On the Shoulders of Giants: New Approaches to Numeracy (1990)

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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 Dock—y 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

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 ~m—W ~ 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~

:~ i, .: . A: : :~ ~ : I::: I : : : :: ~ : ::: ~:~ :: :: :: i:::: :~ i: :::::: ::: : : :: :: : If'-' :::: ::::::: .~ a: A: ^~ o, A. ENCORE ~ ~ N~ Ad. - i~ that ~~ of -~: '<a ~ 'n~:g =~, (~) ~~Z =~$~ (~' ~3 311~ (~t ~~t 3~-~ <~) ~~C =~{ O: ~~d i:n ~~, such ~ man~s and p~. ~ ~ ~ ~ O=:~t ts -em o~i-c and A: mther than Iim:~d an~d ~~+ i~ =~7 tt0= 3= v~ [~w 5~5 ~ =~- ~ ~ t -do not b^e $~e bear:~ on c~. In pm th~s :s became Mathews matins is ~ h~:V )~ and 1nl=~:~ed ~~;~. ~~= chan~ - ~ s~h ~ compl~ and wn<~d ~~-~ that we need ~1 the ~~s we ~ mu~cr to ~~e )t ~ st~V chan~ the -it of th0 ~0 ~~} ~6 t~ It :~ ~ 5~ ~6 ~6 I 8~ of t—:~! mathemat:~7 maim matk>~:~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 vow—the 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~

As the twent:~ centu~: ~~ to ~ closet ~ ~~ ~~t ~ = iCs is I ~~le—ose Id :s vanish. Mathem~i~ ~ ~ ce ~m ~ d - cooping in d.0= my. unctIon OWE: ~~ t~ it, ~~t,00s to science—p~, 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

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 ., .,

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~ ~ been—hem ~ 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

:~ 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

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

~:~ 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 e—t 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

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

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~<enlury FYenob scienlis1 P.P. \~dbuI~s1~:is ore ofibe most cam mon #brig models ofl~in~ited g~o~1b~ S1udentscan sludytbisequalion ~iLb1~001Sfbom simplcalgebn~,bol~b by making Sand bysi~mplifyingibeequalion. Ibe populalionleveI ) = ~/~ is a ~o~leveI: once il is abed Abe new huge, ~+1~> 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~

d'h~wA~~= ~ : (~67 t56~0 '5 ~~ ~ en=ps~e the ms-~:;:~s in ~ sinde Mimetic ~~t ~h as ~ t~e serum By: ~ ~ ilk ~ I: :~S ~~n 1~ ~m pnncI:~- art: ~~s ohe~n ~e ~~e tha~ mnt~s Ohs (~. -I W~ 1~ =n be in~=d and stu~ ~m—IBM of—tues ~ mon as in beam their ~~ of Bigamy Guam 0r y~ balm thev =e ~~ ~ =~- However ~ :s also Cam to derive th~e ed ~e~ ~~e Radiate ~~s, =6 the1: t~nt N-~er~ ~enmen~ The ~~In :~w nisi an ex=~;~nt ~~tv for numeral ex~ pen~s u~g oni~ elemen~ anthm.~tic ~ =~- Ewn elementa—Y schoo! ~~en can f611ow the mIes, yea~ before tear =e into t.O the I'm of Id. The W~t ia~, who" bm1c ~~ is pa ~ i ) ~ m~) ~ p(~] can ea~ly be tmn$~a—into ~ mble or ~ spr=~et ~r eXplOrat~ion ~r van~s ~*~s of the ~~= m. (~e th~ we are *new brim ~ to Amity the ~~tion promotion, whim we prev:iousI~ c~d q' r~r t:han the position si:~3 Stan math some v~e of p(~, sav O-~^ and Mculate in tum ~~.~' p(23, pig, ~ Tn words. new population ~~s old po~n minus the squam of the old ~~atio-n, mu:Iti:pli~ By in of. Far example suppose ~ ~ ~ ~~n the Tsuc=~ - ~~s am IS O295 O416 .~6 .~- C~ OS see initial ~~-~;h' seldi:~ down to ~ specific final I—el. This ~wth is s::mit:~r to- the expenmenmI:~y verified ~~b of yeast and other hot mo~s populations (~e Figure 2~- When ma ~ ~ we get 0. in 0~' 0~9 t" 0~ 0~~ O.~2:!~ O-~, Of 7' ~ ~e v~s in t~s caw appears to osc:~e between ~~t O.6 and O-~. (~n fact, this oscillation even:~11y dies out, but w - -I sIowiv. :t becomes more apparen~t a~t ~n—S. ~ or 3~2~) F:~11v, consider ~ -~~ 4- O-~' O-~, O-922, O-~89y~ (1.821, O.~' O.~^ 0~: :~` Now we ~ no cIear pattem ~ alit What has happened The W~! I~' leads to ~ nch ramp of behaviors lucl:udin~ periodic o-~:~s and apparently pattemIess~ irre~r behavior. The 1~r ls known as ~~. Here ~ s:~91e eXpenment using ~ calcu:~r'bri~s .* A *: a:

