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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
I d4~A,~-~= ~ A~Y ~~ as Shy ts =t su~-~t ~r lit-~;~e and ~~-~s not s~i¢ient ~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 ~~]
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$e fit o~s are su*~6 to mpresent
1~42 ^~ ~ ~_~: ma / \. ~ ~ .^ . \ it/ \ ma: b ~ 1 d ~ / ~ / e ^ v Of .^ . A. \ / ^ FIGUR.E 2. lb~c am only h~ _~; -~m ~~ abed ~ only ~ awns IS of con~nl, _1~ ~1~$ 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.
~.~:= :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 . .
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
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.
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 .~ /
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;,~
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#
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
Add: .. r £ ~ ~Y l - :: : ! . . ~ ~ l 1~ i: ~' ; ~ ~ ^v qua - ~! mI! tI~ the pl=e ~e the sum of Me m~ Of a~es :~ 360~$w wh4 Is ~e =me as ~e tom! num~r Cf dens A~nd =ch ve~~ ~ MA* Copters ~ ~ g0~tA3) a~ 3~A~A~ a plant mth =~ =~e ~d once w(~c 6t )~t <A Dimmer 5~t One of the :most stnki~ ~ ~:t panems of m~ aims is their :* =d this shimmy - :s an i- ~ in their analogs. ~m :~s something ~t wpeats in some ~:~, ~ :s the c=¢ - that Cams that Sense praise* The ~ of ~w ~i.~s by' In pa~ Allho~ some sh~s ~ not at 6~t ~= to. be ~.e Of s~r ~*7 it 1s o~ :~! ~ Mink of them ~ if ~0V I* Or example' I- him ~0 ~ ~= into cite ~t ~7 Dim tt~ symm~es of ~e ~re Id* ¢::s I helps us ~dy the way svmmetnes Parka. Tn pa~icul~ it :~s that svmm~ - is self mn=~* :It is this Of- that we consider Baltic} and that makes svmmet - ~ a*. meantn~l o=~.~:ing pn.~ciple in the =~sis of st=~*~- W~ng ch1:~:n 1~=m gu~te easily to reclaim svmmet~' not only in ~s and butte~:ies-' It ~w in animals, 0~*. :~d utensils' toys' buildings~ and a~s of eve~ :kind* I. can bc Fund almost VIA o~: chi.~n can ~t =~t pleasure' and g.ain ~t insight' by =~ting $~mmetr~1 pat~s and d:, the miss that govem ~A 0~0 Of th~ =~t i~ti~g bit*! I I fit 6~*- plorIng pattems is paper ~- We are . ~ami:liar with the pm1~y
He ~151 Ins 1^ ~ bead ~~ Band gent ~ lad. To slow, Is of~ls,~ pasts ~1- area ~ non =~ ~ ma big ~- Spat #cruets of ~~ amen 1~ of ~De#~.~a~~o~d~~bs,~ Yes of nub _~(s<~of~ aim ~n-~)~,~ by Id un~l~-s. ~61\,~_~~e~ing~^e~on~ bags ~ get ~~ ~~ ~~:~ ~~.~p~n~~ of H~ 2 one ~e; oar fizzes ~ afar PI pages Mule ~im~n~ob~s: ~~ ~~ ~ ~i~= ~ urn ~~ gum Me p~I~ of moron. Kiln pa~iculs<: budding ~ ~~ei-~e Ban ~1~1 ~ ~ Or ~v ~~= ~1e~ As ~~ -logo ants 1~ ~~ ~- ~e~o=~k panels. ~~ USA is much mom gad a toy: in mar Comely. Men one Liar has ma ~ ~~ us: as as ~11~ Milan ~ caliph by 1be Mar ~ used in elm b-1~. Me I is ~~ complex, ~ il 1~00 is Bad be principles of Modern in a minor. em the okay of ~ Hoopla Isis you just need ~ ~1~ Act m~ Ed some d~ Hind Bents ~pl~i~c a. ~ is a In Lao the ~ m~ lamer Boa one Ed, gin ~~ ~ sect ~~ Ah Her. Pie the Beck on a -~> between ~e Sing mat (~_ Ilk. ~ you look in Me maw you ~11~ We was ~ in a ~1~1 anew. A C1tIe ex=~nud~ ~ Boa ~ come awes Sauce loader conb~l~ns man on_. ^~ cyan => in Me Fords of the ~eido=~e~s i~n~r, Sir ~~ B~r~ =a Spew ~ole=- a Knin number of i~nli~I was ~~ ~ ~ cow papa By playing # ~ the min~,ilisnu dUbcuh ~robOd~n ~ Or Erich aa~:esproducethisperR~T kakido ~ pici~ge. By doingsothoy gill babied animpoMenlk~sonin1be ~ dean Judy of e. ReOcctionsgen~e patterns Aid a hinge nu ^ cofsubunits: pad e~nstb~ have Own as Dallas Minor ~~nm~. The rol~ions and=Cecdonscanbeped~nnedon~aherd~otbe~ysl_ingibe ped~holc~appaendyun~ban~d.Fb~n~ly~sucbasys~motm~ ionsistno~asa ~~ Add. ~~ypro~iesofsb~scanbe analyzed ~ sludying1heir Amman ~~u~;indeed, ~rmo~t~b~ ~ cenO~1bis~=le~ basement ~idi~ngpdncipleinibe~udyof~ome- l~.Byusi~ akaleidosc~udentscanu~ndee~ndtbis~n~menlal Ida bydirectexped~c~ilboutma~ngalen~byde10urtbrou~ 1be gland ~str~tal~bmiclangua~in~bic~hi1 is usualkexp~ssed I: 1
152 :^ ]~ ~ \_C~/ MOUSE 12 ~ maps of ~c ~~i~sco~ is dimmed by pa ~~ 1~ bias mains ~ off a~r ~~ in infinity -~ gleam ~1 ~ ~c =~e n 1~ miff is In, ~~ ~1~$ ~ ~t ~~ (~ -~) Ma Add. nOURE 13 ^ pyre case The line on 1be It'd aces indicts 1ba1 1bc c~l~'$ illegal Use lack ~~ of 1be symm~ ~ Be cur.
