 0t ~~~ 8= edit ~ ~ ~ pm to wO~ tc>25 This varier ls ~ ~~= - :~ of: I =d ~ she ~ Foraged ~ ~! ~~. More~ over, =~n Brim =~r ~~cs) ~w I. :~mm school ~~en ~ :~e=,m school tracheas to scientists to Press ~ booty and m~plex~ of ma~s an-d to p~ ~t to {em I? The eme=~ce of ~~s :~ styIc of mathem~ ~es not imply that the Audi em.~s ~ precise Ice of cO~.~- ~ n. orous I~ proof ~ ~ ~ Ids ~~n t~ cont~' Hey r~;n an essential com~nt of the m~l endeavor. Rigor =d we~ cistern are as A to- mathematics as - ~~;~t is to ~e rest of acid =d for much the s~e wamn I pI~~ 5~ Hi: ~t Nbe;li~ing that ideas 3nd =~6S 34~ SO~Od Th0\r 3:O paw of the sub j0015$ lDt0~ C~ =6 ~~, ~ CO~t ~6 ~~ C~. The tm::ni:~ of Iffy mathematicians mi} necessan:~:v cont:~e to :~ui:re accurate Iog}al thinking and ~ p~w undemanding of the =;~g 0; 449~- We u~ of compute~ as "~l t~1~- in my c~ ~~e and mot~*~e :~- ideas and problems' but these expenments alone =~ot provide u:~*nding of urn :he ~~ se~d phenomena happen. Their role is to o~r ~ de~ of co.~.e thy ce~n phenomena do indeed o=~. In fed ~ i:: twnd has become ve~+ *A as A:= gin t~ u~ of computers has develo'd. I~t ~s the did of ~e I'm: atntude, "Putt it on the computer Id that mI) answer all :~r qWsTlons '~ w~n :~e =~r to ~ pr<~m Is, - ' ~ sI:~:~e nu - ~~, such as He failure Io~ of an en~g ~~, aii of one's pr~s indeed do died once that number is ~~. But IVY ~ ices I- ~:~i~tio;n my :~e several :~d diatoms rep~:ng the behavior of the system u~er venous conditions. W: ex~ ~~~ think of the :~w of a:r past ~ spa~ ~:~:~e for Fit speeds, an~es Of attack and atmosphenc den~:~. :~ch ~ cataloguc^ despite its ap~y :. sIZe' lS likely to be inadequate ~r determining the behavior under ad possible conditions T: the system inwIsres three adjustable pa=~:~, as does the one just :~' and each can take

OCR for page 183
up: to ted :~, ~= ~ If: Of ~ ~~ :~s i:~: pos~- ~ "^ m) t~- Tn: ~~' ~x ls ~ ~~ n~r Of p~. s~ :~s I:. - e~ ~~Y 1~ ~~ ~~ : m~ ~~ ~ ~~ ~ -s om~S ~ ~ ~e : ed e of One Millie bait I~ ~ne ~ bill ~ ~ ~,6~ que~.~ is ~~ ~~g on ~9 ~~ - m co~r sand:= to Id. xa~ of m.~.~.~* ~~ : - mput I'm the~ hum~ brain ~r ~= ~an ~ ~e c~ Ho~r the m~ Of ~e =~r ~~d ~m ~ un~.~.. Tt :s :. .~.n ~ ~~ p~alem ~~ aide :~..S =~t On e Ems- ~ th.~ c~ ~~ ~ =ed to =~em ~ ime~- diate sums of un~ to te~ hi- =d poss~e Dachas : Wt~ ~pr~iate sa~, =~wr =1~13;~ns can ~-~611v pmdu.,= ~~s pry. of m~ reSuhS ~~ =.~d p~mare ve~ ~M con~on and ~ `~ de~ ~ human t~t to se: them* up. Baby are fir mom -: aced usu~v w~:~:~e : ~str~ Amway and Thy machine um* More ~an a~;~:ng :* they Con~e -: speciall~ area of :~tics~ "~t it of* ~e =~- is no panacea" ~r r~ns of =~:~on on6-' r:~us proof does not ~:~re prom:~ nenti: :n ~~s *I ~t :s p~ ot tte mat60~75 58~: t00~i~= and it em*: :~ as :~t as it ever ~37 5~t }t 60:~5 much :~s intere~ ~r the nonspecific. AccOrdin~v, lts role has no! been made - ~~.~-it~* ~~ - lt unchains- ew:~.i~g d1~. :- the fa~ that pmof :~s imp ~r the pm~ ma*~- uc1sn d=s not imply ~at the teach1~ of ma~atics to ~ ~~n wd10~e must be limits to ldeaS whose pmo~ are ac=~e to that audience. Such ~ limitation is Ilkely to ma~ mathems110s ~11' dW, an6 drea~ It ma~ of the mo St stim~u fat} ~ and =~ting ideas dep ~~ biggie co-~:~x theories Or the~r proo~. M~nv ma*~atical concepts can be gmsped we: be:~g exposed ~ their Army Amp Using ~ :~ea :s gu:te died mom dwelop-~ng :~- Tt :~s po-~le to "~- qu:te a~d =~s to oh: by :~s of exa*~s a~d experi~ - ~n when ~ ~~:! puff :s too A::* For exa~e, in the thec~ of chacs an tmpo~*~*t concept of "~y to :~l cond:~;~.$' I: ~ sv~,~m ~~s mom t~ similar ,*:nit~! ~~:~, He resulting motions can quickIv become t0~\ It I 800055 t~ city Id ~ id 3~.~0 :~0 ., .,