195 = ~~g ch:~! tO the - ~~ of ~~. :~ th~ eX~e l=d ~ an Ads- range of-~;~m a.~.~. ~~ out —:~= ~ ~~' =-~' or Arid s~sheets:~ 0 =~' ~8 =~7 ~Yz~ng-wh+v t - ~~: '', A ~ ~ ~t I, 5~7 t~ ~~: 57 ~i=, ~ ~ u=~ns ~ 12~=e expllelt =~m te~s Bm thew m te~ ~ Me Wry Have :~ :s ~~c - ~s e~ ~~, =~, ~~ beh~:r prey - b~ ~ simple and explicit i~7 = ~, t: bls p=~i m~) iS my:= :~c :~. t~r nuc~ tu~ ~~e - m Alarm i~, m-~g tt po~e to torque! Day :~¢~ pheno~a ::n ~ am e maim It ~so demonst=~ th~ s~e ~~s c~ ~~e :~:~.d - ~^ It ts one of the mo~ e>5;~.~g Gus of: cu~t :~em~i~ Ah: :~4 Tt is ~~s poss:~le ~at Tactic behavior =~ld be just ~ Hi of the mama and not ~ pa ~ nature~ Per6~54 8~t ~~: pOp~ns do, in am-, dIspl~ isobar owili~ behavior. Fit ~ ~~s =~l ~m on ~ population ~ Fir Tics 'God In ~ -~:10~d =~er a: ~ ~ con=m pronely diet-:~) When the popul~ior} n~s toO- h.i~' them is wo little fO~ =d the hip =e u:~e tO ~ - prope~. ~ ~ 45 ex~5 ~~' ~= the 0~s 6 un~t—and the population shOots up amid. ~e main - ~~l e~t is ~ o:~:~n with ~ penod Of ~~t SS d - - How - ~~, as the t~ series sho~' the ~ i:; Bill the ~~:~ thin Chat :s decimal mmplex. Ma~ of the peaks in the gray aw do~, ~~:~ mom Misshaped than ~~S689064 ~0 height of the pe ~ OrC~0 SO'~ N~ Aft: : A: :~} $# 00 200 300: 46~ 500 ~O 700: ~ ~ ~ ~ :? ~~s 4^ Nanat;~:s in an cxwnmental popu)~on of ~~: 01= S~ow $~: ~~ —r 5;~t t;~t ~ t>~ O; 6010~#~( ~^

I4~r AKRmW~:~ TO N~y ~~. 5~, ~7 it in t~+ ~~: t5e 6m 450~ - or w, ~e chat become more and more i; :$ ~ i}~-~^ ~ :~t g~ ~: I =~- i,~ 8~6 -A ~6 8~S Ot :~:i6C ~. ~ O{~ ~~ O5~6 -I dw to the wpuladon iambi of—~t n:~- ~~:rs m~ due to out~ :~s wch ~ ~~d ~, ~~, :~r ~~:~e ~ the Eli ~ of Ma~ :~n ~ sky. How ~ ~ we tell - :ich am - .~9 T:t would be =~y to assume that ~~e ~~:~e I: p6~$ the ~~n in - ~r size—=e ~:~d to outs:~ ~~s but th~ ~ ~ ~ i.~sin~ A. a0 ~ ~ ~ ~ dw ~ ts due to Som ~2 ~ l~ ~~g m:~h an In: c~e x' TV 0~~ this 855~;= bear 6 in Inexpen~ mth models simile ~ the w i~ show that s~-e math :~ can. p:—uce ~~ ~~ar o~ I- and if chacs5 I: ~ making slight cha~$ to ~ single wameter. I:n At ma~ a~s of the Fit 0~ in.' ivies :.~- -~. =n be -id by simple—teals. -~. can be bmu~t to u:~d the possibill~s ~r com~= beh~r i:n simple ~~s b~ pe:rf~g ~m~1 experiments, :fi~ With cal-~s and I~r w~h co - ~~. s :~n appa:~v :~r data. tor example. Avers ~ time senes ge:~ed by the Wrh-~st lams Or =~ed equat - ~ - * again p(~) =:d ob=~e t~t all tM poims lie on a* smooth ALA ~~v can an:~e the cu - e to determine :~s ~~e ~~:~' -dyer chi:~n -~:~ wok an appropriate ~~a and estimate the wiue of the grow*~*:h Mae parameler m. `~:~e sopn:~:~m ve~-~s o: :~;~s atomic wc~e new oeen applied to many s~s of Anal ~~ for example' ~ the ~~:~- early ;~om fIuct:~ations that occur :-n the ~:~rs of oecole su~ns - m ~ dimple such as mea~. O~n the ex:~:~l time senes appear ran~. But gm~phical analvs:s s~s that ~ simple p=~57 :~sembling ~ d~ce eguatio:n, und-edi-es the -it i=~nties~ Tn cons-~-~= it is o~n possible to set up i:: but rea:~:~tic my that repmduce t:ke pattems of chan~ in the~ svst¢~. M~g on ~ -~:~s ~ ~~ ~*~ Or ~ 03~5 5~311 has all imp t0~-0 t~ ~~ ::~ moldy p-~:a ~~. In this case it provides ~ formula rather titian ~ p:~re or ~ :list of-~umbers~ In the calculu~bawd mode] the value of p(~) :-d not be ~ whole number, whereas ~ :~! Do-: necessar:~ takes on whole n:~r values. The mode! ~s thus ~ con- t:~$ approx:~:tio~n to ~ disc-~e phenomenon~ Chit is ~ common