a: ~3 <Z ~~R,;E 1, 4+ ,iX CUb1,( ~~¢rI:~g =,;O bC .~,0 Heir pap= ~g = Mm of one ~ the ='~'cd=1 Amp: 3~;~33r p~r~$ (~. ~ net ~ t~ ~~e ~~s :s I ~n ably ~ c=~s ~ ~: ~ ~~C =~ ~ t~ W50$C t~ IS t<~ l~t ~ ~C ~~; ~ 3;~ Army ~~ Jo; Age ]~ ~ ~¢ ~~ ~ dI~- t aJ~ ~e dOn~ tIn=~ =d t~ ~ The C~ ~ a~ ~ t~r (~. wlth tM ~ em: d~ =d pa=~! to ~ ~~) )~k ~ ~ pl~ 'of =~spawt ~ o~= 0~ ~= ~~ $;¢r~r ;3 6;~i ~~ S\/r arty his 'tet:~n ~~.~g the p~e =~~ ,v,0u mid see ;3L Gag ,9~= of. ~e =~. The summed of thme~:~1 - ~~s ads to be mo~ Twit =~' but =~y the pnnm:~$ are the ad ~ is ~e tw - ~~ (~. ~7 ¢~07 t50 ~~ of the c~e incises re0~=i-=s ~ two :~ of mI=~: ~~s and. rot~s ~~t thr~ ~~s of .~. Wun~r ~ i=m ~ =~:t do ~~ut ~e sv~met~ of the cud by trying to (Le=~e ~ in ways con$~t ~h ~s Imp Ol~ students can t~ th~g~ ~t the task ~ cha:~ t3~.s ~m.~ Fir ~~],0D, Such demotions aooear in natu:~-~- - ~= thy urov:~e clues ~ the If; ~F0 6~.3~=~~ Age py=~ Berm ~~ ~~ ~ ~ ~~ ~ ~~ ~~ ~~ be an ord:ina~ ~$ ~ ~~t i~ ~~.~s Inn -oh the =~s ~5A' lPhew ~:nmions are consistent h so-so, but not ~- of the Imp of ~e -I The :: ~~r tne add:: :t tums out' is th~ the a~t of dome I- the c~ is I-~-ss svmm~ than its e=~] -=bic form suggests. ConsequendV, the py=e I. is ~ ~e ~h teXtu.~' or ~ decor-~d ~- ~ ~ ~ ~ . O~ne of the more exciting and involve exempts for older st:~ts ~s to make ~ wh~c :~" Me cube :;s divided by :~ m:~r p1~s :into 48 :~t tetm:~" is inked mth :min~~rs or some re~ng paper w-cn as Spar w::n ~ne tr:~-~e belon~ to the c~e space r~d =d the -I- venex ped o~ an entire cube ts generat~ bY the rede=~s Redecti:~ =,~3,{ p3,5~6, `~,t~0 08~3,{6, 0 60~ p8,907 Id.) ~~ .~Y0~~ -~ three of the -I tet=~1 w~s should be =~d so th~ you w:~! be ~~e ~ SCO As ~~t 1, ~ $ b'0WrS h~ O ~ ~ if ~ model o: o~ ~: :~ ~~:~s~ losing - ~~+ If all: we Ieam ~~t Aim:: is to ~~ it, we m~s the Rams point. Sv~met~ ~:s an c~ct not ;~ =uw ;9 wrap st=~s ~~-~rica:~9 Or cxample~ what atomic forces ensure that
154 ~ ^ sit am: \ #) ma. ~ 31~ ~.~ 1 ~~ by using ~~ king ~ Flyers ~ ~c=, gab 1~ ~~c a_ ~1 = c~ vary ad ~1 arias idler Fund Ed lowly un~cd problems: ~ g~ tang tar ~~ In aver Wily ~~ ~ by James ^~= Ed Inca Cam in dabbing Hair distal of lbe Abut of D~^:22 Beer, on Me moaner ~I, s Abut of a demonic size and Cat has to ~ built up ^m smiler units . . He Acing Amens sag likely 10 Speaks Main and again find Hal subunit a~ lily to ~ Plied sym~m~ elects. In older Cost nut builds modular sl~clu~s 1ba1 organize 1bem~ sexes acquit lo Twain ~I=. Repetition of Me Ages lends to lead a~n~menls of modules lbal ~ cat ~mmeldcaL ~l~edm paved a ~~1b of excclIcn1 examples of a~emenls tbal me Red On and Can. Hen you build a cube #1b =~ bow squats by ~1adbing The squaw ~ Deb comers you are coins slowing a shape 1bal Fishes a certain packing armament: it must ~
SH:~^ WN - I::: ~41 :~= l6* CO:~x d~ - a~ f: O~ - m eg uDate~ ~ ~ r=~d w11b ~~.g ~ of ~ ~=,ls~ ~~ - ~ or' ~e ~ awes m~ be jorned ~ ~ beam =~ cOmer. BY i: t~s =:~:~=ion to: Other~, n ~ ~ ~ pO~ (~ 1:~- ~c a~ - ~ chid ~ ~ se~r~a:r go: (~1 5), :n em more th=. one ~d of mg~r palm cant ~ -~and the = vex _ - (share t~ ~! oi: whme ~~= am =:~eral tamales but ~:~ ve~ ~~me~ hem n~ ~! ~ ~~e ~me*~:7 ~e ~r design or ~bmI~oumal ~~ - =~s ~ ~~ =~- -I ~~ of ~~ ~~ of ~=s~ so in Ned= ~ the ~~ I of~ s¢:em~ to I:~ ~~t symmetry to Char =~- un~t stm=~ws ~s ve~ ink W~= are to capsules ~~ con~ mn ~ ~~w agents The =~e :s :~0 of p:~n s~its- t~t ~~p together to ~~ ~ dmed sand ~~n and Cnck r=~- n the =~e of eady X~ ~~i~:~ns ::~to v:ims ~:~e, ~at ~e shells of ma~v views had polyhedra or Whelm! ~~. S~t ~~ ies owed ~M the polyhedra were often ~~' ~ Is su~ed =~t 8,lt~;\rO =^~S ~ th~ 3,~,~=,0~t Of th~ ptOtCl;~j, =1bU.~;~. IBEX; ~~re recedes ~~ models hoe been- found to be ~~=t The con ne~i.~. ~~n pac~g a~ and Semi.! ~.~- in Videos remains an unsolved problem. pm~s ~~h asse 1~d also to n - ' developments in ma~tics. they: ~~e mod to rethink ei;r Monitions and to b:~n the ~~e of Weir invesOg~io~. ~5 Con earlier times We b=mi~:! shapes th~ we ca~ c~s have been ~ s=~e of wonder and a~imtI=.. ~7 do theV have pO - ~~ d~ ~~s won ~~ other n~:~: stmetures ~ not? C- ~~s were the 6m to be studied' at Fist they were thou=t tO be pieces of pe~y - ~~n ice* (:~t is i:~tive I: our ~M "~:~.- comes from the G:~k woM ~p`~, skim means ice*) S~ the s~ teenth centu~, w-ientl~s began to Su~t that the sh~s of c~Is renected an o~, patterned~ i: ~~.~+ -I ~~e the de velo~:nt of mode~m atom:c thee'~ ~ ~s su~ed th~ c-~s are m*.~e of ~~s of aim spheres that rep=~ - the basic pa~.~s of the okatever those might be. ~~er the panicles were reprewmed 85 bias t'~':~5 (~6 ~ 7) Sp60~0 p8~ ~ ~ A ~ ~ "angular:) are stIll ~ m:~t models ~: c~ ~ .,