OCR for page 183
arm Ap^~= ~ I: Be t~s sens: - e and parad=~ ~wior in' ~, ~e Wrenz air (~ Ib) mem~ ~ Hi h~ twO a1~ equ~ A =~8 =~8 ~~ 8~ 5000~0 Of However-. ~ ngomus pmof ball ~e ~-~ Aim- ~~- -- ~.~ :n ~e ~~r th~ =~.~r expen~ S~ 15 :~m o~y ~~d the~ =~.~..~f ~ ~~ per- wn, ~t has- n~ ~ beenhem ~ pi mans and = an acts pmbl~ ~r Fad re~. The ~ ~ V16~t and mn~ Of ~s ~~ ~ m - ~s m~cs win be unposed not ]~un ~r ~~:~ns ~ ~~ tots bm ~r ~0 tn i. Aid. of ~~. ~~ a~ us ~~. M=~ pounce ~~ss ~$, a~d <~er decision makers mu~ =pe oh world. 1~ must appremaw~ :how ~~e ~~ ~~' -~: -ace Flea out~ a~. ~~ ~s ~ t~s ~~ ~ ~~= mashes of ED ~ ~~- tton of w~ ve~e ~~.r mm :~ Is to ~ tn chndren ~e :~f ~e ~~ I~ =d ~ ~~ ~ ~ polnt of <# We ma ~~e b~ the tm~tional ~~h of an~e I=~ to ~~bm =d ~~ce to ~~. In the desk of an effects new curn~um7 one impotent comma ent ts an -~ of the n.~w WE that -3,[C deV~.~ ~ the f~ of re~gsa`wh Yet lhe 0~=A=~= must be while ~r i} child=~, no ~~t for ~~e who mI] become restitch sci~is~. N:~ e~' n~ k::~s of mathematics tat are evoking at the :~h teve! ~ the s~le for appl:~ns =d reduction :n the ~~. Thus it :s important for teaches and edu=~ at ~] I - ~~s to und~d the Gil nature of thew new :~s and the k~s ~ que~s the: ~s :~ ~~ The mathematics of change can be viewed at many I - A* The b:g pi~- What are the possible twos of chow :~cifie areas of m~ti~ technique* H~w are the equations ~? Genera areas of application K ~~ ~~$ tt~ bite 0: 8~ 30.~8 ~~tion Vaw Aim t!~9 Individual applications. Hi ~ chem:~1 rea~r to produce Bite tt0070ti:~ OK ~~ 60~ ~ p0~ oscillate? Mathematicians operate Ace ~1 of these Ievels b=~e lnsi~s obtained 8t 0~0 i0~l 870 0~ tt8~6 t~ 0~: i~5K {~ mathematical -~ tmns~' pattem.s are no! tied to any pa~r area of ap~ A 3't]~K

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:~ Eli ~~ t5~ ~~ ~= It 0[ I, Ire 4;0 i~ t~ a~i=~-~. ~ 0~07 8~ 3~8i~ 0{ I. Id. ~ 2.5 of no direct use in the Body -of mng h\~;tier in w:pe~nic =~- ~In :p~1 te~s t~ pendulum went o~ -who ~ grand- dock. simple exa-m~ have the:r uses. - v pmpare us ~r He =~xit:= of ~ I:~* A pendulum ~~ :~ :~mponant ;~es of ~~on ~ acce~e then ~~d ~ ~~:~ic mo~! of ~ vIb=~ al~e ~ Il~e ~~e ~~s we w~ use some Sofia ~~s UPS' the DOW - e Of mathemat1~* ~:~e q~s b~ been (kt0~: 0~: ~ ~~ ~a].8 t~ t56=Se~eS7 I bade they marine Moselle ma, t~. ~ How~ do ~~\r'ing ~~s ~ den ~ ~~s -~.~.e - ~.9 Why t~ ~9 e fm of :~= qum:~s appeam to ~:~1~e chid The Phi ~i: ~3 ~ ~ ~~ ~ K ~ n~~ mndom. ~ t=r is st~ - , ~ icopard :s nm, a~ -abler the twin shall ~- In ~~ the qu=~s are aL ab~ ch~ of so~ ki.~* ~ ~.co.~s reallv p-~.~ into the emb's at=~m "~t :~ dom,- or do~ so-meth:~ ~re s~:lie behind -ad appe~ce ~ the Bind my? ~ made -bird u~r does -~m ;~t ex~ as ~ -~c o~- :t d~s - m ~ -single Unpegs cow Somewhem a~g ~e line of de~~nt the ~~es 6~t malce their ~~. C=~ ~s the =~:~:~ t60~ 6~ 6~6 0[ t6~ I q mP~:N ~ - ~~s If we ~t ~ Rev bits on an uninhabited island, pmty Soon ~ere wild im more ~its. On the odor hands the Huh -~t =~ti:~-e Abed or s~n the~ wand be more - bits t:ha:n island. It f6:~s that shad in ~ -~:~on is a~ed Sir fbOth lOtO=~] 3nd CXtC= ~~:~" How they combine to -infl~:~e cha~s in the populat:~n is ~ good example of mathemat:~ m: that can be studied at many d.i~t I-~-~. Limits tO omw.~h WO - ~ city t6C It C85~. ~ population -~SiStiOg Ot ~ S:~0 species ~ ~~h ~ con~:~t (~d thewfom limited) Cod supplv. Figure 2 shows typical e:~:~al ~ta ~r math of such ~ populat:~:~- T:ts - ~~l S~shaped cu - e is chamctenst:c of many math phenomena ,^i

