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Suggested Citation:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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:"Dimension." 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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Page 60

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13 ~ ~ ~ ~ ~ ~ #~ #>a ~ ~~ : 1~0~0~ One budded Id E~ year ~> ^~hcb Corbel (~= 1~: Name of He ~ Kings Levied a ~1 of Is 10 inlay dug children to nations of ~~e1~ in ~~ distant dons. His philology ~ clear: if cb^=n Bug ~ stimuli lo Uses geometric objects dam ~c eddied stage of tbei~r bunion 1bese ideas would came bad to them amen and In duff 1be fume of their ~ __ ~ _ _ __ ~ _ _ ~ =~oli~, Aging ~~ each new lees of ~phi~i~=don. The Edit meads association of sb~ _ Aces ~ ~ buses shoot 1~1 would become mom bang as quints Tampa nag 5~111S 1~ antis magic and mcasu~m~nl and 1~ in mom ~~ ha and geometry. in off lo peplum 1be im~nalions of gist hug sludenls, F~~1 present 1bem Tab a sequel of Then Sects far lbeir play in the Cbild~n's Ones. Only tear gold 1~ lessons of 1ha1 directed set of play evidences be hauled in10 concepts and An Iaer Realized info malbemali^1 expressions. The impo~an1 ibid As 10 introduce SUMS 10 WAS tba1 abed could app~bend Ed 10 enough them 0 obsess and recognize oboe Ads in all of 1beir Evidences. In lbiS any Abe could Aver tube Sillily of visualization, so im.po~n1 in applying matbemalics 10 both scientific and 8~iS1iC pundits. F^bo1 bean mild offs Am the mosl concrete paw of matbe- matics: -balls, cubes, and cylinder. He proceeded 10 ~ hider levc1 of Lion by p~senling 1be cbild~ren ails trays cad ~ palmers of 11

12 ~~ . ~~ i: i) , ~ .~ a: Ed: ::.::..: At, ~~ IS S ~ :: . :: .. MOORE 1. Comb ~I, in of kin, urn ~o~dc OHS 10 M child~'s minions. files. dew be Id ~~b~or in10 ~S1~C1iOn ~ inking collections of stacks of ding lambs, 10 be plow in daisies ~1 Weld ul~m~ely ~ flamed lo Humor palmers. ~ . , ^ ~ , we In ~c~lze some o1 gas 1~^ in matedals 1ba1 we End in 1~'S kinde en cIass~ms. gem ~ stiff h~ blocks far Skins am film far Saline pagans on walers. ~ oPen, bong abed flays a~ IeL babied Ben chid pass in10 1be Odors Todd em abed ^ ~~ maw ^ ~ urn ~ ~~k e=~ ds=: but a student is laud 10 me albino Dimensional glean kindenga ^ n ju~dorb~ghscho<J A1fha1( melee ndgb1bea brief mention ofarca otplane Dgures~ohe~ merely as anilIuslration off mulas for measurement. Tb~en lbe student must gait untilbigb school before any fu~berl~bougb1~is Ghan lo the ~ fld oSplasc Mom Cry. Tao ~~cmlions~golbe~ba~dy souls So made itibrougb1beyearof formalgeometry~ropennilled 10 ~e~nlertbetbi~d dinnensi~oni~nastill more fonnalized semester ofso~lid geometry. Then cu~ndcula cba~god.