: :~? t~ =d ls ohen used ~ ~ max:~m popu~n s~ is: ~~y 13~- Then ~e chan2e C=~d bY:: addln~ ~ ~~: :a single Ike. ~ ,— ~ *' 4 u~ is ' - ~~v :~' ~ ~ ~,$ 'pomade math :~ sing ~~t eas~ ~ ~#~5 ~~ {~8' ~~ all' ~~ 8~ ~ —~~ ~ti^ COe ~~ ~0 ~ ~~s of h~er ~tem~4 ~ Hi: eqU*~n :i~s n~ ]~t van~= such ~ ~e p~n but ~so mtes of ~an~ of van~- The mte of ~~ge of ~ variate h re~t to nme is t~y ~~ - m' dpldI* The sl:~t I.. equat1= for pit ls ~ TO In-: gum ~~! ~ mp* ~~s -~es thal ~e bate of ~~ ~~! of ~e popul=:~n ~ at ~ ~~n ~~t :t is pmpo~ion.~ to ~e p~n ~ ~ that ~me ~-~t, whew ~e c=~nt ~ ~~o~*na~y Is ~ Tn In* ~ ~ I~r 'p~n pr~s p~.~v -~e o~: ~n ~ ~~lerone* ~e Solomon to th:sdi~ential equation ::sp(~) ~ p(~-~=t ~: 8~ Thetis ~0 p(~) ~ ~—07 brim is ~0 00~ti~5 *~5i0~ 0: 0~ - ~~. ~0 p0~ti0~ 6~85 I* ln -I oth~er ~-~rs must mme imo pale to limit the ~~. As Aim t5~ ~~t am;? ~6 =~ t50 ¢~ti0~ 5~ subtmcting ~ -* np ~ (where a: ill ~ =~d constant* ~~! ~ my* Apt. e -Gina of this eMm te~ ls that when ~ is small^ ~ ~ ls bile in com~, so ~~ the mention berm np' has linie -God in ~~s =~e e obtain (~) exponen:ti~ Oh* How*~*' as ~ bec~s 1~? ~e term *~*np begins to domInate the 67~=i~? Ill* reducing the mite o~f growth. :I:~d when ~ Babes the value min. ~e rate of cnange of ~~e p0~7 ~~' 5~*~5 =~.*~* Wade Dim ha no ~~er Oh t~es plane* So min =~ts the m=:imum popular* time Using techniques of -I, it :is Missive -lo find ~ fo~:1a the solution* ~e ~~h of this 50~i:~7 ~~ 35 t66 ~;~` =~ 5:~5 the same 3- as the expenmen~ data on ~~st (Figure 2~* it~ =~ ~~* 05 I.—5~77 ~ri~:ic*-S c53~c - o: the dies -**-I Ve~:~t I~ is absent from its continuous analogs - itch vie:~s only ~ smooth Stashed curvy. Ikis shows in ~ pa~i~Iv conv~:~- :ng :~:~r that chan~g - m do models to =~tinucus ones' of* |t i ~ ~ 0t ~ ~ 5t ~ ~~ ~ K ~ Examples such as these raise impotent questions ~~t :he :*~tion between -~ti-~s and di~e modelS,~, relations wO~h exploring in :~:~tiC$ (~S 3t :~* SC~} iC*~- The continuous mode: permits -~:*~enmen:~! data* to be 5:~ed to ~ :: -~*-~tical cu - .~> an-d th1s o~s the way to prediction of ~~e behavior. ~r c:~e, if ~ Iog~:c cu—~e :s §~ed to the population of the Un~ed states up to 1-~-~, :~t p-~*~-dict.s that bY the yea:* 2000 the populatiO~ should

i~) '0~ ~ 870~6 2~0 Ink ~~= =~:~0 ~~ Eve ~ pm- :~ected I ~r ~e ~r 200Q ~~f 260- m~, ~~t 3~ h~. So ~ ~~pli60d approach dOes- $~ns~y ~ h I d~ (~f ~ eco - ~~, ~ nanon, or ~~ ~~) c= the ~~C cu~ ~ th1s ~u to.~e ~e Con,8 to predict I- INK =d ~ '=d The behavior of meteorites ~s ~ sm~] cart of ~ ~~:~ problem of ~ ~~ 0; (~ emit ~~' pi=~' =.~, DISK ~6 remnants' or aims Polaris' ~ ~e mo~= of ~e ~~ have Pro - ~t history b~ ~ ~~r m: ~r ~ nu~ of -~* It s n~ ~~t ~ maner of ~~n with me- m~t s~. ::~ortant do~ —h pro- so ~ a: and navigatmn :~aw at venous t::~s depen~ upon ~~e of ~ mowmen~ of the stars and ~~-~- A~:~om-v is ~ nch a~ ~r fi~g, ~d cIassmo~m =~:ivit:~s about chan~ the pha~ of the moon, the eldest the app~nt motion of s~' the chan~g ~~.' ea~h Allis* Anmher po~i~V is to recon ~~t ~~il0075 expenment:s usi~ b~s on inclined si~ and the Iaw of motion in ~ unique gmv:miona! beld. Dau ~~ed In such en~s can ~~i m=y nc~ ~:~ A:. ~ ~ ~ ~ it. nI~.~, our un~ding of su~ A.. w.~t ~~ A. em! s~es—:~ I- emp~ models' ~metn=1 m~- els, I m~Is~e~e culminate :n the I~s of motion did covere6 b~ Isaac N:~" But thew I~s o~en 1=d to - **I th are ve~ hard to sohte~ 1~v can be solved any ~r ~ s~m of bodies, whew t - I- predict elliptic orbits* The w - ~~ of cele~i.~! motion ~r ~ sy~em of three bodies has—n notorious ~r o~ ~ centune-s ~r ~~s apparent intract~li~. With mode~m compute - ~~e can see wh.~.~+ - ~n Styli. - ve=~:s—for examOle~ w~e =e bodY has n~151e m=s—Iead to complex and hippy :~ar behav:~- C0~: 936~S O~ $i=~O pi8~7 fit t0: ~5 Of t~^ th~, ~ more bod:~. Children as young as ~ ~ or ~ ~ =:n use ~~e packages- to experiment ~th the bob~ ior of the =~ar elliptical emits f ~~-~> sty ad the 00-~6 ~ bod:ics. :13y us:~g these p3~87 t~ 08~ g8~ =~0 idiot t~0 t56 ~ And? 0 Or :mot:~n than T=~c ~ e~ ton did in ~ :~time study.

