Cover Image

PAPERBACK
$24.95



View/Hide Left Panel
Click for next page ( 140


The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement



Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 139
A': ~ ~ ~ ~ ~ ~ ~ of ~ ~ :~ ~ ~ ~ we encount:~:r . Ad: the time - ~~( - ~~. in ~e s~n and n wo~, in a: ~~s =d video lmmS, in omamen~ design am ~ =i,Or in tTi)~C p81~' and in city abUity to Pi inte~' and create pattems is the kale to d~g wi~ the ~~d around us. Shamir 8{O p3ll0~. ~~C 55396S 8= V15~ ~,r-160~! TO 6~- 50~S S~S CiOVC~ ~~, C - V$~^ S6~' p~. 0~, Bile 0~t ~~SiO~ ~~S ~ ~ ~ Id =~:~$, ~~C highly ab~= and a~le to w~ ~~ "~e incre=~g p~t of puz~s =d ~= b:~d on the tn of ~~s and :~.ti~-~s lllu~s the ~:~x- ~ forms =d their relations :~ld Or maW p-~- ~~ g=~er Branko Gmnbau.~- "~s are - -A :n the simple W~tion of: ~ mund' ~ motion' or ~ ~~nc ~:~e, as I~ the intricate asse~:~ieS of ~ ~ x ~ c~s o.: cells into higher f.~-~s of bite or Iess other examples of o~! 6~:~087~ Mimetic pattems can se - e as rela~y aim- =~$ O: =:~7 t:~5 05 phenomena, and their stu~v ts possIble and de~:e at all levels But despite their ~~! impo~' students Iea~m veW Illtie t 568905 it - ~~) K ~0 8~> of shape tas h:~v been wL sumed u~er geomet~ (li:~ly "~h measu:~-' Bike for a 10~g time has We~ domi:~d by po-~.~/ Biopsy and theCrems of EucIld. # 139

OCR for page 139
I d4~A,~-~= ~ A~Y ~~ as Shy ts =t su~-~t ~r lit-~;~e and ~~-~s not s~iient ~r =~y, so Euclid is n~ m~i-~nt finm~. U~ wholes ~n all t~s =d p~$ Emit wrom ~~t ~e concepts of Comets ~ ~ ~~w and ~~ be c~d t~ m~ the medicos ava~le to Lima ~s he d:d not vmm ~wt ~ ~~ of :~s, networks, or flexible ~~, all of ~~ are of central :~e ~~+ is ~ ~3 =~, =6 ~~ t~0 lO Ice ~ n~ tO ci=~ ~.~w bm WAS ~ -I 1t tn ~~, me= inn and m - ~+ Pmpedy devoured, ~ nudy of sham cent~ =~:t of m~ education' ~ co~t the d~ on a~ c~= to n~ cays m~i~ but alm ~e ~~s and maw 015~: )-~nt concep~ `45~= we 68~t ~ prems~y w68t `486~- ~~ pa~v ~~e ne~t ~~s of ~= are ~~s bong d~- We assume we know - ~~ sh~s are, =070 or 1055. we know one when we s~ One Or we see ll mth our ~ or in o~ tma.~:~. But we ~w much more than whist We know that sh~s m~ be ~~ in some ~xYs and dissent :n others. ~ ~~ll is not ~ basketball' It both are Oh ~~d su~' ~ man~ -is not ~ ~07 6~t 50~ 8= ~~;~sY We ~~ ~~t Is may have dot pm~. ~ treadle ~ ~ ~ Y K Yin ~ ~ ~ ~ ~ ~ ~ :0 ~ that ~ ~ ~ =~ ~ d Kit K are ~~ - our shadows; -arm ~~= they -~ in size and c=~r throughout the. :~ ~ ~ ~ ~ 50 0 - Hi I 0= ~0 0t A" the ancient G~k philosophers. to disbar si:mil~ties and -I &m/~g ~~3Kj ~ =~70 th6 =~ of An; =d tO r000~26 IS in. ~~t it0~. ~~i0~7 3~57 8~6 :~ =~Mi~ =0 0~: t~ p~3i t~. 0[ =~07 these tools are dosebt into so d:~tinct:~s among 0m are to some extent a~-~ficialY Is ~m-me~ ~ too] ~r dass~fy~g patt-~s or ~ to~ for ana~i~ them? Tn ~~' it is both. N~nheless' it is helpful to days each of these ~~:~.~N O~e of the =~t ach:~:~s of ancient mathematics was We d:is e~ that them are exs=~7 ~ (~x, thwe~imensionat 5689~5 ~80 5~5 8~6 00~ 0{ t6~{ p0~57 cite the same nu0~r of polygons meeting at each -my ~e shy known as Be rehear ~ ~ ~ ~ ~ ~7 ._ ~ ~ . 4 . ~ ~ po~ are $~V:0 in tIgu~ ~ This Is ~ 68:it06 t~6 imag- :7~0'~: of the ancients that Plato made these shapes the co-~e -of h:s -theory of matter (~e his dialogue [~!'y; 8~6 ~6 ~~]