¢~ t~= :~ mth all of:~.mic ~~) wew su~ posed to be ~1 mto ~ -~mgIe geometry -cur- ^M too ohen ~ Did ~~w mmpo~s ~ I; me~ as ~~ taw t~= ~ ~ ~~ -I ~~o h~ ~ bit of 1~ t:~. -go w' ~d ~~w ~~ ~~ - m ~ ~~ - Am- :.n ~~.~. In. the prepay msh to prepam Bus floral them ~ A- ~ m~ :~ -am me~ of ~~T ~~e of o~ ~ ~~= ~ now ~~ ~ =~ ~~hy tO -ap~ec:~ ~~t d:~s At ~o Us. Altho~ Our ~~d ts t - ~~, ~~ Of ~ media, as lt harems, am ~~+ bb~3~.s b—is m~ -telev~s -~M =~:~ screens. ~ ~l ~ ~ Cat deal of e~ Iea:~ng how to intern - such planar:-~r~sual ~~:~, off In: orb to hem us ~ - threed~s ~ I~ 1n a~ Woo we do Eve to ~~w how tw~m-~! ~~ imerad. ~~= WE # ~ pmvIdes ~ Bees., miud.e to u.~.~d ~y -~.r ~ din ~~ :t I ~6 ~~ ~ ~6 ~ 0~{ )~ - ~ idiot: t50 of ~ ~n i~ ~mens:~n—the Dew number =d Amy—intermix :~ the me :~e =d powered! walk The ~~ ew of the -I line t=~s b=~ti~N I-~-~o ~e ~~' bmb in :ts cTass:~1 ~~ and in tibe an~:~-c Am of Air pam. ~e S10n ~ ~ as into ou.~-~.e ~~n mtt Awed inside- ~e Bin 8~ Here ts an eXcl:~g theme t~t -is w~.~h a, and pa~ ,0n tO our ~~. ~e momentum t~ bn~ uS ~~m one to two =d up :~to thwe dimensions does not -stop that The -in is ~=r -mere a~ ~~r d~:m=sions waiting to be 0~0'6# Ma~-em~ics -is the ~y to the el~r that m~s them accessible. Th0 ~~t di=-~n, in p8~7 iS 0~0 0[ 0Ut ~~St I :~ as we leam ~ good de~ ~~t our ~ ia~ and cultism bY stu-~:ng the hangup and cult~re of -other countr:~s, ~ we can be~ ~~ tO 89~0 Large thinks ant =: OWE "~ ~ WO~d ~ -I that ca~Y ~= am to the ~~h d:~sion~ Alth-~h we cannot explore hider Pi I t~+ 8~¢ 80~Itti6 tO 0O ~ ~ k ~ =~= technoTo~r -~0 8~0 :3m07*6 t~ 00t V1SiO as w?~:~:~-

l:4 A. ~ N~y I. into 1~ ~n m~= ~' .~ a~ tn~ is ~ of I.~g ~- 13~' ~ ~ mIT .~ qut~y Setde mm ~ A P~5 of =~ 0~ ~ leakage, ~e~c~vel&t -A de - ~t of .~r pO~.. if ~ ~.~ ~s ... e~V to odor I=~ he or she mi! expertly :~ch Amp :n i~w ~ s~ - ton~. M~ ~e ~e :~ tme~ mlh~ -,~t tO ~ emat~ ~ Tf we wall anti! ~- thaws develo - ~ ~a meat hi K ~ (~6 ~ ~t =~ =~60~) -I ~ e~ra~ Hem to tHnk ~ Hi -A and:~e IBM n different dimensions' ~ m~ ~ ?~iVI~g them of :~e ~e ~ K ~ Get ~s O~s shO~d alw~ be ~ A~ :~f wace =d - ~:e sbould ~ ~ ~ continuing ~ Of my. =~e ln - :~ at a] I~:~- :~ such as ~g quantity= and relating the = mn mme in gmd h~+ But t~ shy m~ well afar e ~e - en ~ child first becomes ~ of ~t ~.s off m~umment. Too oft~' the fim time ~ student is cncou=ed ~ ~:~:k atom what wI~e ~:~s its the same k't th~ be air she is ~n Imp.. ~r Me voluble of a.. sphere= or a. cone. ~ cncoum~ pu: +n the I~-of ~t-a ~ need ~ ~od deal mow "~- thm~u ~ the s~.~l ~ ~e ~ =d ~t ~d include pm mIld as well ~ W~:~= ~:~ - Froeb~ =d bls mile~=s created ~! offs ~om As ;~e tO t~, pu~y wood, p=~' and ~ roVe on. t~ ~hs in ma~V ears-svith i~:~c =d v¢~- w:~h =e and me not to mention with the powered! c;~:= ~:~:~:~. ~e ed~:~:~*~s te~ "~nipuiatives-~m maters_ ~es on new many-en w*e c~ put in ~nt of a vou~ ~nt ~ too! tO =~O OCt O~Y Si~C ~m;~.S butt. ~O the Arty gCO=~ Of hams;: dimensional ~. T:f we caw adbout ~ our ~en w~ the ~ ~ ~ ~ $~= U} 811~g :~ u:~S I've ~s ~r our - - MEASURING VO=~ ~ st~s never item ~t Volumes because they do no:: :~e It :~t p:1~e ~omet~.~. l50$C - O 00 I- rC&~t caicutus 5v ~ 6036~g msh that Ie~s :litt:le or n~o dime ~r the ~nd of ~omet:~1 thinking on which c~1~s thrives" Ca.~:1~s is :~. the time when stud~s should o~:~g ~ heir fi rs: seno ~ ~::b ~ ~ k: ng abo At: ~- ~m e~ ~, Ra ~ he~~ ~ ~s:~1d be