I:.: undemanding of planetary- mmmn ~~s—m work ~ tM incare ar~ ~e Bum of (X~32~ tO i887 ~ 0~ i~ Ot In o~ - ~ Awe ~~ 25Q~ crowns ~ ~ answer to ~ quarry q~:~on ~ =~- Is the solar - ~m s~ ~ see now ~~ Pointed :~e ~ ~ enamor tum:~ point ~ ~e mate t~ of ~~est~ - Am* Scientists cad ~ sew sta~ if ~ d~ nof charms—en pe~ed by s~ dismrban~. It :s untame ~ s~! fL:sturba~3~ ~ to become notified, I=~ to large ~~s ~ by ~r example' ~ p:n iymg on ;ts s~ is ~~' whereas ~ p~n ~~ed on its t~p ~s unstable s:~= :t w~l always fin :~ (Fish= 5~. Aimed ~ Revere s~d ::~uition abed t~ not~= of ~~e =d un~ ~~s, =d bend ~r ~e airy co~exi~ of Chemical s~s ~ e~ ~e behmwor of Van~s ~.~.~ "~.~tive t~-~-—m~;~ Emmy raced m~, grow :~-r em ~ wOm wI~ ~ ~~-C ~' a=-=—to swing =~r the top of ~ se=-~*d Belt Tf the two ma~s h~e -Gina ~*- IanW, Am. the ~~*~:m Is stable in in d~ po-sitmn' ~ the Io~r m~. But if the ~~es ~ :he same and vou ~ to hold the pendulum ow: ~e Iower ~~* it tnes m- m~e aw~. ~e downward po-~n is ~ any—am the chDd can ~ ill Sxperi:~ts of th~s ~e, usually camed out in ~ ~~: ~~ way, am cu~y chamcten~ic of Ah. day Ace- ~~ ex - ~~*~s should be =~ed out fin mathematics dass~ as ~D, ~ an integrated pa~ of the de~ elopment of imunion ~ ~~ and motmn ~ f b: Alit itv and chacs. At I~r *A ~~r ~ child's intmt~:a ~s b~r d~ oped, Su~ eX=ric:~=s =n be ~~-d with ~~te ~~i*- =l ~~:~. ~ - ~ him He~ b i.. ~ ~~..~ i; .. .~ . if ............. .... . ~ ~ ~ ~ ~ ~ ~ ~ ........ ..... . ~ ... i. ~~ .. ~~ ... .... ~ · ~ ~~ ~: ~~ ~ ~ ~ . : .. ... ... if.. : ............. .......... ... ~ ~ . ~ ..~ ~ . ~ ~~ ... ~ .... ~ i i i ~ ~ ~ ~ . i ~ ~ ~ . ~ ~ ~ ~ ~ ~~ ~ ~ ......... ..~ ~ ~ i .. FICURE )- Uns - ~e and st~e ~~= of ~ pln~ Ruben balanced 0~ )~5 t)p 8~ ~0 ~~l (~= ~ pl~ ~ ~~} - ~5 Keen =~s 0~ ~ ts 8~? *I ~5 p—uc¢ 0~v 5~-~*lt (~5 {n tht posit t0 of the plnx

=~A - : TO : a: is ~ :~V in que~. ~~n al~e mu~ not ~ ~ ~ Its h1~t mu~ be ~e, ~ ill md ~ om Of ~e s<. =t ~5 ~ 0~: it ~~ ~~ t)p - ~t 0~- its 5~- ~~ ~~t ~7~= , `. ~ ~ ve~ cOm~ed plece of 6~:~ICS. H~ do ~ ~ow ~~ the =~D -is ~9 ~51) ~l ~C p:L8~$ =~313 ~ =~W it :~01~ ~~t -mount o—as? ~~d :P]um wash :mo the s~9 ~~ld the e=h Gamer o~ :~o ~e mId of ~ cum planets? Them =e w~ Awe problems - ~~e Anew am ve~ ~~t to disco ~~:~ ~~,t SO~ ~:~g 0$~S plum t! ~ tOO 5;~. 8~t ~C made =~h ~ dent ~n ~t Wat he ~ Foamed ~e pn~ a~. ~ do it :~e inwnt~ ~ n~ b=~h of m - ~~.a,l~ n~' called ~ ohen ~~n—as —ber she~ =~;=W~ ~~.r is mo~ pmpedy de~ :~ as the ~,.~= Of cont.~' ~ the ~~v of ~~ - m~ cha~s, the sc:~ce of ~e unb-;~:~.+ D:~ominuit~? in i? ~ :~n =d dmmatic_~s where ~ tI,~N,( chan~ ln came eno~S cha~ in And:. T~ =~ motion of t~ bod:~es—~ universe co;nsi=ng o:W Of the ea~h and the -~:~, -my—:s - ~c. :t re~:~s ~r and ow: again, on~ ~ (~ ].s, ~e defin,.],1;~n o{P`~a,[~) Th1.s periodic beb^~r immed:~ prow that in such ~ miar sy~-~nmi;~:~ng -amp ~e ea~ =d the sun~e ea~ ~~:~d not I. into the sun or wander o~ i t~ ~0 0~: :~$ 0{ i~.~' ~: :.t it 6~7 tt ~6 6.~ t:0 ~ t:be sun ~- ~~:r or wander -ad to infiniyv even y=~. ~ow arced ~ In: you =n do mom t:~n once' and ~~v ~~t I: 1m yea~ so they -adder wid. I-n other wo~- =~<l:~ty ~es ~ ve~ use~: :~andl-e on .~illty. Tn our real naivete In-= wi:~:! -. this sl.~e scenano new~' pen~i-city is still i:. Under g=\r Airy two bodies behave s~y they both move in Pal - ~~s ~~t their arm: center of ~:vits^. Three ~~:~s behave in an unbeli—~~: compl:~:~ed manner~ ~n if the problem is simplified Mar ~ K ~ tt ~~r :~0 Mom then th~e bodies can bead :o area W(3~C Or t3*11~i~g IS 8~ ~~;3mipil~ 0f ~8k>~ p33~idt33 =-~ti0~r~ A _ ~ . _ ~ _ . .. .. BY Y Y l! {s periodic be cau~ chit same act:-~:~s are Domed -~-~:ir and -and against and :~t must bee stable s:~e ot:~ise it wYou~t BODY Juggling two bod:~s :s Am: ely simple, ju~ng more quickly becomes Ale compl:~ed 1f teaches children to ~ u~, theN will leam qu:ickI:V about the complexity 0f d>~i031 57~;~> Thwart ~~ 3~ ~6 th~ p6~0 p~tg>=i 0 juggling fir Why is j~i0~g 5~9 Want is th~ 3~ 0f Or ~cd:l5~:~:?