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190 .: a SURE 2. ~~ in 1bc sit ~ . . . . . `~ 01 8 ~ ~u~1lo~ ~~1^ I~ ~ an ~i~monl Limb ~ limited ~ apply. ^~.~\ Cy _~_ ~ ~~_~- ~~.~_~_~.~_~ Similar cuts adse if ~ measure truly Abut of a sit Ids opting _is~r Sample 1be bow or fit of ~ ~~ Wild. It is Emma in Cam Babies lo Scow He fiats or weals of china as ~~ Ha. ~~ char ~ be Baaed on wisdom gals ~r Prussia ad Mason. we ~~ of Bum cbid~n in ~ Sue as ~~r ~ =~ of one or lag gas ~11 illume ~~r ~~lb. ~c her- on Me cat toad Ink Omega ~11 lit cloak to ~ ~1 liner Ho~ the complete ~~ rod of a chUd Cam bit 10 ~ Id alibis lbe chic ~ so. fiber he inili~ abase fir We bad pram is lions. Early on We Bulb is app~im^ ^~n=~li~; lager i1 Buttes ~ it Babe ~ contra Blue. Cbi~ld~n So baa Scoped Lab cut ~ ~ inl~du~d. 10 He ED Shaped cuff eilber as an embedment Saxon or as a ladle of numbs. A got else fir guilds in midge school is -lo use evidence ham Beat cabins Flab cuts label ah dog ..... as older students, they On let boa lo ~p=~n1 abed cures ails Joules. Cbild~ can be encou 10 analyze 1be main Genres of Ibis cut and ~ consider ~ 1bc cut ties Thea. Sup~= Lice is 1 ~011~1~ ~ 0 and 4 ~1 ~I1~ ~ 8. lf gigs ale Olin- Gel c~ ~ ~rs-bo~ tall #11 she be al gas ~16 or 24 or S2? Kansas: 7 ^1, 10 Deli 13 ~1 ): Eon your Milan an see 1ba1 these answer a~ nor credible. Salads aunt Tb~e malbemadcs ~ Cne, bul lbe model-line g~th-~ inapp~phate. Foal: When you use malbematics you bra 10 pit ~ sensible Gael ~dn~juslcaIcu~te numbers blindly. ram Heir own bald 10 plea 1bei, ~ Dull beings. Pals of Anemias ^ study of ~pu~l~ion Dumb can ~ cawed on ~ sea lately 41> numerical, graphical: dynami~l-~ilh lbe so~phisti^1ion in= creasing as the children become older. ~1 description of 1be mast Mob cut sbo~s a population tab increases may al bra bul lben ex~ne~iaDy. Tba1 is, 1be breeding population increases by a