DO - - 1:5 :* :~t am .. to ~e baw a~d :~t of the ~. ~e =~ of w~ of =nsideraticn of Scrawny ~=- t.~+ ~n ~ ~m 6~b =~s ~e ~! jum6~n of the fawn ~: the V^=C 0; ~ =~C O: ~ Sp60~$ ~ 0 ex~' ~:~g ~ promise ::~t ~ ~ ~ expends he ~ she :has ~ m~ cones =d sphems aH ~e way ~ = ~ ~' begin: :n k~- ~'s y~ ~ms spent ~ great deal Of hme I wMe:r so D~ shako. c~: ~d ~.t :~^ ~ ~ smack: ~d ~ :~ c~on ~auc~sh~m m~t ~n arm.; ~f wading, A.. down* W: ex.~amp~' bOw ~..a~ conI~ =.~s ca~upw~:~t t~ ~ Ace. ~ a ~ of su~ c~S I. 2~, .~ ~.t c~ perform ~e expenment* The I: ~s ~e wps. we =~ test thIs owt ~ over again ~ *I hm~t ~ ~ ~, ~r ~ ~m 15 ~ili.~r m ia~ of ~' n=d this :~tiOn:~ be m~ in te~s of: o -~me ~ one~ of =~r Bill Ia~r, that wIMi~s~ c ~ ~ ~ ~- the volume of ~e cone- is one~ ~ area of I,5e 15~e mO,1~l,,~1,0d b\t the height. By ~stI.~e ~t wi~W shoulder -I. - n o~ ~ din the sh~s :~e square b~d p~ a= be ~ w7~h He ~d - m one Squ~ m:~m of ~ ~e b=e ~ h - t (~e 31. ~n i: ~e b~ :s flare this relationship is tme. We don't ~ hew to h^e the =~: of the c=e o~ the =~r of ~ base' a: th~ t~C 5850 CVO~ t8$ ~ =~t ^~i thIS understanding, c~ we place b - ~m the ~t h~ - ~n seen ~ - action, Iet ~e ~ nu - ~r trike =* - Is W.Da ~6 HA ~t )~ ~ =~= 0~9 6~8 t~0 psrt3~$ - ~= base and: h:~t mawL t~= of th~ Use. if: bus mIatICn botds eN~n ;~ p~ ~0 £~ Art ~t ~ =~;j3- ~ =6 =~ t~ 6~£i bY young, chtId,~n,o~.~t by p£~g wallet Or =~-

Aft: w~ A - ~' :~~ ~ ^ ~e ~ ~= ~ Add. ~ vol Aim: : CJ1~ :~e ~s :~t ~h ~= ~ ~ : w:W Ar^~= TO N~ J - ~ : ?` ' - ~ - - 6~t mow suwe a~ ~n m~ t - ~e is t6e ~:~p the s ~ on ~e ~e of A - ~- -of ~ :~] 6~s Ceil |~ ~ (~ ~077 tt0D ~0 ~0 0[ t50 t~ )8 tW~5 t00 Wium~e of ~e Ceding- ~ Illu~e t~s -~e can show. thM th=e wh~s * ·f ~ -~e hIl~ m~ tM ~* - m twO - ~= ~t .~ t~ wher~ i. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ <\ there at* vOlu~s o! I~ s~o O~ :~n be ~.~d W~ seeing tow *~ -I t~ ~= ~ Very are co~ submenu. Tads l=~ natu~y t~o -~e nm:~n -of density as ~ we~e bail ¢-e n-~n of a:~= can be ink by -ok= mth ~. ~ 5~ p8~5 ~} ~ tt~ 8~6 ~7 (~ m-~=e thetr ^~:~es and :*~te them to the ~ o~f tom* ba~~ The he:~t d:-~*~-~n is "~d out:- :f :t is the ~:~e :;: ah Ale. In th:s ~;v ~t :s ~ ~r children ~ ~ t~8t t50 8~08 0 3* =~t t the ama of -~e a~d ~ an~d that the =~a of a walene tnan is half the area of t:~.~e ~:t asmct~ pa~-~s (:~= $~+ ~ . ~* .* ~ .~ FIGURE ~ B~ ~ waLer lnto shallow pa:~b ~n =n r=~:v c~= ~e a~S :~f di:~t -~:C :~. Of ~ 9~ ~ ~ ~ > ~ ~ ~ rI~= O. tOur nest *A 1n ~ $~m - ~e r - en* ~ pr~: of :~e P~:~: th=~. ~e w^:~*~e on the hypO`ent*= =~Is the sum ~ the ~s on the Ic~ or the n=t tnan~~