OCR for page 139
mare -A ~ Kit ~ :~ - ~ FIG:~E ~ ~e :~e =~r - ~~ mb Is =~ - of ~ sI~e ty~ of r - ~r =~ m~h the =me n~of ^~;~s ~~ng at =ch comer. Th~ t~on' , and :~r<~n =e m~ ~ ~;~, ~~e cu~ :s made of -I a~ the .-~ Is made of wn~ mu~ ~ ~k X][:~:I of kits ~~ to the~r =~^ 175~v ham :~on none of Belt ~~:na~on IVY. ~ m<~ ~ ~~e ~~ ~ ~ Am. ~ fiche -I f th ~ d~:~: may of the regular polybedm. In its time it was a mace ~t of mathem~ )n ~0 tm pi806 it ~Ct ~ -~ ; ~ =~ ~~ ~ ~~ ~~ ~~ ~= the Bier 2~ $~: ~ C0=i~ ~~- ~~ in ~~ =~t t00~0 ~ Me ~~s have pmpen~= that Ah themm miter ~~ts and be ~le to ~~ze - or dist:~shing - -in ~ ~ambi~s ~. Seco:~-~- you must be ~le to u~ these cntena to tnd out precisely dice o~ts ~~ them. ~ one ~~s just how the ancients made their d:~, but It is eaSy ~r yW~ -mailmen tod~, e~eciallv :if they have ~~ar pol;~3~-S to pl~ Ails to convince ~~:~s that the Bit of ~~: polyhedm lS complete (Figure 24* The k~ i- of math~] cIass:~=ion wew ~~ in use ah- of arms ~,~. by ~ id t~ obj~=s :n that class. What ho changed thmu~ut the :~057 8~6 ~~} CO~ti~6 tO C~' 3~-0 t6C ~~ Ot C53:~:~tiO~$ t58t seem impotent to us an-d the Ohmic that we use for enumerahon" ~~e 3 chows several cIasses of ~~=s that ~ ~ ~~ed together Am ~ mathematical point of IBM ~5 such as th=e =n st~e student d:~n What properties chametenze each cIass~ A:re :-m ~~t wavs go- clas,~ 7~e o~;~ what mb ~ ~ ~ Bide cla~9 We mention hem ~ fen of the -~$si6~n s~:~s that ~ twe in mamt appi:~atiO~$. -~~g~ I Si~ ~O ~~S 3~0 -~; it t60V 8~C CX~ 80117 I; Berm tO t~ i8~!;it 6~;~11 X~ {God their position :n space* Cans of tomato mop (of the same brand) in ~ He. store~ square tiles on ~ 0~, and has in ~ -~;~t pattem aw all ~mi:~r e:~s of con=~t 6~-~. Two o~-~s are similar if tt0)r Bite O~7 it pO section and wale. Similan:ty teems to be ~ w~ ~:~dam-~! In:. Preschoolers unde~-~d that I an:~- doll clothes' and pl~v houses are all small versions of familiar Hi: The :~ct that even such young -~il~:~n know ~~t t50$e fit o~s are su*~6 to mpresent

OCR for page 139
1~42 ^~ ~ ~_~: ma / \. ~ ~ .^ . \ it/ \ ma: b ~ 1 d ~ / ~ / e ^ v Of .^ . A. \ / ^ FIGUR.E 2. lb~c am only h~ _~; -~m ~~ abed ~ only ~ awns IS of con~nl, _1~ ~1~$ gut a ~~1 tba1 ~n ~ gad up to maw ~ ~~` ~~1 ages. H~ ~ ~ the Me a~n~men1$ lair ash big completion as ages ~~ =n ~ ~1~ up ~ ~~ me aria gyms. saws tab they 1niu~i^1y unwind change of ~1~ Build and wing ~~ scale gels o1 1~> an_> houses, sagas of ad kind Me 1be chi~l~d~f a~ alma h~ Map of ibis if. \~: A squaw is ~mme~cil: if you ~c 9~> 18~,27~>~36~~~. ma. il bas Cur lines of mirror symmetry across Daub you ^n Hillel it onto itself Chum 4~. It is gag ~ think of other others 1h~ bee 1be same s~mcldes, or selLcon~en~s, as 1be Guam: Abe Rod Cross sy l, a braced ~i1b Cur buggy speed ~ads, ~ circle of Cur dancers and a Leaf clover (~i1bOU1 i1S stem) are a ~ examples. Summed dodges o~ecls avowing lo 1be a~n~mcnl of 1beir con~ilucn1 pads.

OCR for page 139
~.~:= :c nwRE 3:. ~~= of ~~d ~~w I: tn~ u=~ c0~- ~t -I tt~ ~8 i~ =.~t 0~$ It ,.~ (~0r~:= ~ . ~ ~ ~ : ~ or:' >"" ":~ ~ ^~ .w:.:~,,,^ .' ~ ~ \: ~ . ~..,~,.~,,,. .. ~...~ ,~- _~ "em N ,,rr~x... ~ <= rile At' ~ as.. a.,, REV ICY his ~ ~ mher subde, for example~ the ~ polybedra :n Fume S:b :~.~ the same ~~s - . Al Just as commend Ieads ~ dry (I'll ~s ~ =~:h~ n~e ~ =~7 ~ 5~. 6~8 ~~ t~ At "~e ~~:c ~ct of ambetlc~ ex~- ac=~ to an h)~onan FIJI. Gombnch ~ "~s th~ ~~t IleS sO~:~e - ~~:n ~~ and. con~ :- ~~s this lS o~= Of ~e reasons w~ ~s =d o~= ~F s1:~ar ~=s a:e ~~t 8~> :8 HIS ~5 t~>r~= ~6 ~6 ~ )05.0 ~~;~. Self si.milanb~t has r.-~y b=n reco.~i=d ~ ~ ~~d concm in namre~ The awa~ng ~ ~ Ned pnze for the formation of "~al:~ion =~" and the cu=~t wo~e cmss~iScipllna.~- instep in ch~s shag md:~e the profound implications Of similarity and scale Ah* s:~e and mat:~tics~ ~e ~~ of ~~g MS ~im:~d (~ b=n im*~ 'by) the stu~r Of - ~~s a~d Other selF5i:~:ilar ~~. ford* ~~! pm~. : and sl.mi:~:~:V =e metnc co:~*- Or if, bC Hi b: ~3 4t _ Am. '` - `,, Mar 4tb f 5_ ~ ~ f _ _ _,~ W ~ _ Ala r ~ ~ <' ~ `% s :1 :; a: - ~- f . .