156 1 I! ~ ~ ! 1~.11 it. IIII+II<~I I ~~i~i~:~i~i~ ..... a. . ,. ,, , ,. , ~ SS'~S~S~i~:~S~ ::$.:: :: :.::::::::::::::::::.::.: :::: :: IS ~~.~#'~; ~~ AS ~~ WAS Sail - \~''>'`~:~'~ SS:SSSS S~SS~S~S~S SO ~~S~S~S~.~S~S~SSSS~S~S~:~S~S~S~SSS~S~S~S~S~S~S~S~:~S~S~S~S~S~S~S~S~S~S~SSSSS~SS:~S~S~S~S~S~SsS~ \~S~i~ ~ ~ S ~ S ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ S ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ v~ ~ \_~7 :~:::~.SS.~SSS~SS.SS~SSS~:} ~~S~S~S~SS~S~S~S~SSS~S~} .SSS~S~SSSSSSS:SSSSS~S~S:SSS:SSS~SSSSS~S~S.S~S~S~SSS~SSSSSSS~S.S~S~S~SSS.SSS~S~S~SSSSSS^'SS~S.SSS~SSS~S~SSS~SSS~S.S~SSSSS.SSS~SSS~S.SSS~SSSSSSSSS~SSS~SSSSS:S~ ~~} .~31 at' ~ !!!!!!!!!!!!!!~!!!!!!!!!!11 11111111111 1111111111111111~113111111~11111 !!!!!!!~.IIII III lIIl!I US S SSSsSSS: ~11 ~~S~'S SSSSSSSSSSSSS 1 A sssss s s IIII IIIIII! #11 1 ~# 1 1 I1 1~31 ~1~ 111l~1!1ll I!1~) I!ll!~!IIl~lllI!!I~II>~!I ~1 ~1~ ~~i~i~i~i~ As ~~S~l~ 1 ~ 1 ~~ 1~ 1~ ~~ ~1 \~ ~~/ ' ''''' ~~~ ~~,~:~i~s~S~ 1 ~~ 1 #1/ ~ ~~S~S~:~S~'~ ~S~SS~S~S~:~S~S~S~:~:~ :S~S:~:~S~S~S~:~S~S~:~S~ SSSS~SSSSS~SSS S~S~SSS SSSSS~SSSSS~SSSSS~SSSSSSS~SSSSSSSSSSS'S~SSSSSSSSS'SSSSSSSSSSS S~.SSSSSSSS Ss:s~sSsSsSsS~S~S~SsSsS'Ss~sS:S RE 17. An ~1822 Icy ~ USA Bout in Mica Atom ~- ~$ 3 in ~ Wing ~il1~ airy =~: backs. Wh~ber ~ use spas or backs, He impose idea is ~~ of an on dedy am. 61 us expl~o~ this a li1de ~~. A one~imension6I is a ~1 of spiels equal spa glow ~ ~line. (~1bou~ ~ can dam ply pea of able set, ~ assume Abe ail ~ on amp.) ~ one~m=~on~ lattices we essentially Mikes ~dng only in 1be spacing between trials. Bul ace am loo basic kinds of l~imensi-~ Was: onc in IBM be agings of 1be m~ lie di~lly one Knowers 1be 01ber fig IBM Hey am shims bo~zonuIly (see Figure 18~. Eat Bins of a Office '4~cupicse a can ion of 1be plane, the Won newer 10 i1 Man lo am of lbe orbed latli~ points. These anions, calm Didchlel domains, display 1be symme1~ of 1be labia in a co~ing brick model. Me O~cH~ domains ~ ~ ~m~n~1~ _ q~- lale~ls or browns; alibis gab Iallice lho MOODS ~u1 each of able points am comment. Slices Escape lbe underlying symmeldos of pa1~1e~s. 0~ ~ onto Men on Akin on 1~0 or 1~ divans s $ of lain paw and
* * - ~ ~ - ~ ~ * e' * ~ ~~ * :~ * ~ * ~ e' ~ ~ ~ * ~ ;~G'cRE ~ 8~ ¢e ~~ cf ~ * ~ ~ ~ ~ Aft - ~~t i3~ hi: I'll ~~ b~ th~r DInCh1:~ d~~ * ~ ~ * ~ * 44' * 4F ~ ~ ~ : wIV~s centered at ~ l=1j~ wIm which An- ~ - ~n ~0 p~ne cl~r lo the coc~=d lat~ t10e po-~:~! t - ^ to =V ~~+ I. 4~ 4* 41: 4~ * Its =~7 ~ ~~) or e, -* ~ ~ ~ bexa~, ~r ~ ~~n la+~e th * 4' ~ ~ are ~t c=~t to =meimps lattices. ~ mI] Brie d~er that ~u =n ~~ ~ Emmett+ -A the -I by ~~g ~e m:~ ~ ons of the mws you. -=n :~k the It 5v t0~ting t5C Airs maim No matter w~ ~u do' tihe ~~:~Y w~ always be of one of the hve t~es sh~= in nose ~: s. it its an important ~= th= ~~W ~Y-~i=~5i0~ =~ting pauem' whe~= it ~ an a=~t of points -of elll:~s or po - Ad* ~ 41~r p=~' or an :~:~:ike ~~ng 0{ t56 pi8~07 08n 60 item 85 ~ 00~:~ati0;n 0[ the Wnch~ iet : asweia.~d ~h ~ lattice that ~~ to one ~ t~ ~.*e Ammo types. This <~e emits ~=s ~ air of ~~=ting qu~ti=~- What ~~d of p~i.~ a~S =n wee create if we m~.~e the Willis 'I other shape~ ~~: sh~s can be fiber whether w~m ~~s to Arm O~W p3~ ~,~t ~ ~ =~: b) O-~V ~ W =t 370 t~C pOS~ ble w~s to extend a=~s to th=e dimensions? It tums out ~~t ther are on~ ~ small number of so:~-uti-~-ns to p:~-ems s~ as these, which -~:ins w~ the "me na~s =~r so often :n 0~r~ 5~$ If* dial- lis-~, h-~mbs, w~' tex:~:~, and t:~d Flu-. Three~dime;~! iatt:ices hew ~~:~ used 5jr m=hemati-~.ians a-:~-d 50~' - wing in tt~ In (~ 7 t~ t~ t~ 0~ tt~ 87 ra:~:~s of atoms in (.~# l~ ttttt Ins chew are l~ Or ~ ~ ~ v ~ ~ # ~ ~ ~ ~ ~ ~ ~ ~ ~ v ~ l tt ~ ~ ~ 0;~3 On~t t~rp05 O! ~~C OO~, It is fit to - ~~timate the imp:- of p~v with cubes and Filer blocks. ~en one vear oldie* Aim building taller and taller towel e~ and watching them ~~t down L: chi:~-n use blocks to build