- 17 ~ ~ ~ as tags of unit tbic~: ~ ~1 did in . ~~ ~_~a Ala by Its ~ a~ minip~c~ Dig Reck. 11 i~t I -h =61 Edges baa lemma Taut ~squ~ r=~ beg y can ~o in ~ill~ion of Me Dawn Ream (Firm 6). Can p[~ ~~ Tic buzzes mat illumed ~ompo~Jons ~11 Ed il mob easier lager on 10 Pew ~1sj ~~ ~~ ~~ ~ ~~ -at is abet ~ ~~e awe c=~ ~ degas into Be Bead pieces gong ~ ~~ ~- cube (ad ~S)~, Jo as a ~ is -~0 ~ ~C0~1 Inns ~ Dine ~(~ 7). I ~^ an. noodle S. ~c dinged music tion of ~ cue inn cadent pyramids =n ~ illume by begs built ham desponding Splat. ~ ^ ~ ~ _ . . ... ~~ 7. 1 ~~ ~~- ~1~0~ of a ~~ id I amp td-~$ sew ~ ~~ Id to ~ singe dam Iron in tags mend. . ^

~ - - HO=E 9. ~ -I ~n of - ~s 'three .~ of ~t ~= but the ~e volu.~~ Decomposition modes :hustram deeper ideas than do mmpans-~ns of wIumes Aim. t~ n~ Add- ~ relM10~s ~t ~m why these ~s ~+ ~s showy; Ally come to see at ~ geometnc ~ ~-ed on Rudy This ~iCu~.tion Amp- office ~ =n be ~ bu m.~.~- i - -big 60~0 it ~t 00~0 ~ ~t 0~= =~[ 50ii68Y' I th~ ~ d:~-~) Owes IBM ~ We :~:to two Agent tnang1~ t~ ~1 deco-~tion ~ ~ recur solid 01 usllaDy ~e con~t pmmI~ (~m :~) :~h th pans ~D ~:3 h;~ve the she volume ~ :nm the- Emu shade Tads Den. by pounng ~d into pi ~d c=:~.~. but ma~r I: comes Em ~ did model-pl~g carols. Think of ~ pyramid con~md of th:ck ~ar cards And a~e the base If ~ double the th~1~s of each card in the stack~ then the base stays the ~me File both the Alit and ~e Wright 0t the smck (~(L therefom :~s volume) ~o ~. ~: ~ ~p :~ ~ and the th:~ of each =M the same Hi double the Ie~^ ~= the wIume Act- doubles. ~-~ an~ ~e As: cau~ the Flume to dou:~.~* ln genemI, m~:~ ~ stn~e di:~io:~. ~ - Y nu - = w~t muI:~:~y ine entitle v~.~.~e by th~ ~e n~:~+ This procedure e:~eS us to obtatn the ~me ~ any -ad by ~ di - Al d~mpositi~ of a =~1~ ~d-that ~^ of an~ os= . . ~ ~ ~ ~ ~ ~I. ~ ~ ~.. ~. Y ~... I .. ~.... ~ ~ ~ ~ ~ m OS;0 tOp VC~g>< iS d;~;~\r O\~: ~ :O f the base Fu~-~r work mth ~y,':~midY-shapecl bloc~ mI] qu~y show Ha; Or py=,~t,~ ~th ,8 ~;~;~f 6,~yity~ I*,; h~ h%y-%y-~t Il~ (~y~ -I_ a_ I: ~ ~ ~ ~'~ .,~ . ^* . m:~S o:I tnI:s swcIm ~' ah mlb the Same :~- ·* t ~, ~ I. A. YY - ~'Y Y by Ta1~n together, ~,~,,~, I, ~r0i,~ =0 Of 3, pv~mid Fir ~h 8, =~,18;F b;3,,8-C IS one~th1m tte vo-~O Ot t60 =&kit =~: prism w:th the mme base and height. Expcn~s :: stackS of cards or thin ~s :~n lead easily to ~ pow C~. id.~3, A, tO mathematicians as QavalienY's principle ~r sbea:r t =n sf osmatic n3 Y ~t 0656 - 0 5 0W t60 same set of r~s t h at h:} ~ S ~ pa ral :~mm will also fit! ~ rectan~e with same teas-e and height- Hence