7~ ~ 4F ,' ~ 2:~: .. ~ a, ... \. J ~G URE ~ ~ POm~ $ ~~c ~~h to =~;~`r~ :,{-~C ~;3,~ Of ~ ~~= d:~S cIo~d Inky In: ~~= (jams' ~e SO -m'=t'~ =~c =d 'arm ~~~ ~~= ~ ~ tt~ 6~= 05 p~= 50~ti0~.87 8~6 he ~ ~. . ~ 0 the ~~v :~ ~ ~~d b-v ~ mpol~ m.~* ~e some Air ~~m of t~e ~e -hem :is :~;n some p=:~r e =d that ~ ~ ce~n time Ia~r ~ ~s ~n :n the -$ ~~n :~t :~ reread over =d ~ acme ~~ the -~+ -~ that tOok ~ - m .~ ~e ~ to nSelf* Retiming I- on~ to ~ pr - ~~s ;ne pe~ in ~+ detail' ~:s ~e essence of penodIc -~. T~' enters when this ~ - ~s m~e ~4 ~~:~e th~ the state of ~e svst~ is descnbed by the is of ~ ~~nt in s~e hid ~mens:~ =~' - :~h sc:~ti~s c~ ~~e spate- As the ~Y~-~m ~~' this pOl.~t win m - ~' tmci.~ out ~ cu - e in ~~e spa=+ Tn oiler ~r ~ system ~ retu:= t~o :ts :~;~! ~e, this cume mum cIose up i=O ~ IOOp (~,~;~C 6) (St~\r Of -~h,6 $-; thUS t:~S tO "'Whelp does ~ cu - :e 'em ~ clo~ 10~- The :~n a~s nmbi,ng ~~t ~e 85390 0: 51~0 0: politic ~ ~0 l~7 =~ t63t lt 60 01~. it ~ ~ que~n ~r-~^ ~-~s the -~e of amidic miut~s £lepend-s on topol~ prope~ of the cu - e th~ -I the chafing ~~e of the ~~em in phase space Phase ~~e is ~ ~~ m~:~-~cal spa~ lath ma~ --maps that wprewnt ~) po~e vanabl~ the ~m the state -of ~ svst-~' -/ /// ~ ~ .~ _ f .~ ==:. ~ /~ Am, ;~-~-RE 7. Sxample o~ ~ ph3= po BAND dI~: cu—~s mpre $~t pOS$~ *# 0~: ~ $~= .~.: 6-~#~: t~),3) :~* .~ ~..~~ .