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~:~ Go: ~~r :~: wCc~e penodS uladon b=~s =~ Ia~ t:~ ~ of Swab ~~ d0~^ ~~ t~y IeN:ng o~ at ~ steady m:~m v~ue This Ve~ ~~: ~ wrely des~- lt :S muw- ~~t - v the wp - ~ Anal de~wn ts help~ ~r genem! : K~n ~t ~~ ~t ~~: a~ o: ~~ lt tmply ~e ~~= Of mown.. To ~;~te the et of exponential =~and to I: insight in.- ~ w~ ~~ c=~t cOmInue ~~hi.~n =n ~ to~ the fawn Ace- - ~t the Am: rots ~~M ~ ~ far count~ ~ pe== performed ~ iamb msk ~r the Terror, and sbe was Used to of a cam two on the ~0M, By. ~ur on the next, then el~' and ~ on ~~ .~h 6~ ~ ~e em~r ~s n~ w~e l:~ und: be worked out hen the num~= ~.~ ChIld=n can dO ~e =me mth ~ c^~or ~ ~ i. Wu:~r A a 60~8 Without ~ Gina ~ num~ ~ folding ~ ~~ of paper r=~te~v in halC How : tim0s can Vou manage bc~re ~u ~t Da~ on ~~- ~ an:~mals~ ~~= of b:~, Ache of tr=~ num beds of I=~es on pl=~s, ~~-, can be g^~ (or p~:~) in the ~~ of num:~:~! t~:~" Children ~ Ioo:k ~r pat~s 1n the nu:~- Are they- ::~9 Wc;~7 Cons:tat~ They =n c~te d:~es and In ma~ t~' and :~:k for - ~~. NO - ~~s ::~d :~31:~v to graph:~1 re,9~.~. The ~~:h cu: - e prm~:~es ~ aims picture of the wad fit which the two v=~' ~~:~n and timer a~ =:~. A-. ~ ~~, sometimes. :~6 ~ t:~ =:~, r~s numenca:! ink Fit ~~+ t is the Limit example of the geo-~n of ~~- ~e idea ~~t :~:~:~= =:n be ~~:~d bY the po:~S ot- polnts' and cha:~g numbe~ b~ cu~' :s the ba~s of all ~~c methods :n the maths mat:cs of change (=e Figure 3-3- hi:~en ~~ed ma:nv opponu~-s to- :~m that :n mathematics ~ p:~= is :~d I ~ thou$~d words. ~r wu:~r child.~.~n - ~~! wo~ is most a,997.~. Thev =n cou~t the number of e~ pmduced ~ ducks o~r ch:~s, measure the Beirut of ~ ~~ plan:, means= the temperature each - ' at noon, record the pos:~io~n of the moon in the s^~. sY ~~.~g t~s d=~` ch:~n can s-~=h ~r patterns of c-han~ am discuss possible causes* Odor children can ~ set mow ambitious tasks. the water l - ~l :n ~ p0~$ t350 ~~: 0t i:~5 0~ ~ bush' the :movements of the stock market' cxper::~ts Am ph>Ys:cs and chem:~oY iaboratones. Us:ng data ~ m :~e al p:h ~ ~ ~ me na ~ ~ ~ ~ ~ ~~: :~ ~ ~ Nt t~ ~ ~ t03~e m at ~ ~ ~ at ~ ~ ~ into other school! su~-~" Algebm students can ~~o u~ mathematical

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192 ^~ ~ ~ OX G=E^L ~~ :~ Dig OF ^ ........... ... ............. U~r AGILE 3. Imp Elba many divans IS of Ebony, Wilbur -~ Bar fit lit at. Spasms and Nebulas lo gene~le 1beo~lichl data, lo look far ~pal~=s, and 10 sampan 1beo~ mild ~11~. D~_~ Stems Ibe neat level of explosion is to Angel not Be pa1~1e~s in 1be numb beg but Be ~ 1ba1 Ages dab 10 these pallerns In 1be traditional

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193 Spew 1bis it lots ~ drench equals ad pus guiles Aim. Bu1 Moor ~s~ili~inc=~i~y a~=i~ in ~ he Of C~~lC~-iS lo ~~ ogre gains of Calculus and We see ~ lb~ he War of chums in a portion is Awe ^ ran continuous. lie ~~11mC~ ~ Uncut in.~e tar seeps, f = ~1, >~ I I1s nag rue )~' + 1) ~ ~= ~ aged 10 i1S currant rue b{71 by C1~C ~~ 1~. Ibis 1y~ of mom is cam a ~ {~1~ ~ln loins Captions, unchecked bandit al a =~S~1 ~C ~ ~^ rosponds1o a1~ ofibc fin ^ ~7 + 1~) ~ ditto exponential gnats: (I) ~ ~(0)~> )/bere~(O)istbeinib~ pppulal~ion. Abel of flied grow ~bidb a110~ forlimi~ imposed by kick offood or ~ ibestbis Ire by cycling s coercion fa~orlba1 redacts 1bescIimhs: Ace+ I) = Age _ ~[~]Z, Bra ~~ and ~ are contrails 1ba1 depend on tbe~p ~ icular circus stances. ~ is equation, known as lbe \tdi~a~S1 1~ Unsaid aider 1be nin~een~ is O$ es are aD subsequent valuer To Judy bob population compacts Bulb 1be cull level, ~~ can expr ass ~~) as a~p~opo~ion of ~/~ by cba~inglbe unils of m~easu~e~menlLyletling ~~) = g{~) i/. This leads10 tbc equalion ?~/ + 1) = ~~) - ?~: Oboe g<~) expresses lobe population as ~ ~ clion oflbc cuff level. I~nslead oft parameters ~~ and ~> ~~ nor ~b~veju~ one parader 1er ~, Limb mats the mathematics mum simpler. Bemuse g<1) is a Lion of 1he colon population, i1 mill be some Onion Careen 0 and 1. OiEbrence equ~ionssuch as1he2~bLul~ 1~ a~reide~ fb~compuler calculation, becauselhey express a simple nopeti1 ~ procedure for de- schbingibebobavioralth~ ne~lin~an17+ 1 Foam the bebaviorno~ al times. acompulerv~can easily o~culale ~~utionsof1be discreet \trbul~ 1~ v#ibou1 knowing a fo~nula forlhese soludon~