OCR for page 139
1~ ^~ ~ ala o He x r ~ ~ o ~ hi v ^ i: ^^ ^^^^^^^ ~ 1X U X IO . : nGURE 5. ln =~ h^~ me ~ sides of me bow ~ i~nliBc~b~ ~~ ~~a~ man ~= wand ~ ~ an i=~ ^~ ~ ma ~ ~ gas. ~ you ~1 Limb ~ gem sills are guilt in tog _~_ ~me? propels of saw ~^ ~~= under sum Lions. Or example, the numb of Is and Prices of a Plan ~ ~1 Plead if ~ So or bend 1bc pomp. Thus able 1b~ b as of Fame 7 a~ al beacons' An about may are nailer content nor simian beg is a~ cloud top of six line Seals. Bets a began a~ Idly AL 1be combinalod~ proxies of a shape ~ 1be tub we can gum Ad the ~1~ a~ hoed lumber. Bus ^~ be combinalod~ point of vita lbc shapes in Fiche 3a a~ e-~- Ien1> d~ each has 6 guess 8 vertices, and 12 ED connected lo eacb giber in lbe He way. ~61~ problems Hen involve c~bin~o~al problems. Or ovate, if ~ ~n1 lo demo a lining system far 1be compules in ~ Filings ~o ~ Unaged Al wilb boding 1~ poodle a~n~e~ls of lions and nodes 1~ An outride Me conneclions we want, and only then need we consider bog low me -1~ HI bag 10 be. /~ ldpolo~i ~ equi`~ienceiseven more genenElban combing narrow eq~uiv~ence.~F~om the ~andpOin1 of 10pOl~Ogy, allpolygonsare loopsand allconvex poRbedr~are Mike. Piggy a~guedlba110~polo~cal concepts occur pdor lo metric ones in child development; a Old gnat ~c~izz a low ~= disli~uisb.ing among kinds of loops, such as age. ~~a~ papery of shape. . . ~ ~ - ~ ~ Cupola in school is own described as gabber sbee1 Omen.' I1 yields maw excellent examples 1ba1 can enlarge a Tildes concept of Be Darius of shag. In Or ~ mew ~ Ages of Fang Sc am indi~inguisb~le because each can be defied ink lobe other. OlS, of the boy and Id SCOUT widely am an excellent subject ~r bandwagon s~dy.l2 Cbi~Id~n can leam lo play Dime on a 10=S and

OCR for page 139
145 an_ ~_{^ .. aim co saga_; ~1~- Casio fly games Pied H~ Pa Vernon globe ~ user 1~. Opus ~~ _~ he ~c~of~e~n, ibid lbe ~~= beech ~ ~1~~ ad ~ Pus Ed ~~ endue ad ale, ad abet cap of ~~: gab ^~^ adzes he JO divans ~ a -~~ ad ~ lass Anus zeal ~ gnu oaf. ~de~di~ cab chic ~~ y ~~ ~ Menu ad ~- as ~11 as Them ~ ~ . ~~_l~g Sb~s nag naves. 0~ of ~ mom Ames as of Anal is ~ asp names 10 gags. Ping is a phmili~ ~n~pl ~ is ecb~d in ~ gabs ~ ~ as in ma ~nl~ Pious panics. aria is the 6~ lo ^~ Pier il is lo age of a aeon or he name of ~ sable. ~ Ant 1bink Soul slopes (~ bias d~ fir tat Myers or exploit our ideas lo oafs if ~ do -1 he naves. Baling 1e~nic~ ages is Regimes did as ale auivily: but such optics miss 1be pall. ~boi~1 nag a~ usual no airily; 1bey envy ~o Angus Amen in Cab odorize lbc An me ~ naming. far eared in En~^spea~ng counldes IN nap in~~1e 1be Dig a~ b~1~ des~se an ~i~1du~ in ~ Big. Thus Jones is a peon a ~~ To is a member of ~ Jones fly. He ages of shapes sea similar -=ions: s 1e1~b~~n is a memo fir of 1be an gaily, a ~~n~1- of lhe Sicily of boa Bleeds lb~ bag Cur aces (see ~~ 6~. When ~ use 1~bo ~~ f ~^ I: ;""''"'' ~"~ {~ / > \. ~~ >'1~ a\ // ~ ~ ^ ~ ~ 1' ~~} \,& ~ / `/ )~` If / #< 1 #/ ~ ~~ ~  ~ ~~ V/~# ~ )\ 2.~<`~}~\~34 ~ ~) '//~ ~~ ~ i!'''' ~ ~ ~ ~ \ < 1 ~ . ~ ~ t ~ ~ / ~ ~ ~ 1 -I \? 7'j'! ''ail. ~ .~# ~ >at} ~ ~1 ^ ~2 / / ~ I / 1 Atom ~ t j FIGURE 6. ~^ Jonas is ~ Labor of Abe Ones a... . -# Emily, and Abe 1e:~b~n is ~ maimer of 1bc ~lybed~n Emily.

OCR for page 139
146 ~WA~ go: ~Y Awake - ~- ~ =~e ~ sh~' we =e ~e "me dme Agog it y t.~e a~ (^C-~ g 1.~. 1.~. a. m..~.~l ~. Ion-fires pmci~' Beret ~ no I- "r-= :~o cI~ she. ~ are cask ;~.~mH~ad:~ ili= ~ many ~m ~s,-Brig = ~.e prances us Wt example,: t.~.e do. mt the o~S :~f ~ ~~6 the win me- e11~' and ~ c~^ ~utiontzed-~:~..su2dy~.~-astr~y' - :m ~s ~t ~s Do. - ~s ~ ~ donut," But oh =.~r =d elhp=s a.w chic sechon.~ am. :-. th~ lo.= a,= ~e bow. Age pout ~ chew of ~;y'~ ~ ~stmenon benders shapes the encIose red 1.~e b~' and sh~= ~m h=e holes In them' I::ke bagels.' :s end; wide ~=e bmad chased ~ ~es aide But ~ f~b~ doer ~d n~ be happy m!h ~ bas~b~] as s~, nor ~d ~ b=~! blat ~ whim to- :make ~ - h ~ bas~' bemuse the Al kids of b~Is have c~ It pro As =~the:r e~:~e of c-~si:~, arching kid that it ~s implant to b-ui.~ houws th~ =e sturdy, n~ hous~ th.~ might wIlaps-~. This mn== tmnscends o - r ~s tMt h~s :~- commonI~ d.~, such as ia~ and s~' Bile ~+ cr ~U~7 t08~' or Ink.. CI=~tio-n ~.s dwel~ ~^ w~Y ywng chiffon team to wco~ize ~ =~ mam-~ s:h~ without be~g f~atly ~. emit wodd :s Am: ma~ of ~^ sh~s thet ho-Id ~' such as ~s a~ bags and ba~ ~, shapes to A- ~, such as balls =d p~s and 5100~' 55~= tO- use' such ~ Axis and s}~ns am b~. ~Ms r 5~]r~ children team FI:~:S ~ ~ Th^~e hex~S th8t am t~ IN ~ it~N :t 0~ i8~ A ~,~0 p}3 ~( I. ~ j),) 0:~-~,= ~ ~ ~ 7 ~ ~0 ~) leg) tN0~$ In (b I: and (~) a:= tnt~n ran. ~t ~ 3;~0 -~;33~t N0~:5 At ~,.~N;N - - ~Ntr~$ I; 0~t ~ 008. <~t, ~f\,[;85~N~$j (3,33 0~],) ;3N$~;0 CONS o~f thew 3~0NNp0$ \/ / ~ ARC .~ /