:~s ~ - : 't ~ . '' ~ - ~ ~ / ~' ~~,~:~= w: ~~y In, - - ~ / ~ <~, nwRE 1.~: :~ =e §w combln~ ~~ of ~~:~t ~~s t`~ ~0 ~:~$~ ~~+ ~~¢ 5~C ~ 3= = - iCS$ - ) ~~ t53D ~¢ ~C lC $ - : aw ~ ~~t :~s :' hou$~' cou~' a~ other -I Wung chosen are likely to have wO^'6 bu~ o~ but theY =n use Sm~ cu~s to build Iarger Ages ~ sm~t c~te c~he is made up of g Baker =~s, ~6 ~~t ~~ is =~ ~ 0[ 27? - - g~g ~~ t~.~5 =~= (~ti~ ~e chid As mme Averse: o:f vol~. Older - I a Amman ~~o I=m a. Iotm play1.~ w1~ cubes Cu~ ~ the prototypl~ Ah:. t:~, and man-v st ~~$ ~6 =~-~l =6 =~, 87~ t=~6 00 )~. ~t t~ ~~ t~ t~ to bulld Lyme out of cubes. :~r example, POP ~~ng ~ rouser oc~n by ~~g w~ cub~ t~r who ~~. ~e I~r y~ m~e wur wgar =~edron (if it isn't toO m=~' the closer: the -Ed ate ~~;h ones 13~:~g po:~h~am cubes- ~s thus ~ s~hi~i~ed te~= :n ~~me measu=~. l! is i: that -I 60~77 t~ t)5 0~0 ~~ ~7 - ~topes'5 re~ to the cube o~f anY dimension as ~e "~re p~.- (~e wo~ "~oW tope" I. to th~ hi.. an~s of pollens and polV h:~C ): :~n An i:mpo~t pmb-~m hi many Fig: is how to divide ~ r~n into compa~s of ^~s shapes. An arch:~:~t or deSiwer p:~.~iti:~s the i:.: of ~ b:~:~.din:g 1:~:~o moms to se - e Gail ~~. we al] t over the most efficient Boa to ~~ck ~ suitcase or lhe t=nk of ~ =~~ ~ ~ ~ =~= ~ icing ~ t j C~ ~ ~ OC~ 35 ;3 p] ~ ~ ~ O ~ BOIL lo; to ~ ~~~ ~~S from ~ si~e =~! t:~- in the earIv st - s of =~ divided :~to =~- =~:is that ~ a~d divided again- ~e sPtud>Y of how dividing cells organize t:~s into tiss:~s a~d then Into organS :s
FOURE 20. ~Re~il=- ~ tiles 1ba1 =n ~ alla Elba 10 ~~ Splits of 1bom~1~$. Hey build cilia ~1 a~ ~1 similar arid 1bal. like tbi$ one. may baa no lattice slipup Such tibias a~ of Hal inle~sl leeway ~~ 1bey spa mop S1~C progenies Fib amp neat diamond Caroline malehal$.
Ace: A: ~ ~~Y s Is ::: T~ ~~ I~s ~::~ mwr Fey ~~:~ met a~ s~ e m~ ~~ p~: Char some =~: d:.s~d ~~.e moments eased for p~= of p=~ms ~ ~~ By l~ became thev pride ~~ ~ far ~ ~~e =d - ~ Is ~ =~- ~:~ ~-~' m~ ~ w=t -m bulld ~ =~h~ Is ~ ~ Aft, aches. W~ Shim we set Be~w cu~ em: ~~ns ad try t8~0 t~ t0~7 i~ ~ ~~ 0~t t~: p=S]~. ]0 ~ ~ ~~ ~] the mimers' ~ ~ ~ be tn=~, bec=~e .~ ~ buDd ~ m~ ~~ -em one pon =d Ah a=0r ~ each eden If =:r htst ~~n ~ ~~, ~ wo~d m~n om O-f p~ sow we oni h~ ~- so t.O bu~ ~ ~~-~ anach ~ man~ ~ = of our hm m=~e (~ 23~' and -~en tO ma~ ~ cIosed ~~mn mu~ ~~d up ~e ~~ sO that the other 0~:~- his *~ ah:= ~ ~~s Of the A.-. m. OCIWO~ Oi ~~Of -~$- We can ~ Gil :~m ~m to d~ ~~:ieS (~, co~) thew t=~5 =~t 5~7 t~t it ~5 imply to note ~~t ~ have ~~ made an IBM d~s=ve~. ~~~ tetrahedron :s ~ wmb`natartal ner w~ - or all: simile wamnIng sho~ that there ar~ two com Nimbi types of pent~, -~m with :~e ~~s (~u ~ 5~d them i:n Flare 6)K ~~= 870 0~ $~en Wpes of he- (~m 2 1 5' :: of :~e ~ cube d:~:r - v there are no mo*~- \ / ._ P I. ~7 \./ ~ .f A/ // {t is ~ cha-~ ~r Students to / :: / ~-~:RE :~. ~~re ar~ s^en com^~:~! S:y~s of oeXah d* : - ~ ~ * 5U ~ * ~ ZO t5C ~ ~ ~ tt *~ ~ ~ C~ S) OO3) ~~O ~ V~ C6~3 -
161 K ot 5~.= 3~6 hi == Get -A their :~:.~c p~ies. If we tm to bul~ ~ =~= .~ - t~edr=~ 0~ 0t 50~5 ~6 ~?~ ~~r s~- ~ ~ ~ - : :~: ~ · * *. ~ . ~ I: . . .. ~ . *, ~ such Zebra ~ =~;~.na~ :.~. It~s belt~ to know th~s :n advance, ~ ~w "W~d spom mv~ po=r ~~;~ ~ ~~t s~r bal~l th appe~d to be :~de ent~.:~ly of hexagons. ~e de~er ~d n~ Rife she h~ dr~ an 1~-~e i. ~e ~:~:~ ~~m cf co - ~~ Chip ~ podded :~s Eu14~m,~i~is~d~ ~ ~ 1y ~ - ~^ Am. w~ Of ~e ~~= of ~~ plus the =~r of ~~s :s equal to ~e n;~= of e~s plus two ~~s c~ be ~~n s~y Fat ~ ~ ;~2 whem ~ :s ~e number of ~= (~r cam) -of the ~7 ~ is t~ ~~ 0{ - It 8~6 ~ is t56 ~~: 0t 66~- (~t i~ 08~ t~ ~~ -I equation Pith she n-~3ks :n F:~ms -l, 6, l-9, ;- 27) Euler's Theorem is eas~ ~ -tLisc~:: (with gu:~, =~' to te~ a~ for more ad~ed stud=~s, n~ dimwit ~ use. ¢e thec=m and ~S m~ =~= and ~~liZat:ions ~ lmpo~t tools {~: ~ ~ ~ mmb:~} pmpe~s of ~~s ~ chid -important wo! in the AWRY of same is rep~:lYon~ I:n ev 6~? i.i~ 35 ~~: 38 in It mathematics, an-d 8~$ ~0 ~~ ~~t 0~.|~ ~ ~ Ems ~~ ~~ ~~ ~;,3~\t ~5 of rep~tions of mode Is p~:~ drawings. The tools of reprewnteti~ 1~e ~e ~~v to und-~.d S~e m~' to read masse` to u~r ~~ ~~f0~57 ~~, 8~ p~7 :0 =~t ~~/ - = t~0i-t i=~5' tO 6~ 800~V 3~6 tO ~~ 00~:10 -king i:~W -+ Ems ~n each caseto determine :~e telat:-~n d:~t l:~s of the ~~e between ~ shape and its :~ma~ or between The simplest ~~on of ~ shape is ~ mode! of ita t~<it Aft 8 a,~;,le scale ~ 8,phen=,: ~~e is ~ m~) of the -A or of :7 0: 0t 8~ pl8~. ~ lobe is not an cxac~ rep:~:~a of ~~e =~7 but an approx*$~ e one that d:ispias~s ce~n ~~:~-es of :he ea~t q~.a:e weld It is approxi:~e b-~-~e it is per~y mu*~d,~ which th~ ea~h {5 00~^ 805i605' it is (~7~6 0~: =~ ~ 5~: ~~t t63t ~~ 0~: But eve~ ch:~d crowds up in this ta.~t cilia appear as linY dots.