~2 - TO N~y which ts itsel: =~d ~ ~ ~~e pOlnt in phase spa= - ~ ~ ~ sm:e ~ h t: es tra0~:~ 0~:t8=~ Ot0~ {)~6 I of ~ ~e 00w i~s t! t~r :~s called the ~ ~~t of sy~! ~e 00w ls typify tndic~ ~ =~ imes': = to ~ tl~me evolution of ~e coord=~es of VanOus :~al pO:~ts ( ~~e 73^ Arrows mark ~ d^~.-~n of mot:~-n Of time. Bed =~t of =~hi~ the ch=~s :~n a si~e wnable' they =n be ~~:~=d to- ~~se ~~.~ Tread of plmun ~ ~~ ~ ~ ~ ~ ~ ~~ ~~ A ~ Mar =O ~~ ~ - ~0 Of ~~S O~f WO ~~t I in WO coo~in~e Erects Such ex~eS \~ll dewIOp insight l~ ~e mul d~ ~~ of cha - . ~r yo~ fin th~ va~ m~ be the height and ward of ~~g =:~s or ~~e ~~- m~e am rainfall p~ d~. Olk ch1:~en wuM =~er ~~! phenom~a su~ as ~e pOsi.ti.~- of the sun and m=~' or m=~e ~ts ewe on an ele=mnic clr<;ult' Or obw~s of ~ pen~m, or p=~0 ~-~5 0; t~ I. 0~ ~~$ 0~ t~6 ~6 It market. The osclllations of ~ s~e =~ Drowse ~ MAY ilIumi.~ine ~~e :~- ~ phase po~: - ~t =~e tn full - ~l onI~ ~~r more advanced ~~. Me t=~! Ah to the pendulum is wnte down an ~~;~:te equ~on whose solut~on :is ~ s:;~>*e cu—e. At 660,3~= $~ t005~S O: =10~1~S cannot some the t~ equ=~n ~r an eXact mode: q~ ~~t does |~= uSCtUt p=~;~CS O t~C Sl~¢ = - C ~S W0il ~$ ~ ~3 ~~f t50 period ~ ~ :~m - me sv~rin~ aw At;. Ho~, th:s traditions aromas is :n so~ re~ts un=is~¢~Y S)~ce the a~:tions employed are mmIv Jo. It Ieaws the u:~d i=~Y5i0~ t~t i80\ 0~t p:~0i5Yi0~ :$ 30~0 :~ emit li~StC86 tC icy O: I 0; It ~8~ leg) 399]~~tt ~ \<~016 ~~ 6~.*~t =~l ~t t~0 =~ 05 3* -I K :t :~5 t'0 t60 0908ti0~* v~o'sc, whem ~ {5 velocity ( 3~6 it 3~0 I;; 3*~*6 ~ :s t:50 8~6 t53t TV p0~;~= =.~Y I; t50 ~:~31* It sketching this ~~:~y of cu - Yes one :n e~ct dorms the gh~e po~:t (Figure S)- All of the motions of ~ real pend:~:~, :~:~g I I ~ 0~ :~5 - Ad. :t :~5 Aide propc:~:~er can be seen in th~s picture with this 811~16 3~h, students o:~:i:n ~~.:~:~` ~Ya.lid pi with the s:~e fixation, a~ accurate

~3 ~. '.< . X X ~ 4~. 5~n ~ . ~ : xr oxx. in. 7~6 %- ~~ ~~t O: ~ '~= ~ ^~ ~} ~0 ~~$ 8= $. molly no Id ~ 8~ \~t Swirl pt7~6 ( ~ of ene~) I=:~t t~t ~ - ~~r ~~v tO th~ abound the pendu1~:~9 The Id equa~ns ~r three bodies cann~ ~ solwd by ~ ~~- :~' but they =n be put on ~ =~r =d id numen~1~. Such moths pmv~e ~ ~ me=s of explon:~g the w~g e~s of re~;- onance on ~e mot:on of dvnamica1 systems. : Resonance occurs when : : :~ti:~:~s :~w :~s that are tn wme SImpt :~61 relat:~p such as ~ ~ ~ ~ ~ ~ ! ~ 3 . 2 ~ an ~ so on ~ Wt exam ~~, Titan 7 ~ 3st611ite 0: Id 535 8~:~ Iced p6~4 that is cIose to 4~3 resonance pith that of anot1~: sate11~, Hvpenon. If I :. takes 2~6 6~S tO- (~=piOtC O~6 bait 3~6 71~ t8~S ~ 5~94. l~C stir O-: 56 iS i*33377 =~V C]~ tO t60 Dim 4.~. Older children can u~ ~ computer pack~ to : planeta~ dv- nami~. They can ~~v the motion o~f the moon or of ~ elite in trans:t ~m ea~h to moon. Thev can nudy the ~ :n w~ct :~s satel1:~$ are locked into =~t I. They can I the m~11~d : acing, where satellite$ (~r space =~nies) can rem=~ in sta~ 5~0 p05iti0~.~7 60~0 8~ 0[ 0t 50ti~6 the moon. Th:s too ~s ~ kind of ~ .. ~ ~ w ~ ~ Resonances a~m espec~1v impotent in dynamics~ ~~y :: ead to ~ nch and subtle ~~=g,l~r th3,:t ]S, almost unbelievably -1: {n :F:gure ~ the Iar~ c:~10s : reamer rnot:io-n, seconda~ "islands~ bet~n the c:~-10s ..: =~, tenla~{ i. sI~l mo~ delicate multiple resonances. The spa~g>~ti-like cross:~ repwsent chacs. The ~~e repeats ~~;r o~ smaller and smaller scales. . .. .. .

2:~ V:~;x A~= Do: ~~Y - ~~ - ~ :~ /~ : : = FICURE 9- Fm~: ~~m n~r ~ ~~;:c o~x ,51~s ~~) r=~= of vanous Icy ^~e wn~s - want ~~s :~f Hi~ school students can easing seawh astmnom:~! t~es to io~ ~~ power 0 ~;~0 city [~/7 d~:ls7 =1~57 and 00~. T1; ~5 t~ ~10 t~ this can pr~= 6~p :~8 t~ ~0 - :0 i00~ 8t tt~ ~~.~ from ~ mathe:~1 I- Resonances o~n generate chaos~ Firm ~ has ~ panic~ar I g~ O; ~~ C306 it 6~S ttC =~0 CO=~Xi~', i- Ache 53:~6 g*~34 item ~5 85 t6t 0~titC I This 60~;06 =~ ~=,)~74 3~6 :$ ~~t =~t =~6 I n~tmam4 Tics ~~t w~v h~*~5Y, 0 =~t 0: =] ~~= i]34~> ~ 080t 507 With t~ ~ ~6 i Aim 05 :~ ~ ~ t:~) 800~*Y, :~ ~6 =~0 8~-~e to children around :~e a~e of :- Ache you~r - ~ . .. ~ . . . , :, Y .^A The topic =n be i. using natu- m! examples coaS:~' icaves~ ~~s, ~~. NeXt, computer mOdels of ~ ~ as t~e ~74 50t ~6 *~*~fl~e and dragon cu~$ :~0 60 ]~ 3~6 t50~t p3~5 8~81~* 00~5 0t [~34: 5~0 and sel:Lsimilan-:v can easily be deve1~ed—m these examples- HYPE young cl~ildre~ can apprec:~e the -idea of I dim-~sions-—which need not e nu*mber*~"