OCR for page 139
tat? nG:~S ~ FOur natum! ~ (~) ~ ~ the =~ palm (~) homs Cf ~ : ~) Act: ~n Aft: ~ ~ colonng and 1~' :~< the ~ :~* e ~OD 553' S~=S ~ (~CO Ct=~ =~$~'-p:~ ~ ~ J:~- A ate~ =~ 306 ~ ~ ~#. = nam~ ~r some of them, such as I: In: pol~#~-~, a~d some e po~h-~- s, in our ^~Is :~fi~on and cIass:fi~atio~n of shapes usu~:v ~ just at ~e point - ~= they =n ~i.n to be reaLv mtem:~tin~ where they ~ to e>:~re ~es In thr=-~onal ~^ How maw ~e T~iz:e th~ even ~s that are not h~ =n ~ ~ng and I:? :~ molecul~ have =~! shapes' but o~n these po - ~s am- ~:~ =d their Con~tio~s am t~ :~~ to -I chem:~ pmpmms (~m 7~. Bes:~s fiche ~ns and poly~ns ~ 60~ ~ (~-7 ~= 8~: I- ~ ~ ~ 5~ By broadening ~e defi~;~n of polygon to :~ any cIcsed Ioop, we may also ~dy knots" :In addition to ~r-`~us pectin importance ~r tying, things' knms enter into the design of reworks Cub as cIos~- 1~ and are helptu! in understanding the ~:re of: some biological molecules So~ bubbles soap ~;, ajar f~:~s am also endless sources of ~i:~g ~:! pnncip-~. The ~dv o~f pol-~ra can be cxte:~d from s::~le shapes that are easy ~o =~: ~o othe~- such as star polybedra' that are more corms 9~- ~V I'm 870 p8~55 5~05 35 ails 0 t~0 pi8~C, t~8t are beautify.! as well as use~. The helix and the spiral am ~.~! ~.,;- - 4;,~

OCR for page 139
1:48 ~ Air ~O,4~ TO N~4= to t:~ an~ a~ AL web as~ to~ ma~em=cs. ~: 8m wen tm day' ~ "~.~.e,1.iX~ h=~ ~e ~ ~ houseboat phrases. ~w p - ~e reach; ~~ Isa ~ 6_~' hit ~= =~ =~ :~: M::~a con~ dis~:~: ~' ~d ~ ~ (~m if*: :~:~s~d :~m ma: ~ day I.: - - ~s Atlas ~+ Awake-~ ~ we Ace: ~ 50; ~ if ~ Our~ ANAL ~ -~ t - f id, ~s ~ whM~ ~e unl~ ~d :.: h~ ~ ~ axis wew ~ 5,~mest : ~ :~ ~ ~ i: I'll ~ ~ i: 'I ~: ANTIS : In; aver to ~;~=t and awe pa=:ms >n to-ts :mage-pae3~d I :~t IS: n~ end ~ ~w =~= s:~= and ~, w~e ~o ne~ to an~:;~e that ~s ~ us to He ~ As ~ I~ sha~s are butit -A s~-ones and ~ Awe =~ms =d thmr pmpe~s . ~ ~ Wan chi.~n m~e sh~= out ~ bloc~ or ~' th~ oDen imitate the d:~e com:~sit:~s ~ ~ see ~ar~:~em (Firm 91. Nature too creates patterns. Like man-made In n~T ~s a~= ,~,t, Wagner I. art ~ 0~.;~*i706 into =~ ~0 ~::~;0 ~ 0~i~ iOt0 0~5 i3~:6 =~57 ~ti~ oily t~= 8= 00~n the $~U~itS of ~ h:i~.~ :=re: o~i=ti~.^ Warm =0 -~=i~0 p-~5 =~97 ~ {in:] i;~t t50 5~6 fats . and a~s Bear ~r and over - ~7 - ~ when the Elects _: n:~:~E 9. Many ~^ ar~ bu:~t :~m ~r ones~ Th:e ~:~g beams a* ~ bn~ :* :~w wpeated p:~:~:~s a:m USES }~ ~g 3~6 8-~-~' 3$ t~ ~.~' t~ ~ 3# ~0 out Ale p3#-~S#

OCR for page 139
E t0~ Young: ch~= =n :~e the ways :n - ~~h 6:~ t~r ~ t11e a. p~e '~0X . ^_ invoked are wo did ThIs is not Just ~ mincide~. ~~ -of mo~ oat=:~s :s Herded by ~ By few baste ~~s of fo~: ~~' and d~t ~ ~ :~k ^~m In ~~w ,0: Peter St~= :~s s~ wa~ ln which n81~] ~5 8~6 ~~ 5~6 35 ~~$ t=~& =~' p8~tiO~' CiO$C ~~, 3*~6 C=~^ `' ~ The results ~ thew m~s Of ~~=ion .~.m rem~y If.- - ~~e \,~.e vane~~ of mate~s on Bail th~ - o~e (~ee FI~= 8) ]~t a~s of pattem -I can be grasped ~ -~-~g the ways in which cop~s of o~s -me be pa~d toge~-~* Stu~s q=~V hisser ~at ~~re are onIv ~ I- ~ ways to -do Whist delis ndamen~ pro=~y of Am can be studied ~ ma~ I~^ For ex ~~ it can be smdIed i: and "~ when the o~ts BAN ac~d ~ ~ ~= ~ ~ tA quadniateMs' and hexagons Cadre iO)* Ol~r ch)~:n can expen ment with ie~ re~r -amps and discover some sumnsing th-~- such as the ~~t that am~ quadnIate~, Been one that 15 :~t co:~' mi! t:~e t:~-e p:~ne (:~re ~ ~ ). (~ is ~ Su~nSing but ve~ simple cO-~egue:~ce of the fact that the sum ~ the i? of the an~s of ~ -~:~:! is 360 -3 Him ^~1 i?? 030 51~ deeper pmpe~ics of sphere P30Li~8 AXES Ii1:~' =~5 8$ :~Ci: a?) ~.~ - 3~6 t0~ rhea can be gener ate~- (~:nxbaum and Shephard,~ ~~ and Patl~~0 is the definitive resource for *~rial on tilings. 3