INS -ah Cat ~ $ _ \ .~ ~ \_ cy =~ _~#~# ~ ~ -~ $~ _ _~ ^~-_~ _- _~_ 1/ ~ ~_~ ~ ~ ~_ _ . ~ ~~ AGILE 22. ~apm~= use many divans memos of pr~ecling 15~ Ida to cream 0~t maw. The choice of pillion Beings ma of Be map's Matures.
~3 ~~e ~ un~ that ~e - e :s ~ a- of ~e earth. Mm! i: m~s are ~~e in ~e sense t~t they ~~ Coax (LemIs :m awes to =~t ~y~.~S mow:~:~.c]v ~Ma~ a mo~ Baits m~g ~ Minion of: dice `~s am to be ~~' ~~ : merits ct~m ~ion. ..x Inte~g que~m abom Me ~~n between ~~ and Adage ~w, ~r crumple, fin the ~~ of ~~+ ~ ~ ~~ use both glees =d :fl= amp? ~e an~r is Simon* ~~ are u=~T ~ ~~t pu~. Alt~ ~ Awe =d ~ 6~ ~p :~=sent ~e Age thing, }new ~e 0a - 7 ~;~> ~~ IS pt~$ ~~ ~ent~- Flat maps ~ r~=~:t ~~! Anions of we earth qude why Am pm of the sur~ of ~ sphenca1 JO =n be closel-~ ~~oximMed b~ ~ ~~:~- Bm the rep=~n ~s ~~ and wo=e ~ ~~ ~ ~o Ike the ar~ =~ - b~ ~ m~^ The =~ti~ - ~~n Awes arid 0~ ~s :leadis ~~y to w~ ~Ma~ne~l~~ que~io:~. You can,: Mae ~ sphere b~ ~~g up ~ same of p~ so to ~~e 0~ ~~, ~u :: to- p~t ~e hose in s~e ~. Mapma~n use several distant p~on methods (Future 22~+ Thins- ~ Oman of ~e =is an Marx: o~f ~ sphenc~ worse, an ~~im=~n that ~~s wage =d ~~ ~ t~ Notion of the Abbe ~~g ~~d is- ire -~. ~~ 0at =~ ne=~:~v C~5057 =~' 0: 6~* Like me cons:~:~g mow g=~! kinds of :~, ~~ map- meek me compm=~e tar dt C:~~ Age ~ ~6 M+ am mo~ imp for particular pu:~* Shadows ~ ~~m 8~#7 =~5 t~0 =~t familiar =~$ O: mamas' =e : th ~ btIc ~~= ~~ dl~on conto~= as! as si=. ~e lnte~g question is ~ determine ~ AM ~~s of disunions =n ~~, and whv. Wu~g children can Ieam ~ ~=t deal m! obsewi~ ~~r own sh~. To c;;~e ~ s3~, you need ~ :~:~t Muir= (if vou are o~, it is the sun), an ~~t (~3' and ~ screen (~:he ground or a ~~) Th sh ~ I is your p~n onto the ~7 3~6 ~~t t68t p~ i:~5 i~6 depends on the pcsi:tions of the light, ~e ~7 8~ t6~ 5~. ~~: ~~:~ts =n expenment~ v-awing the positions of the :~:i~t, the screen, and the o~t that blocks the flint to pmduce the shadow-- ~m this they can discover which Naperies of shapes ~ p=~d and which
t:~ ~ so' ~ ,~ i' \; ` `~` :: ~ : ~ I'm I: xq~ n~E 23:x A: ~~ -- ~~' a~ a~= ~~:~t bye ctrde on An. == )n dl~= ~~ =e :^t ~~r is And promo Fir 0~7 ~ ~ 00~10 =~5 cam as prom shadows of ~ ~~e ~(Fi~ 233- ~M ~ more advanced :~ ~ -in the of shows ~ m~s in wake onTy ~ oWIme of the map ls retains.. ~m ~S Sperm the p:=,~ = ~~n ~ map of ~ e~ ~ ~~r If. s ~ ~~t be~g mappers ~7 ~~' ~0 56~¢ 5m in =~6 pt6~6 ~~. ~~S, stifle p~' mI~, me and mm~s are o~ ~ few of ~ ~Is ~~~ch Ienses enter our liws. Inhech$ I~= :n --I m=y pnnci~ of ~~ ~~t mn ~ ~m ~ we~ t~1 m ~~= thro~ Chit s~) ash b~- l:n ~, c- and ~n eve~ ~ areas how ~~ with the pr~- Tem of represendng three~:~ shapes on t~ sur . The solutions they have ~nd are, :n maw =~$ ttC ~0 35 e of the ~~" W: example, in Firm 24 the akin is ~~y m~ ~ m~ of the shape he sees be~m thy. Thee device he ~s u~g is ~y to ma~ and =n be used in the ctassr~m with ~d res-~* :~er~e hosed is another example of mapping. -~ ,~ .~r0~,0, d^IaWiOg W~ ~dClV Cam. sashay ve~, few ~~le know how to draw a=~' and, = se~, ~~y no i=~r notice th~ as carefully as they on~ did ~ f~ y=~ a~ there was gee e - ~~t (~r ~~d : have bard when ~~o ~~m dis;~d the Me :~ Io~ of the
1~65 ~. . ~ '! :q _) g: ~ 1__ : ~ ~ A. ~ ~ ~ . ~ A. ~E 24: <~ :~s sket~ Off an a~t m~= ~ m~ of ~e -~w he == - ~m' ~ ale: Mathem~ AsmcimIOn of *MO0D='-t~ app~d on ~ Of its pu t:~ti-~, Is ~v d~ IFS 25~, this e~r ~d Amp de* A At; ~,l t~ ~ was wen ~ ~ ~5 of mathem~i-~^ If aims ~:~ is so wed ~n among, thematImans' ~= ~tiOns *I ~e nsk of w~b believing that the I:ve in 8n E-c:*~li~ imps ~dd (~re 963t I:n his am-on tJic m:~ imps-n Gril~*~* presen -a small ~;~e of his mIl - :~n of b~y ~ tex~k fees (~e ~ ~ 27~* Woki:~;g at; them w ~ : on-e each ~ cringe (~r bo~. At how m~ offs c~d ~ betted Indeed, he merit teachers of mathemat~ =~* ~n d;~ ~ :~e cube? These Ales presumablv -~Id not have occurred were authors and gmphi-c antics mone ~.~:~r ~ the pnnclpl~s and pmctice -of Amid 80:6 p~ - idly ~: Aft? ~ t - bitt dim has As* =~ tO =~05 in t50 ~0 8~6 i*? ~ ~ ~ [=t they a~ essential for ~! stud=~- ~. ~ em. ~ . ~ at, ~ x`N ~,~. ~ BL - 1 &> F `O ~ ^~ .. ~ &~ FI0URE )$ Tne 1~} loges of the MathematlCal Ass=~-~:n of Am¢~ :he old one 15 ~y OX Lhe new one ts accur=e Th~ error ~d detecLIon ~r mam, ~; ~ ,~ >~ ~ ~ ¢], ~ ~ ~ ~ 5 ~ Wry .~9 ~ Hi; ~ ~ ~ ~ I; ~ ~ t3~t ~ ~ 3~ ~ [might Aft. 0; 3; A; ~ ]~= (limit 0#~:~.t ~If '$ $~6 5~\ p3~) 0: 0~¢ t~^ ~ ~A ~
166 ^~ ~ \_~ EGGS 26. ~ ~ ~ C. ~# ~ ~ ~~ _- on ~ claim of I ~ ~~cnsi=~ ~= on s ~~cnsi=~ pig Spew ^~t 1958 ~.C. Mar Guam, ma, Holland. ~ -~ ~~ ^ Lee ~ons~bon lf lhe ~iSl,S problem is lo ~p=~nl a 1b~e~ension~ sb~ on O~ so 1be views problem is 10 Size -~ shape the imp is supposed lo ~1. A vish lo an am Blew is an Eloise in imps consl~clion. So is the physicians task of Lading ~ X-~ or a Bag SC~iS1'S ~~ of inle~li~ pbo10~pbs of the surface of Ace.