- nGURE 10. =c ~ ~ of ~$ ~1 ~ ~ ~c ~ ~ a Spasm. of Me ~o ~$. ~~ ~ ant of Me ~~ Id ~c ~~ ~U$1 ~ ~ gem. Mar was man be ^~ (~ 10~. 19 . as ~11 as in ~ pee. ^c see ~~ of ~~ ~ ~ _ t ~a~ ~~ a. am. ~ over ~~ an be Primed 1~ Con of gum ~= Phi_ a cedar one (~ I 1~. guano go Bogs ads of ids fib fits of blast ad stag of ads tang heir ~ stb-1~ ~ airy mom leek 10 un ~ ad apprise the pad patent fir cab elms in calculus classes; giants go ~ giver Doug parries of albums -unJ1 Hey ages ~ calculus ~ =1~1 new ~ numb out of heir cadence. ~ n~ sand ~ ~~ dew of ax Eli F1GLRE 11. He game ~ of Ha ~~$ ala owed Braid In ~ =~^ =~# to ~ ~ Stead Said of ~c amp bag. basic, and volume.

2:e =~:~= ~ ~~y ~~:s rea~ for the Die techniques :~d ~r a~d ~~- mat:~. We sh—d be ~~ ~ c=~ ~r -weir =~—pwp~tion wells ¢yramm r[ M:~v chdd~ ~e Acid by the ~ ~~ ~ ~~+ Th=e on~ summing wonder of t5e anment wona wew m~ ~ images ~ t60= 6~7 =6 ~ ~= I t~. ~6—StU67 e mon~s of anment ~t scan -~e ~ mu~ ~ -~:~man~ powers of ~ m~s ~ ~ ~t ~:me:; ~ =~.~ti.~s cf Car -Gus to ~e m~ sophisti=~ed achievement of =dy mens~;~the - ~~e ~~:~a - - ~~ f.~m Of ~ ~~ - pym.~.. children =n Aim how to :~e Bobs of the paranoids. A ~~e of d~ sand or wet s=d ~ ~ muaw b~ whims one example. Spiels m=~s to see what ~~s of I ~- ~er m~:~ts of diadem ~~es Dim simile -is m~t and challen~ ~r mns~ion. ~~t ~~t ~e bun m-~dS of Am-~=n Indians or other =~shaped s—~59 What ab=t M~n pyramids, with the~r st=~li~ --? What about 3~ 7~57 07 Add? Ah ~~:= pmv:-~s f=~es that I-~-ad to interesting mathem~i~ q~est:~7 Rich the stu ~$ themselves =n formulate and Elf ~1~ mathematical notion that ames Ail: in the -~v of mon~ uments is s~-mi3~:n—-jet expres~ both al~ly :n chic or p:~-~odion and ~~y 1n shado~ and s~e ~^ C: the foll ing -I mat ~6 Ambmse sent ~ snap~ of his t=p to ~" pt. He is Bins next ~ an -~ =d ~ =n see that his sh~ow is about Emu as Iong as the shadow of the obelisk" That's ~ pren~ b~g cofumn, o~r 24 ~~ h~- ~ ~~w, that b~e my mend is 45 ~et tall~ Ihere is ~ pyramid in the pi~ru:0 tOo 1E see that ~~s shadow is Hi: ~~t past the edge of the base. Who additions info~n would ~ need in odor to 6~= ~ how h~ the pyramid is' ~w can ~ measure ~e a~ that the slanting bide of the pv~m:d maw with the ~9 Such questions can be dis=~d at an into~l ieve! :~g ~~m the studen~ deal both tna~es fo~ly in geomet~ and MY 1:* abOrOt the p-~mids =n chow how ~;}~-ms ~n deferent din me:~sions can illuminate each -other. C~ the In of similarity' ~ eaMI? =~late the volume of an in=~te pYramid ( -~e ~9~*. one of the most ::~t problems in E~-~n m:~s