Yes s ~~ pmm~ ~ Ah-: a~ ~:3 ~ ~s in the esteem ~' chit is ~~ly Ted ~ o~ order q~n ~~t meteont=- Mo~ astem:~s c=~ between ~ omits of ma ad Jup:~ ~~k a few mme mu~ cIo=r to ~e mn. How- —-~:~' the asteroid dig =e :nm ~~ un~ Sweep M~ =d Jupiter. ~~ radii ~d to d== =~ m~ va~ and sky away - -I (~w lO)* ~~ Kirk:~rood, an American a~omer —o ~~ ~~on to t~5 ia~ 0[ U~ in ~~ut i8607 ~0 = -ticed an mm~g ~~ of ~e mon pmm:inem game of an anero~d were to odd ~ so in -me of thew Irk gem then -its o~ ~~ nod I donate m~ that of Jupher. ~ - USA* Remade mth Jupiter Avow perils =y b - ~s ~:n mch o~:~, Aim ~e kind of Rabid: ~~ sweeps t~m ~ to hi: at—rich r=~ce no longer O~$* =C ~~ ~0 of ~~r is ~ ~~e s~e it ~s masswe :n mmpar:~on win ~ mher Nan~. T~e Gus are o—wns {~ t000~t 68~3 Brim ~ ~5 2~, Al' 4~, 5~27 8~6 7~. ~~ e other hand, ~ the 3~2 r-~e the= is ~ ~ - of ~~mids' ~e H~^ =~. ~ Ability is not ;=t ~ I* of resonance ~ de~n-ds on ~e t-~e of =~-~" Th~e questions mmmn ~ s~t of intend i*~i~ti=. R=~nt compuw: =~ations~ show that an astemid orbiting at ~ distance that w~d su~r 3. ~ wsonance ~h Jupiter =n either Mow mu~y circular path or ~ much Ion~r and thinner elliptical p~. If the Ail of an antacid is sufficiently, clone it crosses the omit of Mars. Elves t:~e it d's ~ Am is ~ chance that the astem~d will come sufficientIv come to Ma~ ~~r its emit to be ~~v p0~# it ~~} CV-~7 CO=C tOO ~~SC -am ~ $-~Ot O~i~tO SO=0 tOt8~iV -different o~. The 3~: Kir - ~~ gap is thew because Mad ~~ it cIean' mther the being due ~ ~~me a-~;n of Jup:ite:~. W~at Jupiter does is -~e the :~= -mat =~ the asteroid to Rome ~ MA c then Ma~ kicks ~ ~~y into the =~d and ~:~. oueni.~ Ma~ ~~. ~ err Jupiter c~s the The ~~e mechan~m that ca:~s androids to be w~t -no by Ma= =n al~ =~e m-~s to reach the o~t of the ea~- The 3. ~ =~= -~:.: ~ \ 4! 77 34~ 52 2,1 32 11 nGURE lo MANIAS and dumpS ln tte 6~n 0¢ 8~65 teveal #id. 0~ ~ VIA

20:6 A:: ~ : w1~ J~r thus ~~s lo be Dim ~r :~w :~= —m the =~;roi WIt In~ =~h o~ t ~ : - * ' Bosh If Shy hit :~6 ~ irk AQUA game, prayed amusing the a~ -~ ~ M~ =d ~~' ~te~:~nes hewer o~r : 0~:~g cosmos ks—and ~~s mmeh:~-es :~wi~ c~ :mo Be elms mo-~he=. :~t ~~d ~ haM ~ find a mom ~a~c cxa~e of the essential u:~:~W of the emit solar Korea or ~ Bier example of the I:~:~s of cha=~ ~~ ~~*~! :31~d or e~ dare ~e ~ fearful sly ~~d ~~= 8~$ ~~g t~ ~6 6= ~~ ~~ wo~ "~- m~- in ~ technical wn=, tt t~ms out ~at He behavior of ~mme~,0 ~~ ~ ~ d~t beanng o~ Me stnp~ natum of t~= Sv~ :is bas~c ~ oar w:~6c undemanding of the :3 K' ~ t60~[ shapes ~ ~m :~:~ne many of their pmp—=. Ma~ natuM f6:~-——m starfish to m~^ ~m v~:~= to ~~:e~ ~~* ss:~- Man made o~je-~s ~~o tend to be svmmetnc. ~~:~ p:~s =~r plates, squ=e boxes, sphen=! bo~' bexa~al stee! ink ~~t ~~:c causes' haVe w~ mmetnc e~S is ~ Iong~d pnn~ c~e in ~e {~:~1~e of mathematics :~bvs:~^ :~e Cune made the c~e 5~6 "-i Mali (~8 p=~06 00~H 0~;*~- t56~ the ~~- tnes of the ~.~s ~~r :~n the ~~-s pro~d _ - I. ~ ~_ _ _. __. _ I _ _ ~ .~ . ~ ~ .:_. ,._ .... ~ . ~ i. . .. ~ ~ . . T~ pnncip ~ ~ b~=b na:~al Ale-—ou: IS :T T=Cf 1~e que~on :*10s ~ S~ie one ire ~~.i~ not :~t t~e meaning, o: =~mmet~5 6~t aiso t6~t ot "~- ~ ~~ K :~Iv sc:~*:tists and mathe:~i~ns :have ~~-~-e w~=e that' :n ~ i=~t ~.~7 ~8 Basement :~s ~~+ ~ :s po~ ~: ~ ~~: SV5t<~= t~ bCt3NrC {D 8~ 88)Y=mOt~C ~shlOn. Tads phenomenon' ~~wn as symm~) b~ :s an 1:~. ~*. underIvi~ pattem :f6~:n :n Oman p-~ systems hem a~v to :~. ¢e ma*themat~:! theo~ of sum:; breaking p~-~es ~ powe^} method ~r analyzing how sv:~*m~e systems behave and a*~:~:~s -$ t5C entire ran~ of scientific i: Cune WasY Right -~- At aim ~~- Curie's statemen~t is "~vio-~>Y~ t=~. If ~ planet in. the Sh¢e of ~ pe~:t sphere acq~s an ocean that ocean wIll surelV be of unions dept:~' hence itself ~ sphe~- The sphenca! Mama of the pla:~t :s ref1~d :~n ~ co=~*di~g sphencal ~.~met~. of :~s ocea:~.