OCR for page 139
of 172 ^~ ~~ ~ \_~c'/ PI ~~ ~=d#~ ~ gait ~i~$,~~=, ~ It's paw ~1~ ~ pit ~ ~ ~ ^- ~~ ~ ~ ~ SO Ha; fit at_ . Ad_ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ... Em: 1~: ~1 ~1~ ^~ldi~ Clans as .. . amp I Ed n Bait ~~ ~1 quad Ed fits ^e~ Di$~i=:; ^~^ awls Glut =~$ Guitar ~1~= act Al Tiling tbc ~~ fit ~ Is ~~1 Aims Same ~ ~ Nazis Cadge; =~n Gamete Elms Is e ~ bite ~s~^ Id of ~~ and ED ma; Amp tb~ _ ~ ~ I DO Ian a~ using simply mans Ski Clef ~= and 1~1 cut I atoms ~ 3~0 -- ~u~ ^o ~~ ~1~ of 1~ abut; potions; mat land iamb ala; ~im~~ibl: h admit tiv~d~ ~cb~i=1 doing; ^ ~ 1c ~~ Talent a~ micmsco~ Has mat; tbc ~~m Bang I ED card Tuna ~~1~ Gaping On ~~ 1bc Cog Spurn Aphid Spur P1GLRE 30. An a~n~mcn1 of lOpiCS Flared lo $b~ abed pates I Ad cam ~ -~ ma apse as an milt collection of quite Isle loci=.

OCR for page 139
Ad: I: -:v nGU~ hi- Me s~= ~ ~~e ~ ~ lt X ~ ~ ~ ~ ~ ~ . Th~ 15 ~ tM ~~ Of diamond asks ~= ~~s ~ - = ~~s x of ~~:7x - :~5 0t :~t 563~5 - ~= TV ~~s come eas~ ~ ~~t The SiE=n -my Which. bas trans~:~d Me indu:~li.~d wOdd lD iust ~ few dec~s, is bash on ~ ~~re that is ~ camer Of tucre~ly mad circuits. ^~ ~e circuits shed are complex, ~ ~ *~x=~ that hou=s them his -a s1:~e mo~r stm-~. ~r -~ample' bins silicon Is built of I~d z~g ; noes (~ 31)' ~~h am =~ to make end ~~ ~~ ~~" : the silicon ~~= the nn~ aw i~:~ed to fo~ ~~-~:ke pol"~+ ~:~ schoo! ~~n c~ ieam to build and identify these su at middIe whoo:l children =n Ieam to p~ them to~' a~ heir school ~.~dren can stu~ the ~~:ion b~n the all-icon st-~= =d the prope~s that make ~t so use~. The ~T scan =d other ~~s of =~ed 1~ recon~ n h~e revolut:~:ized m-~: dig:- in recent "ma. Whim hi 8~ 05: ~ 6> ~~> i~ 80 exer~ A seen is an ~~e in wc=~:~ting images - m t:~cil* cmss se~. t::~e the c:~uit~+ cn ~ s:~-~n chid the Ames r=~n used in this t-Cchnolo~x 1s ~ com~ex pr~ss ~t the simple~ ge~-~al ~ - that underlie it are easily unde*~. Hew ~~n ~ 6~d that the same ~~c pnn-ciple$ are =~M to manly ~. W: example the con~:ion of shapes ~om sections =d ~~s teas been the task of architects and builders ~r --. Wh:ile it is not be to bnng ~ C^ scan :~ch~:~e or ~ co~tion sne into the classmom, many p~ts suitable ~r school can help students under~d the relation betwec:n shadow -of cross section and shape. 5~ ~~pg~:,g,:~r tt~ {gg~;~h0~3, 0~57 8~0 0~:~=i~" ~6il6~ 0~ ten ieam to make paper snowpakes in s-~^ an cX0'rClse that =n easllv