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^~ ~ ma ~~y asyaxisoflhe offing spb~;for~n ~npedt~ sy _ ~ ~ ~ ~~es~1 beg pf~ed~lbe~ pod~ionsbei~pg gifted ~ ~~ ^~ess~s~ tbespb~dcal dk~l~.~#esgenc~dfDnnof ~ bucked ~bowevpt ~i~betbe _ e gab Ha. lcibissepseCude'spdndpl~ ~~p4rb~s~idfor ~ i#~ii~^~c~ sy~n(~bichisnoc6~adk ~nped>~)bulnolfof~in~ideil~ed ~ den. B~uhcrtban ~~emplinglo ~su~nx~ Cuts ph~plesin this fashion: bong alit gums pads ~ is to underhand tic mecban~ism by ~4~cb ^ . . ~ peach Bed sym Marc sys _ s pro t~bevio: Dab gas ma ~.Slbisisoalled <~ In* lt~ms~tober~pon~b16 0~ 1ypesofpa~ornfonn~ion~inn~u~,snd~basave~ defined me ~ru~u~ tbat~can bound ~ u _ ~ ~ dsucb phoceses. ~~ case Mezzo In latest isle pa sag And mug oe~ cuney~n#~llnas ingest m~dc~1ems sad brave tic ~al~>bulhoftied 10 address ~eirslibEity.If asym mafia ~~ebecom ~ up 1~ 1ben1be system ~~ldosometbing ^ . . . ^ . ~ ~ _ . . . . . ^ . . ~-~n~u~ some~l~ge~1sec~nn~ he ~~n=~eidc. Cod d ~ 1besymn~ry~get~lo~? ~Veens~1b~ quesdonLyanex ample. Theca~opbe m~d(neinv~ntedfor ~ ~crdiR~ ~~ 1~1~ sbo~1halsym ma ~ is DO1 SO mu~bbrokenas~p~ada~ound. CbEd~n can m ^ oneand expedmenl~i~lbil. Tbeonli~ grope Raceme gas Seal Unmet abou11be center line. ltyou bean to ~~1cbibc fee elashc>1h~ Item obeys C~ude~s principles and Sl~yS ~symmeldc; lbet is> 1be disk dam got agate {~= 13~. But as you ~1~ lbe elastic ~~> to disk Openly 1~. Dog lbestate off Salem loses As ~Declion~ sym ma a. lbesym- me1~basb~0ke~and Cude'sp~ncipleshave Wiled. \Vb~e~ b~slbe~mi~i~n~g ~ ~ meld gong? H~oldtbeelas1~icsleady and roulc1be disklo tbosym methcally placed position on 1be underside (Figure lack. Ybu<~illind lb~ it ~ mainslhc~. Instead oft singe m mc1~cs~1e ~c gave 1~0 \~< ~~ ~ales. Tb~isisagene~lE~1~u~ ofsym meld bowing. Tbesy~em canexisl iD seVC=1 slalos~eacb bin Tom 1be Abbe by one oflbe sym- me1~ of ~c f~l~lsy~m. Fbrexample~tbe buckledspbedcalsbellin Iigu~ 11 b~akssym meld ham spbedcal 10 ci~=ular>and1becircula, ~nn~t~ occurs aboulsome pa~icularaxb, Heady vi~bleinlbe pig u=.Inibe~perR~=sy~em snyaxisispossible~bulallbuckIed gales bavelbeidenticalsbape~and1bc~y diner only fly mo1~ionsofibespbe~. Cbild~n can explore ~ ~ meld buckling glib simple experiments. They can compass plaslicru~lerlo End open ~ ~ a, . ^ ~~ ^ ., . bo~ilbends

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2~09 A ~1 ~ 1 sI a ~1 Do {!~I~ ~ ~ 1 1 ! ~ {forte!< HI / ~ ~ . _ IN ~ : - 1~ .~ \ ~ / ~ ~ ~~= BO^RO ~ MOLE EAT . ~ RUSHER 8~0 CROWED DISK THUS PIER SHEAR \ Amp 12. ~ -=l~~ptC Mobil- ~ ~ ~S1~ easily on of =~~ Id ads. flab ~ Eclair dig of ~i) ^~, of tedium 3 cenlimete~, 10 a ~~ unions ~ dig an a~ ~ ~~r Labor. Pa afar dams pin near 1be rim of 1be disk gab as dial upend. To ibis pin god go Epic bag, of Haul 6 Maim u~=l~ lab. Fix me ~ a dial ~12 ~ntimc1e~ gum the crater of ~c disk, and ~1~ Me ad of tbc #~r ~ lo ma along the center line as goon, fir example, gaping it 10 a mail ~~ you an ma by band. They In use a~spdng lo hold ~ ~ uphill, Cob 1be lamer end Basin: on a male, 1ben add whirls lo the 10p add walch it sway or buckle They can Mae a abode- Tom a Deducible meld Hap, pu1 whirls on lop, Ed ~atcb ail collapse. Older ~ud~nls can analyze lobe be~b~ior of 1~ dad rods joined by a spank bind. These models lead nalur~ly lo mom sublIe questions Claim symmetry s~il~ity, and ~n~tinuous cban~. How does a ro~lIing bay mom if its Enter of gamily chaffs? don ~ Hips capsize? The analysis of m~el$ of such champs bangs in a Deal deal of imposing