It ~~1d appear b~arre If, ~n ~e amend of =y ~~c Laud the ocean ~61d dec:~ to- bu~ un - only. ..... . . . . On me- our dam, if ~e planet ml~{ng the sphen=] saw mu - and ~~ ~ b~ catcall 59~ - ~~t t~C =~S Ot =~— then t~ In win 15~e ~ Me ~a(Q[' p~ - Y=g t5C Ci=~= ares my* Tsn't Cat ~~! of b~ s~met~ Chases I INS. Carte Was- Wow ~~- cune's An;- Am- seem o-~:~ous, Wt ~~v must be :me~ ~ ~~11y Indeed, ~r thew ~ m=v ~~c Is w-~e be Maria ~ led- syrnmetnc than the ~1 sy stem. W-:r e~e, if a berg Gay ~ ~= me~ ~~, is compre~d by ~ s - ~~y 1~= few :t m.~! bu~s -I buckI:in~ is not ~ co-~-~= of iaCk of 54~ - - - - ^ . .. . . . . . meted c.~. by ~.e ~~. - ~n ~ ~e fome Is di.~d perjury alo ~e =~s of tM t~, prese - .-i~ ~e rotation sym.~) ~~t ~.~t axis' the t~e will st~ll buc~e :38uckIed ~li:~rs cease to ~ c>Ylindn~— ~~s wan "tackier mama ~~, ~ computer ~~e of ~ spherical s3:~1 buc~d ~~' ~ ~hencally s~m:~c com:~si~ farce :s sawn in ~0 ~ i" 05~ thst ~6 ~~ of the buc~ed state 18 -I mth~ than Athena. I't lS IBM.; tO 0,~-~d th~A~! 1;~50 IRISES ~0f SY=~t,~'r ~,~ these—~~ tems is not ~~)( ~ con^~ce of sm~! i:~ctions asymmetnc solut:~:~s wall e:~ - ~n in an idealized pe - -I ~vmmet:~c math matinal s~Y~:^ T:~' such ~ "~< 57510= i=~ 00~:~5 sYmmetn~ Can b~. Ho~:r, lmpe~ctio~s play- an important We In seeming eXa=~V wh~ ~: exa.~, wYh.~n ~ pe~t synch. such as the ~:~e in ~~e ~ ~ buckles' the axis of c:~r ~~:mmet~' =n ~ FIoUpE ~ I~ S`f=~,Y~r~b,.~3,~.; .g bU£~.] of ~ unI~] sph¢:~! sh~l su~ - to ~~ ~ ~ ~ 0~) ~ ~~$~ ~ K :~ $~) ~~= {n ~ cv>~\ s\:~:~C ~^

^~ ~ 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(ne<~^ urn l23>inv~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

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

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 ~ ~

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~ <'

W~ I: ~ N~y ~ c:~= ~~ Bog ~h o~y ~~= ~~- id ~~n ~ ~~ ~67 i0~ t0 ~~= ~~ I' M—e~:~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 $~¢s Ok to ~= ~~ more famous as one of the ~~r Fifes of m~m ~~. Tu::~g ~~w -Gil chem:~! c3~s prodwe ~e vari~i-~-n~s in =~ ~e chemical remissible ~r the stn=s need not ~ ~ actual Bong Or :t is =~0 Il~)r 1;O ~ ;8 ^CO~' ~~ed dunn^;g misgiven =~V Swains Of the tigers dove- which later tr:~e~ ~ senes of chem~! changes to create ~ Heresy Ho~ the ~61 deta:~-me of which r-~n mnt:*~mal—ante -~m Ian he=. Our aim :is to -~e wale simple and ~~M m.~ti.~ :~nisms Or panem · ~ Y ff ~ ~ tIo:n tn ~ :~r con:~t Tams ~~e day equations ~r this k:~d of chemise ~9 H[e I ..~ ~~ ~ ... . . ~ . . ~~ ~ .~ ~ ~ ~ ~ ~ ~ '. .. . . ~ ~ ~ ~ ~ soWen tnem numen=~ and then ma~ .* o~f ~e results us-cd to bunon~hole ~~ =-d shy them 0$ 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

~ ~ ~ /~ / 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

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

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' ~~s—in 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~:~:~tive—angelic 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~

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.

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.