$:. = :< ~ ~ / I:/ ~ A: - ~ An . f ~ I; \ ~') . it_ I' A fit to'/ ~C~\ MA HA ~ A_ - . W f~ BY FmU~ 27~: ~ ~v ' - no. - ~, ~ Con, ~ =~d ~ - god. wl~ ~'A 3~ ~ 5~ The Ant the X:~-' and ~e ph~= =e maps of Capes which we need to be Abe to ~ "~n =~- Th:s so cIosels~ rotated to the problem ~ ~sual:~Mion' is of Hat :mpodance in the stu~ of shaper but tt has not ~ Om:~d tn ~ w=~r th~ =:n ~ used tn sch-I. :~= is ~ ~. ~ bog to stud eras of mad e weallb of mateni1 related to sh~, =~ss sections' and --I t~5 A }~ chip t~ t~5 ~.0 t~t is ~ (~5 8~6 ~ ~ ~ i ~i =~, ~s could Ieam db~ c:~tena [fir deciding whe~r ~ p tion is pmpe~v dr~'n (ice-, whether ~ diagram :s in ~t ~ In th ee ~.mens~ I Whist amp. ices. t~0 principles of the stermsmpe and - t s~ic pairs ~:r thre - Dim- to us. They c~ ~so Ie~ to- deduce Gamete and topol~! pmpen:es of ~-~sion~ ~e - m 115 tw~.~ wpresenm:~. ~ diwussio-n of claim: illusions at ":m~ib~- h~s =n Tead to malls' impotent India. Those familiar w:th the co~ina:~ pmpe~i~ of polV506= =n ~y their ~tion do.. pew= opal and can tm their s~ at :~ting c~ding thre~i~:~sio:~l £~. The computer is not ~ substitute or real 4;~dimens:~ mod Ace. ~ ~ She :~: =~, Arch ~ so~d 3--rD images, are meani:~fd! onI~ if the viewer has extens:ve pnor expene:~ce with th~- d:~sio:~ st~" On the other hands computer =~-~s c~ be ~Y0i~8rt]~,.g t~ ~;5 3~.6 I;, ~0 5~,5 i~tr0~$~t 7,0. the stud~ of ~Yh2~" ~00 ~8= i;: thus te invaluable in the stud> of shape am should ~ 0,5~d wYhe:n appropnate" .~:~, Aft ~n SHOUT know so:~ng of the geo-~~ ~at undedics computer Chick al:! coordinate £~ ~ cntte~:~- In summa~' the creation of :~s and the reconstmct:~on of shapes Am their images :s ce:~l to the stud~ of shape. All of the :~v
Ace: A~W Ar:~= To \~:~Y fads -of m^~n can b~ I ~~* t6~0 00~t of =~* ~ 3~ 0~ 0= ~07 ~~' 56~' ~~ :. sew ~ lense~ dams pr-~=d b~ dream ~~ Army by ~~' Ad :~ Id ~ the =~e or pJimo~c army -A ~~* As incrm~ngl:y deta~ Wages of ~ w~ }A ~e Act sm~~ am ~e Army hidden alit ma~ vI~le ~ m~ techn~y~ e :~d to understand his bm~d Arced of m~g ~~= ~ ~~ ~~. M=~ng :s ~ m~or Verne of co~ :;~emancs bec=~ :t pride ~ =~l =d icing wad ~ Amaze r - ~~s amo sh~s =d panes -I very ~~ ones)- It aim beds -as to image our class: systems pmm=. C~= =d I -=n be described in the Adage of m~- W: =~:$ ~6 5~5 :0 ure 3a c~ ~ t:~ed ~~o om another ~ ~ mewing t~ Ads ~r combin~ ~~:, t~ sha~s in Figure ~ ~ related beams ~ ba-w ~e same s~ of Ares (whim am -matings of oh's ~ jam on ~ themsel;~' the shapes ~ :Figure 3£ C8~ 60 t7~6 into one another by ~ mapping that is ~ =~tin:~s de~* STEAL Msu~:izat:-~n is ~ bnoad s~= w:~ I for m~ a~s of our :liv=. It is =~ imps to ~! of me and he been ~ thmu~ut hi~*,,. M~atics made ~ ~at advan~ mth the i~~:~n of nurnerals~ which am visual :*~ti-~s of numbed. Ce~ainI~ o~ -of~the m~or mathem~i-~1 achi~men~ -~he iast -l bundred ~ was the d - ~~.~t of analytic gecmet~, -~ endued us ~ combine :1 and forma1~:~ thought. OW:ousIv, visu~ation is w~ i- ::n ~e stu~ of shape. But :t ~s also I for all of ma~emat~* To ~;~y 0~07 ~~ ~~ t~ 566 it' t~ 8tU67 ~7 ~~ 0~=i~0 ~8 Ibid ~~ti0~~ ~t t~ to =sp t~ concept of higher dimension b~ dra~v:ng pitches and ty making morels* Even ~e p:r~ies-~f n:umbe~*s can ~ :~um:~d bY irlSUdl ~iO32~tI18t iS Wh8t 1;~0 number line :s ~~- But :t :s not t=e that -we inst:~ctivelv How Moor tO "~- 8;DV =~ than we :instinctivel~r :~w how~ to swim* V~n :s ~ too] that must ~ cultivated for =~l and intelligent use* {t may be he;~-~! to :~! ~ Afar 016 51~ about ~8iiiCO'$ Id0[ mounta:~s and cmters on the mom* ~ dlscoN{r0~' that 5-01~. t~ `~= -my the way we-vie~r the un:~se and our place in it. "~owing ~07 8~5 of the Middle Ages and the -I bell - ~d that the moon ~s ~ perfect sphere- the protot:~aI sh~e not onIv of the visible p:~:~-~ts and stars tut of the entire universe,- -~:~-~lain$ the historian Samue:! Ed~n ~