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210 HI MODEMS T~^ Is 3\ ~ ~ ~1 1~ ~ a: \ \' =~ 1 1# :f ~ i~ / / 1 ~ ~ . ~. L/ ~ ~ ~ Us ~ // ~ 1~# ~13~ ~ ~ ~ ?~} <<~1 \ to 1 ~~# T:^ ~ /< ~ ~~ I TENSOR nOURE 13. the abed gad is $~1c~ be _^1 antic ~ be pin (~) ~ un_~. ~ Me Mao em_ on gem side {b) ad All. ~1 nagger of 1~ ~ be same of be okay ~nigu~lion. I~n IS =~> ~ i~ ~~ ~~F exhumes in nexus, in ilily bag Ma. Comely, ~ example, flanks Ed normals lo ~ cuff ceased of ~- ity, Ad In He l~n~1ions. Or ~ mom boldly ex~p~le' consider 1be how of aver grout Bose Limb Titular crass scion. line 1be hose suspended venially, nozzle ^~, ~~ Plea O~g dewily Daub it. This Moslem is ci~ul~ly symmetric Pout an finis Knin axially Song the center of 1bc base. And Minded if the Dad of tag Waler is slag enough, the hose just Pains in 1bis Venice ~silion, Paining ilS circular symmetry. ~Ho~r, if the ~~ is lugged on ~~: the ~= ~11 ban 10 Edge. In Act them a= No distinct kinds of Cable. In one it Aims Cam side lo side lid a Cadmium. ~ - ^ 1u ~~ Per IT gas Fund and Funds sag Pier in a spit. Similar eggs ~ omen obeyed Ken guilds wash. the Wily c=. ~so ~1= do not Assess situp lar suet Foul a Amid axis: iodide Hey bag il in 1~0 dishing As. ~.~> ~1~ cloak ~ ices obvious out ^~ Emma pommels: symme1~ in lit. The owns slangy Dog looks exactly the Sue al a11 installs of lime. The oscillaling Dogs do not. We time mew IS no1101~Iylos1>bo~-r: bulb - polesaw ped~icandben~lo~ ex:~1ly~esame~benvio~dallim~esib~a~ ~bolenumbermulliples oflhepedod. Thissbo~sbo~ lbocondnuoustempo=1 ~ mew ofa Ready Baa backs lo ~~ lbe discrete symmetry of a periodic one. Symmel~ baaing is im~~anl in biology. Pen a s~edcally sym- meldc Q~ ~ ~lops, it splits ink two cells and 1be s~bedcaI sy~m- mel~ is broken. At a later Ace of de~lopmen1 {~Figure 14) ~ spbed- ~^~;~^s~ . am. ala ~~ ^ . ~ age an ~ ~

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9-11 ~ ~ URE ~ 4, Cr=~n ~ nd deSt=~n of symmctm ln ~ he ~1 opmen t; ~ ~ ~ ~~g ~ ~ By ~ 6~D ~ ~ =~ - ~ ten tr~ ~ ~ a~ A ~ then bll~ <'

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W~ I: ~ N~y ~ c:~= ~~ Bog ~h o~y ~~= ~~- id ~~n ~ ~~ ~67 i0~ t0 ~~= ~~ I' Me~:~1~, Me development of ~ c~ dent d~ gastmIa~ than lS ~~y a=~s ~ tbe Cut. -of ~ ~~ shell (:~w 11~. ~~s demon~es that :smme~ - ~36~na in -butte d:~= phv~ sea: ~ b~ Me same undedy~g my i ~ S=CtU~ ~ e upswing - e of m^~= ~ =~, ow of t~ =~ ~~ 8~6 imp ~57 is (art)' VIS~^ ~~s I=~ 6~v ~o our mm~ q:~. Who ~~s ~ tiger w:~ its ~^ I 87~7 5~ $~) Boys Boo pocm o~= no u=~:-~. Or: Tiger The ~~ - behind the threes serges $~s Ok to ~= ~~ more famous as one of the ~~r Fifes of m~m ~~. Tu::~g ~~w -Gil chem:~! c3~s prodwe ~e vari~i-~-n~s in =~ ~e chemical remissible ~r the stn=s need not ~ ~ actual Bong Or :t is =~0 Il~)r 1;O ~ ;8 ^CO~' ~~ed dunn^;g misgiven =~V Swains Of the tigers dove- which later tr:~e~ ~ senes of chem~! changes to create ~ Heresy Ho~ the ~61 deta:~-me of which r-~n mnt:*~malante -~m Ian he=. Our aim :is to -~e wale simple and ~~M m.~ti.~ :~nisms Or panem ~ Y ff ~ ~ tIo:n tn ~ :~r con:~t Tams ~~e day equations ~r this k:~d of chemise ~9 H[e I ..~ ~~ ~ ... . . ~ . . ~~ ~ .~ ~ ~ ~ ~ ~ ~ '. .. . . ~ ~ ~ ~ ~ soWen tnem numen=~ and then ma~ .* o~f ~e results us-cd to bunon~hole ~~ =-d shy them 0$ p:~:~. On mme there were ~~, -an miners :i=gular patches. "Don't these :~: :~t like the I 0~ ~~ I ~~ 35~7 in 50~6 0~. ~ls ~~i-~OS showed that ~~ms like stnpes or Spots =n ~ -ace ated b~ ~ mechanism of :~litv. Ima~e ~ Oat su - ~e (mathematics bier s~n) th~ =;~ns ~ uniform di~:~on of some chemical. This wounded ~n the cou~e of t:~e pmduce ~ bier of ski-:* -I ~~.~h MEWS 3]l 0~077 ~6 hit ~ =~H licks. 8ut t6~e ~i51~6Uti00 0t (~ icals :~d not remain -~nifo~- it can chan~. There are two important t~s of change. Clbemic~s at ~ ~~n place re=~, and the :~-~*~tion pmducts diffuse Am one place to I;--* These two types of change compete. Reaction tnes to alter the c:h-~mi~ cal m::~. diffusion tows to m~e it the same ev-~*~. The mathema*~ ics shows tha*t when d:i~*:t ::~fluences complete the result ~s oDen ~ compromise. Here the I: compro-m~w is that the uniform dime but ion of chum icals be~ ns ~ ~ ~~ Nell es ~ .inst~ility occults in only