21 MOORE ~12. ~ em (or ad) py~d ~~ a ~~ lo 6~ ~ flue. UNGLUE 13. ~ bit of ~ ~~d ~ ~ apt idea ~c, me ~ gad ~ ~ lo ~~ ils ~a #: =n ~ ~ ups in ~~e ant 1o Id 1be Baud of ~ l~omplo~ ~~ Bean ~~ ~~ ~ sputum ~ Amp add ~sanincomp~1~1etrian~e<~13~.~equ~hies~=d 6~d~enl10~5ndlbea~a A~=ming~atlbe1~=idis n01 p~llblo~ ~nc~mpl~1he~=lOaidan~e~i~b~libal by ^e~ing~all5~laq~ ~d~na111~an~esa~ bilge ~1~/~=(~+~/~. ~~=~+~)=~ /< ~ + ~ ~ 45/~6 - ~) Then alto Philip Tumult ~~e gibe ~p~oidinane~asthedi~=n~ ofibea~asofl~o 1~0S: {1/~ + 6~5 - (1/~ (1/~/~!~)-(1/~/~} ~ 1/237~: ~ ~/~6 ~ ~) (1/~6 + ad, Tbesamc=~b~ en~ksoneloo~cul~elbevol_eof incom- pletc ~mid(~Figu~ 14~. ~a~ ~nibeh~ei~l40~fpe~ oflbe py~midandlhesid~len~bs~andbofibelopand Dalton ~ua~s.If ~bebei~loflbel~py~midis<~+~>lben itS 101~ flute will be (1/~)(~ + 6)~2> ~biletbe ~lumeofl~es=~llp~mid ~(1/3)~2

~2 ~5 ~ (K 8> =~.~g t50 \~= ~~ p' mm:~' lts w)~e :~n ~ :~ as ~¢ 61~C O: ttC - ~~S ~ t~ ~ $} =) }~f p Amp-, by,.: ~ N~''c'>.~ Ad: ~ : ,: %: ~ ~{ 'Y A f~ I: \ £~ :~ :::> $.~$ .` ~ ~ : : 'I i: - : ~ a~ ~ ~ it, _A ~ ~ ~ ~ b ~ ~ ~ ~0 t~ ~ 0: the i: pyramid its i/~ ~ ~3~2~:l/?~2 f ~ ~ ~ ~ ~ ~ MA ,( ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _~_A ~ ~ f ~ ): ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 3 ~ ~ ~ ~ A ~ ~ '; This f-~18, Rich was - ailed tn ~ ~~ - -m !~0 B-~' rep ~ 56~ ~ ti~ p~ its ~ ~ tt~ ~~ 0t ~ 0[ t56 3:~: ~ ~ ~0 ti~ .A ~~t 08~ 5~ ~ Iated ~ =y stud=~t who maches the leve: of midyear ~~= T:~:v I :~s =n c^~e the ~~1a ~r the Flume of ~ inco-~;~e rammed ~n the ~~h d~n o~r :n hi~r dimes sIOn.~* - ~ and Dim The vo1~ Of ~~r in ~ city cylindc:r ts ~ IlttIe more than th~ quane~s of the volume of the rectan~r box tn dice the ~~:~:in~r just §ts (]F:~re I5) Tf we pour the water from the cylinder ::~;o box- shaped no;: of the same height, with square base - ~~e s:~e equals the rad:~s of the cyli:~' then we can §~! th=e such boxes and st:~:~:~- hew some water left overY. E;~enmen:ts mth dissent cvI~rs a~d related boxes mil quicMv sh~Y that th:s pattem wo~s ~r cviinders of a~v rad:~s or Theists The same mt:~, of cour=' re1~s the area of ~ circle to :ts I ~~. Be=~= c:hild:~ can measure poured quantities Smote easily th~ painted 3703$~ it may be easier :~r them to =~p this fib ratio 6~ :n terms of volume and then sutsequent:~v tn te~s of ama~