16-9 The pmblem.~. ~~' ~s n~ m~ ~~e ~ ~~ ^~h ~ a=~ - but ~ exp~n ~~ med appearance of :ts =~, ~~ 05~ ~~" as :~t ~~d At. :~e adherent au~ - :~d ~ - ~~y ~:~g t6~ t56 i~sf~X 5~ w=~ ~ ~~ ~~ ~~ t~ la8~ ~5 of t~ eat amp :;: ~~m ~e ~ ~ =~d Loft transparent ~xb~= mth some mte~ Sons I: am~s of I:~. O~o ~Ln.e3 a=~ ex~:s d =~ion Maxi the seam of ~~e ~~n :s Aim and -~1,~v ~:~.~^cal. as: a: ~e~nx~ of ph0~s 60~0 luff (~ ~~ 0~X~ ~~ ~~) ' ~ ~, '6W 15 ~~ = 3~6 ~) 0: =~ =~ p=~,=5'fr~ Ming 'amp ~~ ~0 - 0 ~ ~¢ =~, ~~ t~ s of m~d ~ep~- - T~s H~ Am: ~ ~~h ~~= - in: ~ ~m ~ :~g thr~ ~ ~~e ~ the In, ~e =~e ~ that :~o his-dis~. Hamot,~ sketches those, bawler, th~ ~e "~]f8~. pro ~ ~ d-d ~ 1~ ~~ ~ X ~ ~ ~ FIX (n 28~* How could it hapmn that HamOt and Galileo' 100~g at the ~e jam: tt(~ :~0 ~.~5'0~, 6~6 ~~t "~- t60 58~0 t:~-~? Few ~0 Ifs ~0 ~{ i, by- th:s fa~ alone :s n~ very Nominal fix ~~ ~5 3f =~ p0~0 =850~ ~00 W35 Of Hi: ~be t~0 ~~¢ 0{ 90~ 306 65~, the =~der~ng of :~t and sh~. Thus "Galileo d~d ~~ed have the n~t theoretic! - ~~ ~r soh,:~g t~ nddIe of the moods `~nge 5~.7 I+~ ~ x ~ he bmt to hIs teles=pe ~ Spec:~holder~s share, (~s S.~- Govern would 53~7 that is, ~ OVCSi=t 86~6 ~ EXAM unsmooth sphere of the moon :~:~d bsY ~e 3~75 :~k:~ i]~- . . .~. .. :-: - ..~. ::: · - . i....... ::::i:: ~ .. ...................... .... .... , OURE 28 H~x~5 and oai-~;~o ~ sk,~s of t:~e ~-.w. in,, ~.^ \\ ~~ ''', . x ~ / /
=W A: Do: w~Y nG:~:~s 29 ~e ~~= ~ ~ If: w~e on ~ =~w ~ ~r :~e 6m U~ *D *.. ~~$ *I - W~ ptO6~6 t: ~ ~~ tODO~ =~^ ~~ ~6 ~ , A ~ , , ~ ~ ~ ~ ~ ~ K A {n eaC. ~ nn~ "~ilec's telewopic ~~s opened ~e em ~ Eumpeans ~~- W60~" =~= ~~. ~67 85 hid -$ - ~, "~= Hamm `~' shed c=~m once ~ was ~~e ~~e Ho=~s o5~= Tofu we all =e moun:~:~s a~ vamps w~n we Iook at the moon. But wand we s~ Hem ::~f we didn't dream ~~- what we were supposed to =09. And w~t ~ we "$~- w~n ~ Io~ at the imps pm=~ed to us ~ mm1~m technology? The ~~ "lbeholdeKs shank is : 0880~] t06~ ~ lt ~5 if ~75 tight. "~t t~ Am;= is ~ aims seen thrum an c1~on micmsc~e, ~ dista:~; ~~=y explored by radic t0~07 0t ~ ~5 ~~ t~ ~ ~6 ~ =~5 0: ~~* xnsla~ imp an :im~- wn:~es Ham Canaan wn Band ail ~ remet we of ~e Sd=~ ~ - n B-~r ~~s on to ~~*.t out that thIs tmnd~n must be done bY the educated ~ as well as by the i:~:~er~ wo~gs of the computer. ~ case in point is the filet a:~:i~e Age of the I benz=e ~.~87 aims ~5 pmduced ~r the 6:~t time in 1988 (F:~e 293- Can wu =e ~e In*? Or do ~'ou see some :~v donuts~ Or sphe~ i03t t=~81089 80i6~ti~5 =~6 ~-~8 t~ fin the tnan~r tm=s of the he:- s~e became they Id ~~ that th~ ~~e the~. : ~ ~06 0{ ~ ii i tn his ap~) tItI~ I'll Thinking ~
In:= -A Ia~ ~ visual t=:~g :n ~~ ~;~= ~~ I; ~~ : the a,~.~s - ~~ of~ Or ~3 Act. ~:~; Id; ~~t a~ 'my ~ -of ~ t~ WO~ 0{ Stats v:~10 ~ the 0nq ~ ~ m~ ~ ~ ~ ~ ~ K. ~ ~ K ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ whim C~~. 5~ drew ~ ==t ~~( 8~O 50~ t\~ In H6 0~6 ~ ~ ~~m do no: read ~ 1~= a:;~d ~.~= I- ~:~ alum enCe. Pe~ps th1s ts m, but the c~:m ts s~; .~" SnoW's w~= tha~t "~e ~~g point" of ~~e ~~d ~ "~t to produce cr=~:~w ~~- ~s tO I~,0= ~e ~~ ~:~:~p of the two-. L~ the view -~=e ~ks ab~ I: but nO one d~ much ~~t it. VIs~i~n is not ~ s1:~e m~^ ll ~ ~ ~ - ~~' pro - v ~e domain of physiology and Scholl =d shy n~ v~ell u~e~- ?~ne~.~, It is ~ to t~h shape as an :~:~ant h:~t Limp :~D ~,~g p~5 0t ark ~~, l ne stm~: w~ To w~. At:* tO Vi$~*~C 15 tO p7 - id - = ~ ~ =~t 6~6 Ot hands on ex=~:~= m~ ~~es ~ many Hn~* *A wnom stu~ of item non ~~d ~~o ~ ~ nep ::n the n=t directing DIE ISS~s Streets chord I== to :~ze ~e pmtems Of shape' to un60> ~d the pnn~s ~~t govem their con~:~' =d to ~ able to ma- easily b=k =d ~~h - ~n shap~ and their ~ages-. Ad; e *I o: sha~ s=ms to ~ been the crams of ~it:~=l =W ~ ~ Knit an~ ~~ ~~,~ ~0'~ bait ma:~$ of the National Council of -I O:f sIathemat:ics rene~ ~ eme~;~g consensus that this situation mum be impmved~ . The stut of some :~ be more than are sum ot :~s pa~, an I: Wand wew Of 55~e =n help accen~ ~e whole sweet One pos~e approach is 1:~ by the chart in ~~e 30. ~ ~:~:~s R~hink:i:~g the su~t as ~ whole provides us with an oppo-~nitv to ~~ sub~:~ co:~s b~n :~e Amp of some an~ ~~e m~e of shape in the real woody We can take senou§1y Ambe~m~s plea four i*~ti:~g a~ and sciences We =n at50 :~0 tt:0 ~- 5~ of our Demo technolo~- The p:~cipl-~s off the electmn micro sc~' the md:io telescope, and ultrasOund am :~ot wholb~ beyond the sc~ of the WI: cumculum, him school students can' if ace wash, Ie~ the fin neceSsa~,,,~ tO u.~.~:~d the action of these and other :~= imaged techniques. Indeed ~ ~~s on Shape makes manly aspects of modem technology :~h more accessible than :s co:mmontv s:~- Here are just three :::
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 $b~ abed pates I Ad cam ~ -~ ma apse as an milt collection of quite Isle loci=.
