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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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Suggested Citation:"Shape." National Research Council. 1990. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: The National Academies Press. doi: 10.17226/1532.
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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 fin—m~. 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~ $~: ~ C0=i~ ~~- ~~ in ~~ =~t t00~0 ~ Me ~~s have pmpen~= that Ah them—m miter ~~ts and be ~le to ~~ze - or dist:~shing £- -in ~ ~ambi~s ~. Seco:~-~- you must be ~le to u~ these cntena to tnd out precisely dice o~ts ~~ them. ~ one ~~s just how the ancients made their d:~, but It is eaSy ~r yW~ -mailmen tod~, e~eciallv :if they have ~~ar pol;~3~-S to pl~ Ails to convince ~~:~s that the Bit of ~~: polyhedm lS complete (Figure 24* The k~ i- of math~] cIass:~=ion wew ~~ in use ah- of arms ~,~. by ~ id t~ obj~=s :n that class. What ho changed thmu~ut the :~057 8~6 ~~} CO~ti~6 tO C~' 3~-0 t6C ~~ Ot C53:~:~tiO~$ 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 as—se 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 ok—atever 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 =me——imps 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$~' cou~' a~ other -I Wung chosen are likely to have wO^'6 bu~ o~ but theY =n use Sm~ cu~s to build Iarger Ages ~ sm~t c~te c~he is made up of g Baker =~s, ~6 ~~t ~~ is =~ ~ 0[ 27? - - g~g ~~ t~.~5 =~= (~ti~ ~e chid As mme Averse: o:f vol~. Older - I a Amman ~~o I=m a. Iot—m play1.~ w1~ cubes Cu~ ~ the prototypl~ Ah:. t:~, and man-v st ~~$ ~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~a—m 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 po—n =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 case—to 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~5—057 =~' 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'$ Id—0[ 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 ~~ m—ed appearance of :ts =~, ~~ 05~ ~~" as :~t ~~d At. :~e adherent au~ - :~d ~ - ~~y ~:~g t6~ t56 i~sf~X 5~ w=~ ~ ~~ ~~ ~~ t~ la—8~ ~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 bm—t 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 polyhedron—we :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! ah—d 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~' p—Ed - = ~~m Iess dimwit and more ~~. q~ mI~p tIOn Ices on the 3 side of the cO1n—the 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 ..

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What mathematics should be learned by today's young people as well as tomorrow's workforce? On the Shoulders of Giants is a vision of richness of mathematics expressed in essays on change, dimension, quantity, shape, and uncertainty, each of which illustrate fundamental strands for school mathematics. These essays expand on the idea of mathematics as the language and science of patterns, allowing us to realize the importance of providing hands-on experience and the development of a curriculum that will enable students to apply their knowledge to diverse numerical problems.

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