OCR for page 139
A~ ~:~.~Y :~UR:E ::~w Bmnc~d sn-~es BYE ~e ~r s-~ of 70e c - ~$ w=~d ~ I:~e at ~ - sc~e bee =:~=i to ~ ~dv of t~:r symm~+ The hex~! Mew of the snowshoe decides an intm~tion ~ the Mm - of polygons' :t :s an :deal submit :fior the elementa-.Y cIa:~m~ :13~ut the s=~ke has mu~ more ~ teach us. T~ the hm: place ~ snowflake :~:ks {~e ~ patmm ~ m~ht see :n ~ kdicidoscope, =d so ~: ls~ This schists ~ ~y of the :~. Quiche. as we hew seen, ~s an Pain of the ponc~;~es of mirror ~* These same refl~:n p:~:~$ un~:~d mntem~:~ teh;~* one need only think of the :~fle=:~on beams of bu~= ala~s and I~ or of m~r and sonar. M:~e schoo! Ohm em; eas;~:~<^ understand and appr=~t su~ ap~i~. At the h~ - ~! i - ~: An- emer~e of hexagonal symmetry fmm aggm~es cf wamr mol=~es can be explo~ and so c'an the C - ~' dendnti'( ~th^A' or branching. ache b:~hing of the snowflake :~s as cha~tic as its sv:~:~ t~ and is equally si~iCca:~:t in the ~dv of shape. :: =~= of the sno~Dal OCR for page 139
175 ~ mile of geometry Is ~ pemnn~ Issue In m.athe~.~cs edum:~n at ~T levers - m By sch~ to ~ - ~~* For Day veam Geoff :~as been the problem c~d~of the~ - attics c~rr~lum. glance ~^ the N~ - Junk of Cached If Demo 1987 Yak By: and Teaching ~:~e~s smug of the ~~ quesi~ns invoked. ~e Aced w1ttL Good: :s due in pm to Iack~ of See on - ~t Comets ill Land ~ ~ amid ~~ :~. Do we saw it m~ discmI~ned ~9 To cream ~~s ~r .,, . = x ~ .*, mber so ~89 Or ~ ~ because ~ew :s implant c~nt tn Me s~=t ~~:~ = ~~~ ~ him saw =~ - texts ~ m~t om exam~p~s of Comae ~:s i:n :~, scmn=' ~~o m' con~ns ~s ever e>;~Iored in a~ depth. ~e Ethos of method =d cat its usually unmeces^. Or Abet - ~~ =~s ~ unea~ =~m,9~mi505 It.. =~7 t=~t t~t mat jig ~~. teaching Dive re=oni:ng, pawing thecrems of acrid :~ problem solving, tea~:i~ visualization, and preparing t~ ~~ ~r ~~* The contin:ui:~g debate indicates ~~t :~e of ~~e goals is pameulad wed ~d tn the ~~em situation. In ~~t ~! of these Chad ~e Aching of Comet - ~~ is defended on ~e =~ the ~t ~~s Be. purposes' r~r ~~n on the ::~- po~ce of the su:>j:=t in its own n~. For =~ ~ wcent micle on si.~:~? Justi50s the I of si:milanty ~h ~e ~~:~g r~ t:iona.~. S:m~lanty :~as am :~d :n ma~ ~~s of tM s~ cumenlum-+ Some Amens ~r rat:~ numter con~ ~~ ~~ ~~ MA t6~ p8~ 0{ tte . ~7 d~S W1~ =~:~S =~Y - = ~ == p=~$ ':~5 SI=~ - 6~sK Rat~o =d pro~;~on ~ ~ ~ g~ ~~ Age p~:~ =~y God ~o the ~~ Sta~d $ 3:~ AWRY p=~ Wor~ pr~.r~ an~= God pam of m=y inwIl~:~e tes~ stm118r gec~C :~S would s=m to prOv16e ~ helpf~ men~! tm~ '$:~r other t~ of ~~n a - ~~. slWatIOnS. All of thaw tC880OS are ~ ~id. ones, but there is ~ Go omission. ~e pnncipal reamn for tea~:~g rim ou~t surely to be that ~ is of pro~nd ~mpo-~ce in un~:~ng sh~. Mean - ~~, Outside the halls of education the co:~r revolution is Lipidic chan~ng the wodd in which we I:~. These changes aw placing :~ew demands on the cumculu:~7 dem=~s that are just ~~:~ni:~g to be heard in the whools. The r~ion :n ~e stud-v of shape and form =860 p05~6 TV t56 :~et =~ts that ~ ~ ~ ~~r compromise for Be but ~ new and coherent mathematics cumcu:~um that :~m shape tnto the ent~= cou~ of stunts

OCR for page 139
0~$ ~ ~~ ~ {-?~ `~7 IS ~~d to ~ o': gm ~~s , As: ~ ~.~t ~ use~ que~n ~ need to am ~~:~ ~t we -want our ~~ to Icnow and w~^ Euclid red She cafe ~~g ~~ shops I cede 5~?5 of de~s ~d a~mph=s and: veer carafe jar. In o~r to a~ze shapes, ~ud~ mu~ key bow Ibex lengths, ~~' a~ - ~~= ~ w~ a~ Shear and ~~ - As* Icy need t0~ ~~w pr~erti~ o~pall~ an~ Ed* l~;~c aid ~~' =6 ~6amen~ property 0t figures" M0recyer.? t~ need to ~~ 60~7 web Mer and ~~~ to =~= su~ flare :ilgures as: It: ~es, far art?: 3~6 -~uar=- ~ - ~ they've Adds ~ t~iti?~ ~~tW cu~ n~:~` ~ it can=t ~ =~d u~r it ~ a 6~:?~{ =~0 07 ex:~r active- The shapes ~~ nudges need to un~d tok, am the I: fat ~ need to be ~le ~ do m~ t56~7 8= too Yast -a s~= ~r ~~. ^~ - thy is mns:~=We daft: ence of As arid purposes Tra~it~ ~~ ~~ms mow ~n i.? Id? {~?5 mt6 s1~ C-~-~+ Tt.s Mae In ^~! 1n s~e re~S :~S an~s to Me cum~ar :~ of- ~~? =~= l~?~?~:?~- ~ ~~nt who ~?5 ~~ 0t ~~?~t i0~5 ~?~,5? ~0~t =6 mme imp his t0~ 8~6 ?~50 309Uit05 t~0 68~8 ~: -I =~= i=~, )~g 0~: 0~ - ~~, ~~ i8.~, 0~ t~ 0~: t8~, 8~? - i05.5 =~ 0~?8 vet =~6 0~. ~~v 8~ t60 avid 06~10 Ian~ages t~t =~le &~ ~0 :~ - ~~ i:~. I~' As Ad I=m both c1~ an-d modem i=~, If:: ~w h^e ~e time ~ oppo~;~v ~r 50~. ~ Wart. - 6~ Alit 00~? - 0~:~6 Aims ?~5 0[ ~?~0 ~~ ~ ~7 598~iSh~? 0t ~-~ is O0t t~ 5~D tO ttC pt05~ ^~ 0; t~0 ~~ ~ Fit 8~6 ~~> 8~0 55?~ ~ 0~?~81 6~li60= . F6t 0~: 20~ v=~ EucI:6 ?5 ~3 ~?~5 5?~6 ~ct 0~17 85 the mmer?~? - of gecmet~ but ~~o as ~e ~ amok of mmthemat:~ 08~=g ~~ 7~Og ~ 3Ki0~5 ~5 5~O ~Itt~.? not ?~ ~ =?~th0~ti0'5 5~ ~0 ~ ~0 8~6 I* ~: 0~$ it W35 ~*~5 ~i?~06 t Arid ~ schisms =~[ my? ~~?~ 0t the r=! ~~d that ie:d to the ~i50~? of non~n ~'0~ :ich ?subseque:~iV became ~~e ~1 too] :in ~~ng the 18~^Y~IC ?: 0[ t~6 I. ~. ~~?~] ~~ ~5 ~~t 108t it?5 ~~$ 50t o~r r~s :~uire that ~ Do i: tt~ =~th6~?~l ?~ of :~m 18.~ cOu~s into our curncu1~:~. ~6 id ~ ?~t tt8t 0~$ ~?~?8 =?~> p?~$ 0: I ?~d 50i 0~" {t 0~ ~ =~5 ~~ti~ 05 p058~6iiiti0?5 ~: i=~?~ti~$ ~-~.