Ad: I: -:v nGU~ hi- Me s~= ~ ~~e ~ ~ lt X ~ ~ ~ ~ ~ ~ . Th~ 15 ~ tM ~~ Of diamond asks ~= ~~s ~ - = ~~s x of ~~:7x - :~5 0t :~t 563~5 - ~= TV ~~s come eas~ ~ ~~t The SiE=n -my Which. bas trans~:~d Me indu:~li.~d wOdd lD iust ~ few dec~s, is bash on ~ ~~re that is ~ camer Of tucre~ly mad circuits. ^~ ~e circuits shed are complex, ~ ~ *~x=~ that hou=s them his -a s1:~e mo~r stm-~. ~r -~ample' bins silicon Is built of I~d z~g ; noes (~ 31)' ~~h am =~ to make end ~~ ~~ ~~" : the silicon ~~= the nn~ aw i~:~ed to fo~ ~~-~:ke pol"~+ ~:~ schoo! ~~n c~ ieam to build and identify these su at middIe whoo:l children =n Ieam to p~ them to~' a~ heir school ~.~dren can stu~ the ~~:ion b~n the all-icon st-~= =d the prope~s that make ~t so use~. The ~T scan =d other ~~s of =~ed 1~ recon~ n h~e revolut:~:ized m-~: dig:- in recent "ma. Whim hi 8~ 05: ~ 6> ~~> i~ 80 exer~ A seen is an ~~e in wc=~:~ting images - m t:~cil* cmss se~. t::~e the c:~uit~+ cn ~ s:~-~n chid the Ames r=~n used in this t-Cchnolo~x 1s ~ com~ex pr~ss ~t the simple~ ge~-~al ~ - that underlie it are easily unde*~. Hew ~~n ~ 6~d that the same ~~c pnn-ciple$ 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
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~:~ te¢h;~* 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<e spro:~t by- a 1~:~! =~.- Then these branches themselves sp-~t branches' the b:~:~s of the branches branch, and so- forth (:~re :~:~- The ~:It is ~ stwcture :n which ~ certain [~:~re- b-~ch::~g-is :~in~' repea~d on ~ sma:~r and smaller 803~# :t this p-~-~s could be repeated i: the result would be ~ self similar 51~765 :~ t~0 5~0 is ~ ~) 3~ 8~ =~), 51~ 0t Its development. ~. * ~· (K ~
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~ '$:~r other t~ of ~~n a - ~~. slWatIOnS. All of thaw tC880OS are ~ ~id. ones, but there is ~ Go omission. ~e pnncipal reamn for tea~:~g rim ou~t surely to be that ~ is of pro~nd ~mpo-~ce in un~:~ng sh~. Mean - ~~, Outside the halls of education the co:~r revolution is Lipidic chan~ng the wodd in which we I:~. These changes aw placing :~ew demands on the cumculu:~7 dem=~s that are just ~~:~ni:~g to be heard in the whools. The r~ion :n ~e stud-v of shape and form =860 p05~6 TV t56 :~et =~ts that ~ ~ ~ ~~r compromise for Be but ~ new and coherent mathematics cumcu:~um that :~m shape tnto the ent~= cou~ of stunts
0~$ ~ ~~ ~ {-?~ `~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~$ ~-~.
: 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:
~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~ $~5 t~ 3~6 t58t =:~S &~6 "manipu::~-~- are v~e t=~s :n the ciassr~^ But too ohen one :3he~ the ::~ament that "if ~ A ~ ~ ce~ art. ~~ Ate ~~ ~ 63~6 t~ ~~ tt0= l8~t 0~- Axis ~~ 2~ =~g t~0 l~t Tut ~~ - t i. assum~l'0~. First, that ~~ss ~1 sh~e is t~he main thing that ~e team - m ~~s ~~d that it can all be 1~d :n ~ 506001* )~ ~0 500~ 0~0 0~ (~7 ~~ ~0 50~- them all. {5i5- 0: 00~' t~ ~~- tt~ ~6 ~0 mat ~ Beet t0~ ~n t~e ~~V Of SrOlUt~g3~ CO~~ Si=:~, Wild =.~: Sit:~- iFhe ~~. assu.~:}~:~n ls tha~t the main pumice in starving ~ mode! :s to deve}~p our p-~wYe~ of abst=~t reasoning' hem the model plays the
.~^ Me of ~~ wheels on a: b~- ~~V ~ ~t :~r ~ - tO u~d the wn=~ in - - a:~ =~ mp~s In: 00~= 0t ~ cube. ~ ~n beet ~5 :s =~$~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:;<iom~s arm ~bst~ns will ~d ~ hands~' pEd - = ~~m Iess dimwit and more ~~. q~ mI~p tIOn Ices on the 3 side of the cO1nthe wicLe~re~ bide that more a~ed studems dO n~ need to ~~v Ale. We ~ not he to A;; ~~r than to~s ;~#~[ ~t - -i60~0 e ~~y of Is belier USED Pictures Solve the once Inso:~:~- pmcIaimed the h=~e of ~ re=~t mMe on ~ - m p of ~e ~ W~ 77~m i: Sclent:~tS who am us1ng the n~ su~:~er ~~cs Sav th~ bY VIewing |~5 I: 05t ~~' ~ I :~ \#0 t~0 ~> ~~:~ ~~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.
^~ ~ 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. ): ~^
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 ..