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~ ~ ~ /~ / 91 ~ OO(RE 15 ~ fig cb~- ill. ~~n Ha i~ilily in one ~1~ ~ ~$; in (a}, ~in~ilily in Id di~on bang up ~ SO inn IS. IRE 16. Fit =- Dam fig ~ anemia Lion Waled ~ Infixing =)e$ of Ion {mica cb=~ tbc ~cmi=1 mix) and di~on (which ^~$ uni~i~ly). one direcl~n, 1ben 1be Males of can one gay and ~c me wipes If a wand inspirit SC15 in along a ~~dicular di~i~on, then 1bc S1~S lbem~se~s dope along 1~boi, lamb and bank up into IS (~g- u~ 1 a). Competing comic in~i~lilies ma ~1 be the ~n~men~1 d~i~=nce~onamalbem~io~lc~l~bel~eenli~andleopa~s. Chromic factions Awl can gentile pedodi~c ~palte~s-~ir~s) gang m~= ~ ~m-~ed ~ ~ ~1 ~e~ Tang ~~ urn 163. Me most Famous one is 1be Locally BbIous^=Zb~ol~inski1 Marion 21 Students can analyze these pates ~ End 1beir m~bemati~ ci1 ~=clu~ (~.g., Rabat ~~ of Spiral is it?. Obey can also use computer

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214 + HOL~RE 18. Coma models ~ Is on animal skins sb~ov Idling Cults. ^^ ala ~~ al long thin adds u by p~k~ lo ~^ ~lion-di~sion Mullions on dismal subs of =- ~ons and see gal kinds of alleys Occur (Hague 17~. alleys fled in 1~ ~m~lilion baleen Faction and division plaids examples of symbols breaking. Me initial unify d.is= ldbution off c~bemic~s teas gamer symmel~ 1ban do 1be slopes or SpO1S or spirals. Symmetry baaing is ~ ~~ common source of nalu~1

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2:1 ): :* ~ Am; el~ :s ~e :~g of s~ ~= ~ : :. : Compmer Id: of h~ * ~~s mutt Office ~~e trees Ace: price ~~le magi (~re 1 81~: g then base ~ I=s ~~e Clan ~~ ~ on~ =d pro - to b~ up Unto scow ~~s Ill liquid hd~ explam~he =~on eased that a spo~ tidal =n have a ~~: ~7 5~t ~ 5~6 axiom ~t :~e ~ ~~d ~- dad is ~ ape* that has ~ ~~= ~~ct on ~hair b~ l~ It a~S m - ~~ ~~ n~ =~.~' and - ~ ~~= ~ ~ 0~= 9~. ~~ll t000~V 0~t unde~:~ng of gang =me mos0y ~ ~e t - ~~ :~ of ~~= =d hs ~m ~~ r~ and \.~S c~d to ~ Amp. Hi - ~ - ~ =~e numerics ** =e wss:ible In~ti~' co:~ se~d to external the techn~s of c~^ ~ ding it ~~sib~ to m~e ~~= ~~t ~~- Me te~ "~-r cmach:~- captur~ ~e she But todays compu~ do mow tban. c~ adds In pan~r ~~v ~ mp~t =d ma~:~e ~m ~l,(~,1V As, ~ com,9.~ - d.~- wd~s m^~s is alsO about ~r mow t~n 3~st num~rs It deals tn ~~mM ~~es m~tidl.~.sion~ ~~' transductions, sha0' ~~sin shc~, pa Am* When ~~us was invented' ~ e~ed h=d in hand w:~ -~ ew ~r the centun~ ~~c wamntng ~s =~-d b)Y more powe~]but Iess ~n~:~:~tiveangelic Hi- The emphasis shifted ~ Nebulas" Lows as we pen~te areas where fo~laS atone 3~6 :~07 t~6 6~5i~ is 55~g t~ t~ ~~t to tte ~~d ~~} reamn:~ ohen asocial with the schoo! t:~:tmen:t of At 6Ut tO t50 ~~ 0 5~00 8~6 ~~=~ t~6 in of the visual. At 63~ 5~5 8= :~, 0~ 35 * phi=, t~ p~ ~.~0 t~0 It Acts t~ approach the He problems* ~ :: and v:iwal' ~~ ~~ ~~? l a~d cX~-m'=~, ~~t and ='n'~' analyt1c and Aim? algonthmic and exi~tial' Co:~! and computatlonal~