OCR for page 139
: 177 inst-~io-n~ building mod~ m~ u~ Spiders Am ~~bse~- ti~ t~ If - = =~ip~i~ ~~ ~~* 85~ ~~ Belt tractmna~ po~Im for enta~:ng me foamy ~ m~emau= In tI~ :n s~ ~~t At:. ~~ ~ ~ 5~ my: ~~$ m~ ~~=s in whim she pies ~a mie aw n~ used to of as m~hem=~s in ~ n~w or ~~ -~* ~;~, pr - limps of ~ze and sca~ ~ not belong ebb to ma~. :~:3:=d ] s~ of queries that send us to ~e ~~ or to mileages :-in other ~~rt~- Could there aver have be~ ~9 Could ~~;~e ]e ~ ~aL as mice? our Ohs sh~ that Mess qum~s =e -I man =r wco~d hi e =~s -~e n~ so applic~:~s of ~mila~. ~ go t cw~ -I be suppon~ by his ie~ f they were exa~Iv similar ~ our :~ - , intro, ~e bone mass ha% to ~ :;: di~-~ionateK~. This -pi malces ~ sm~ of bob sod more fig ~= -it wo~d be ~f the =~$ ~~e . sim~p ~^ PO:~ ~ is ~~ I~ ~ CI8~, ~~ Optics IS ~ Id- Of p~' ~m~ and - ~r t=-~s are -I cone - s of bail 0~7 bibs 6~10 ~5 ad bore tO =~l the M=~*:O Of m~- Even within ma:~tics? she :s inte~i-~ina~* it re~ Wires vista! and m*~:~! s~s, :~1 ah- and many Aver tools. Teach:~g shed :n ~ -~:~' meaningful ~ can stimulate cIose coopemt:~n among teachers Of many sub* ~Skt.6 Dyer 0f Ache 8~8 p~18 Cutting 80~5i;5 ~~l~~;~. W:r example, the sturdy of sim*~lanty can be nicely Implemented by ~ ~~ of ienses, requiring an -~*~rsion into Chow :~en the :~es aSmelated mth many' -of ~e :1.~ of 0~c oW:~s ~~e g ~~s pnnciple) stand as te~*mo~v to the It that Wave discipline boMem h^e *not owes been* so hide It is -~iv be~se we 6nd the same Ago Or; t6~ Fig ~~ t~ 5~i th0=35 p3~ 0f ~ma.~ema:ticti in ala? 6~ ~ ~ 00 ~ tt~t ~ ~ ~e stu~ of -' tS ~ Ia~ Sumac. All of us' Children and adu:~' ieam ~~t shapes by? mating them an~ ~~r~g m.~.~5 Chat ure 33~- As an ancient pmve~ s~' "l hear and ~ ~~, ~ see and i: m:~em:~, ~ ~o =~ ~ ~~' if Arc wis]h to build ~ ~~pe~ cubes ~ 5~} 50~e, 0;[ ~ chip> ~~r polyhedronwe :h~ to be able to cut out ante assemble pieces of the ---I sizes. Th:s is once of the reasons that Chic ~~Y (~0 =~' p8~) It 306 50 forth) remains i:

OCR for page 139
~W /~:~S TO ~~y : ~ 0~ ~~ ~ ~ A 0~: ~~ t~ H^~:^ ~~r $~ ~ :m~d. Co.~` h~ devoted h1s , to, Age, en:~g patw=S 1n ~~x Bu~ mo~' in shiv ve~ =~te Sen~' is one of the be~ w~s to undo t~ =d prayed=. :~n expenmentatio~n is essent:~+ F6r - ampler when we ma Abe m:th our own bands~ ~ ~:n much more Tonsil I- :ts metric, =~bi:~nal, =d ~abi:litv properties than if we jug :~k ~ one. {if ::~d of ca - card ~~s we :~ t~e Me few plastic I, ~~k in b~Is of putty ~ in ma:~:mal:~ the c~e will wobble. ~~ Is ' ~e wand cube :s not ~ "~- model. ~ the c-~, ~ is use~l one becau~ it teacheS so-~hing ~~t Oddity and n=~.litv. {t elm t=~es something abou~t the sha~s into which the Me can be t=~-~.~d - :~e I. its co-~bin=~.~ :. 6~r~ $~5 t~ 3~6 t58t =:~S &~6 "manipu::~-~- are v~e t=~s :n the ciassr~^ But too ohen one :3he~ the ::~ament that "if ~ A ~ ~ ce~ art. ~~ Ate ~~ ~ 63~6 t~ ~~ tt0= l8~t 0~- Axis ~~ 2~ =~g t~0 l~t Tut ~~ - t i. assum~l'0~. First, that ~~ss ~1 sh~e is t~he main thing that ~e team - m ~~s ~~d that it can all be 1~d :n ~ 506001* )~ ~0 500~ 0~0 0~ (~7 ~~ ~0 50~- them all. {5i5- 0: 00~' t~ ~~- tt~ ~6 ~0 mat ~ Beet t0~ ~n t~e ~~V Of SrOlUt~g3~ CO~~ Si=:~, Wild =.~: Sit:~- iFhe ~~. assu.~:}~:~n ls tha~t the main pumice in starving ~ mode! :s to deve}~p our p-~wYe~ of abst=~t reasoning' hem the model plays the

