~:~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~
OCR for page 183
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
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:~;~.$' I: ~ sv~,~m ~~s mom t~
similar ,*:nit~! ~~:~, He resulting motions can quickIv become
t0~\ It I 800055 t~ city Id ~ id 3~.~0 :~0
.,
.,
OCR for page 183
arm Ap^~= ~ I:
Be t~s sens: - e and parad=~ ~wior in' ~, ~e Wrenz
air (~ Ib) mem~ ~ Hi h~ twO a1~ equ~ A
=~8 =~8 ~~ 8~ 5000~0 Of However-. ~ ngomus pmof
ball ~e ~-~ Aim- ~~- -- ~.~ :n ~e ~~r th~ =~.~r
expen~ S~ 15 :~m o~y ~~d the~ =~.~..~f ~ ~~ per-
wn, ~t has- n~ ~ been—hem ~ pi mans and
= an acts pmbl~ ~r Fad re~.
The ~ ~ V16~t and mn~ Of ~s ~~ ~ m - ~s
m~cs win be unposed not ]~un ~r ~~:~ns ~ ~~
tots bm ~r ~0 tn i. Aid. of ~~. ~~ a~ us ~~. M=~
pounce ~~ss ~$, 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
OCR for page 183
:~
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
OCR for page 183
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
OCR for page 183
~:~
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
OCR for page 183
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
OCR for page 183
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~
OCR for page 183
^~ ~ 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
OCR for page 183
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
OCR for page 183
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 ~ ~
OCR for page 183
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~ <'
OCR for page 183
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
OCR for page 183
~ ~ ~ /~ /
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
OCR for page 183
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
OCR for page 183
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~
OCR for page 183
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.
OCR for page 183
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.
OCR for page 183
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