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216 I: ~ ~~y , ~ scam of pawn is If cb-~ ~ Me ^ on ~~= ~ mug ~~ ~~ as ~ ~ pawns of In. mug cb~ We ~~ ~ data is ~~> age Acne Ion ~~ ~ =~ ad m ~ ~ paws. ~1. am, -~ ~ ~ ad D. ~^ ^ Me Z-' ~~ an, an: ~ ~~ 19S3. 2 ~~ V.I. ~ ~ N~ ~ as: ~~- 19~. 3. ~^ ~d#~ ad ma. ~# ~~ ~ ^~- gab , MA: ~~ ~~i~ Pan, 19$9. 4. ma% Am. _ ~ ~ am, ~A: ~ ~1987. 5 ~J_P.; ,J. -- arch, ~=8; at, S. Am ~-J=~ (a ~198~6), 3~9 6. at, ~.~ ASH ~ e ~ ~ ~~ Amen ~ d'= gap (~= ~ gun Cap ~ qua. ~~ 3, ~ 3> (1894} 39^ 415. 7. may, Rae L. ~~ ~~ ~ ~~ ~ to Pa Ha: ~~ 1986. 8. fan, ~ ~~# ^~ ~ ~ age, Eked: a_, 1 9$8. 9.~,~.~^i~,~i~ ~, 1988. 10. am, agog ad Slain, ~ ^ {~.~) ~ /~ ~ am. at, a: a.. ~m=, 19S8. 11. ~~ at. ~ ~~ ~~ N~ ~ #: Villa Pa 1987. 2.~;~=,~;~G ~^- ~ ~ ~Z ,: ~~ ~1988. 13. ~~, Ian ad ~ ~~ #~ ^~~ ~ ~~ ~# ~ as: #H ^~ 1986. 14. ~1, ma. ^~ ~ Am, a: Ah. ma, 19S2. i. -, ~ ~ 4_m~i~ - ~ ~ in of fit alias. Id In, S.A. I.: ~~ ~ ^~ ~~ final ~~ 197& 16. gut, Jogs D. ^~ he 1 ~ i~ sag.- ~~ 2i8 gab, 19S~, ~ am halo ad River, ~ Hi. ~ ~~ ~~ Nc~ as. a: Spew, 1986. 1 S. mean, lo ^~ ~~_~7 ~ ~~ as, a: ma, 1988. 19. ~o~im and Sag, I=. ~ e~~ ~ ~~n,~: Em, 1978. 20 ~on, km a~ Wok, A.E.~. 4^ ~~n's =~e m~bine.- /~ ~~ ^~^~#~ 74 (1973), 213226. 17. 21. Pa It ~ ~ aria ~ gads, CA: ~em=, 1980. 22. Am, 1~. ^~ #~ _. ^~> Eat: Opal Udi~=ity Pa 1 999 . . a.

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217 23. Saw, ~1~. ~c glue of _Dity.~ ~~~ fit ~ c~ ~~ 10 (1988), 31~324. 24. ma, 1~. ~ ~~ ~~? ~ ~~_# ~^ ~~, Elm: ma, 1989 25. am, I=. ACE: ~ Cad Pa Diva 7~ #~ ~~~ aim. aim, lo Enema ^~# 19$9, i#73. 26. aim, lag. -~d~g 27. ~ ~ ~ ~~ ~ ma 109 (1989\ 57-73. Dawn, I'm. ~ G~- ~~ am, ~~ 7 ~ ~ IBM, ~A: ^ b grimily Peg, 1942 28. an, J... ~ gem, H.B. ^~r ~^ ~ ~^ ~~ ad, Ha: Jan Wiley ~ ad, 1986. 29. ^~ A.~ 4~ ~~- bag of m~b~is.- ~ ~ ~~:~ ~ ~~ 237. ~ B (1952), 37~72. 30. ~~> J. -~ ~b~i~r in ~ ~ -cm s In ma, ~~.V.< Am, I.C.; am, N.O. (ad.): muff ^~ Bean, Ems: We ~1 ala, 19S7 1~12~9. 31. an, E.C. 4^ =~mpbc alibi=.- In Simon, a. a.: ^~ 8/ ~~ ~ Mu, E d: hinge Uni~i~ age, 1972, 27~282.

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