OCR for page 139
.~^ Me of ~~ wheels on a: b~- ~~V ~ ~t :~r ~ - tO u~d the wn=~ in - - a:~ =~ mp~s In: 00~= 0t ~ cube. ~ ~n beet ~5 :s =~$~63 m~t 0t ~~-~ ~~li Me ~ iot ~ I=m - m real models* Ideal, the ~~1d be ~ :m ~ boom ~:~ At ~, ~~) schOo! ahd have a ]a~ Were student =n expose shape. ~ sh~ ~~Y Ode ~~ ~ ~ =d = Motion e~, Me - gal duels of m~ ~~, m HIS ~r bedding ~~, =d pit tO dismay ~~. Tf polite' ~ sill :~e~ c~ m~ grapes ~~. As should ~ mp ~6 ~~t ~~, ~~t =~7 =6 ;~ 0~: # programs. ~e ~ ~~ [~r e~:~e It is ;~n s=d that ~~g is :~! ~r ~~w 1~. Ce~- - it :s tme that ~~s who: h=e tm~ wi:~ a:; ~~:~ ~~t 6 W~ l 00:~Og ~0 tO=~ 5~D tS t5C 50~ p811~= ~~; \ hl~ ~ =~s Helnz~ ~~G ~ ~~s ~~S I. wbom!~' phYs1.~:~. We =n uSe 1t to V:Suallv wan va~ quan:~eS of ~~# W0 3n z~O :rr:n On I. ~ ~ ~ li=:~0 ;:. d~ bCtW0g~rD I. thy 8~;d O~t th.: ~ It is o= bust students, not =r weakest onrush who w:~! be using =~- tOtS I,0 St.,U,d,~r 1,5 Sh3,~.C Of If, =,:d, SCiC~ti5C t=~# thy know how to distin~i~ imp - m u:~:impo~t thinks ~n Age :f they bave n:~r ~~d st:~e at At? The s~ oi~e ~~* Students e~-~v working mth shape' as we ! do. In teaching shape~ espec:~y in ~ workshop Attic ~ teacher :s unlikely to encoun~r ~e balk (~: mot1~\'ation or \be resides= that Aims 8~0 ~~.~ gamy I- #I ~0 t~ 8~t it ~ educatlonal cimies One emotive ~ to anew er questions ~~t the ~iUC3tiO~ \~ue of =~ng ~~pe Is tO hold an open house in ~ ~ ~;~ ~ ~~ ~~ t~t I 03;~ 5000~0 Aide 6~;5 ~ gaging invoiced w:~h the :matenal t50~5431~rC5.

OCR for page 139
^~ ~ Ala ^e ~ ~~ ~ ~ ~~ In ~ bme of ape away lbe may of gape ~i~li1~s ~ eyed Bales far labia. far _mpIe~ ampler Pa is ~-l~ion~ing lobe gum Shy. ~ ~~= the supper is Oh maps En cabs so bang come ~ s ~ ~ ~ a~ Peale Yes 1~ moot of us ~~uld~n~ --e a made .. ~ lay verb. Hey no flower ~1 ~ ~^ loam E Ads; in_> gay me paper ~~ me Nils in Acing me Flies of bang. abed Abed ~ ^~ Bus haul nag ~~ ~1~bi~ Blab lbe~ USA and Me nag Pep ~~- ~ be en~u_d by it cupola gal Eva s^- plo~lio~n of shape ~~u~u1 1be enlist Rectum. I Abeam, OR. ~7 ~' Whelm, ^: L~i~i~ of aria am, 1969. 2. ~ . ~H C. a dam =~ an.- ~ ~~ (~- ^~ A=dc~ of , ~ 1#, as. 3. ala, Ma. ~~ Beg #~, MY: Sp^~d~ 1987 4 Coheir, H.S a. ^~ ~ ~~ lea ad, a: Jan ~1~ ~ gas ~1969. 5. Cage H.S~. ~ art. ~ BY: her, 1973. 6 on, smug Y.. Jr. gaily, Ore 'Dingo,' And 1be 'Sag S~~- ~' ~~ man.- ~~ ~ (1984\ 225~232. 7. ^~=dor, hex ad Span, Owing. Smithy: legions al the aisle Oar ha.- ~^ ~~ ~^ C~~^ ^7Z Redone VA: anions guns ~~_ ^~ I~1) S. ^1~. ^~= ~~, 1610. 9. ~m~ E.H ^~ ~~ ~~ labs, a: ^~1 Universe Pa, 1979. 10. Oats. Baa ad ~~, C.S. #~ ~~ ^~ ~~ ^~, a: a. ~m=, 1987 1 1. ~ ~ ~ ^ ~ ~ 58:1 (19B5), 1 ~ 1~ ~ a- ~ O. 12 golden, ^1~. ~ At. New ha. MY: Columbia Lankily Pass. 1983. 13. Mar Jobs. ~-Su_~puer pickle ~1~ Abe -= inhaler age ha. 30, 1988), 1, 26. 14. ~~1i=~ ~u~ncil of Acts of ~~:bem~ic$. { ADZ? Roston. a: tonal Council of Cab of ~~bom~i~, 1987. anions Council of gabs of ~M~bema~. ^~~ ~~ r~ ^~- ^~# Jason, a: Nastier Council of Cube of ~atb~malics, 1989. 16. snag, Marjorie and ~#, At. ^~ ~~ Amber. Be: Onion sily of ~as~chu~lls Pass ~1977. 17. anti, Redone and ask. ~ . age, Ah: Bib. 1988. 1$. knobs, Calorie and beck, At. ^~ ~{ ~~ ~~> Olin P=P8~l10~) 19. Nobel, adore. ~Symmcl~ fit. In Magi, lslvan (Ed. ~~ ~~ 1989. 15. Ha. ): ~^

OCR for page 139
181 20. I, P. ~ ^~~ ma, at: Lit, Be & Id, 1974. 21. at_ ~~ We. ~ ~ ~~ ~ A man. Ibid, at: 1~. ant am ~ Cd~ aim ~~ of ~~ vita.- ^~ 177 { 19i6), 47~?i. ~~ R. ^~ ^~ ~~ Xe~ at, as: ~1 mar, 19S5. 4_. 23 ..

OCR for page 139