- ^- ~- 1~? - - ~ ~ / &# ~ \ ~ mar ~ ~ :~ ~ ~ ~ S ~ < .? ~ at ~ All, A ~ ~~ = ~ =~ 10 ~ ~ ~ ~1~ ~e ~ -~ ~= ~ ~ ~1~ ^ he d~ bat u~ ~ ~~ ~ t~^u~- · ~ In. ^~ ~ _ =~e p~ ~ is jab ~ ~1 ~~ gem ~ ~o _ of ~~ a_ S ~= of ~~ In ~ in Ed ~ using a ^~ or a ~: Awed ~ Hi. ~ 6_~ Bald ~ ~~ file ~ ~ lie of ~ aide of Be bag> as of ~o size of Be ~e. IF one c arc ~ s Has #= ~1 of ~o1~> Ben ~ hang _d ~e Car ~ ~ lace Lund Be smog. ^ gang awed ~ arc ~11 Lund a ~~= sac emus ~ Be radius Ha one am Was. He camp ~= ~~ Be Ho of the ci_~en~ of ~b ash to 1be ~~r of tbo anal is me ~c as Be folio of Be Plume of the Bang lo ~ Sloe of Be spuming box could ~ e limed only mum Her. BY Be an 1~ idea tab Beg is ~ Red =ho beaten e ~~ of ~ dig ad Be Edger of a ~~= is Labia 1~1 Ad sb~ appoint ~~ bead am mention of 1be my~ed~s at. Ibe on Sheen Be Ma and ci~um~=n~ ~ ~ ci^ ~ be easily On by gulf a cable lid ~ pie ad Assembling Be piers halo a nary Ear saw. Tbe ago of ~ disc tugs oul to be ~q~1 the ~e of ~ act Lion gab one side emus ~ 1be adios ~d: 1be aver Bug ~ half Be ci~um~=nce (~~ 16~. Subbing = Ida mom Bias Bald ma the con even ~= _. (~ucb filer students gill app=ci~ 1be limit comae hidden in 1bis demons~1ion.)s Un~una~y, abed scams lo be no sag nice ~~sp~den~ Cheer tube volume of ~ saber gad the volume . . ~ctangul~ bOX.

^ FIGURE 17 Isle cuts. Quads. and Is 111u$l~le Abe ~~nd~=la~ panty of doubling ~lo=: Abe = agent 1bc gear of 2, deeding on dimension. ~4 ~1 _ ~ L~ ~ . . ~ ~ ~ . / ~ _ ~ . _ ~ . _s . _ . ~ . _ ~s . ~ . ~ r ~ ~ 1 a. ~ ~ by ~ i... : ~ ~ ~ ~ ^ _ . __ __ ~ ~ \ Ail. ~ _ IS ~ . _ [:E~ ~ I~ S _ I'm S ~ : ~ [. ~ nCu8E ~16. ~ ~$1iCi~ a #~1o its ~~ pie pat Id Wily ~ ~ Me Ma of ~ Ate is ~c ~~ (I bride ~ ~c ~e) lima bag 6f Me ci~ (1~ Ida Me ~e). KING DOES Baleen in ~ebel's kindles Had cubes sunlit vised cuff Ems Ad Spain ~~' Ad As Elided rods (~ 17~. ED ~~ Has 51 Ever 10 ~~ a 1~~ ^~, ~ as long lace ~ aide, and Lice as him. , , Cur squaw tiles Farber ~ ~~ ~ lam Quaff ~~ as long Ad 1~= as ~de. 1bin m~ ~~ a ~ twice as low as lbe End. CbOd~ ~ ~1 lams An employ Film Begin. Hem is ~ ~1.1 x filled aim =~d, End in papers and bed ~ ~~ H~ is an~bor box-1~ ~ long Ace as ~de' Ad 1 as bag. HA much mom En do we nag ~ he if or gaper 10 char it, or Ad 10 0~1~1 it? it isn't nasal 10 bag ~ Lily lo mu ~~ or or Plume in over lo Dame and bad 1~be answer: lain as much swing, Cur times as ma sheets of aura ei~1 Sacs ~ mum and. ^ #~ ~ : :: s~s~:~s~s~s:sss~s~s~s~s.~s~ss;:ss~ss~s~ss? ~ #~ I q 1

- -A These Ions ~1 cba~s of saw An Me ~~ eat beta 1be cab ~ much Be ~~1h m~-1~li^1ion~ ~~ Rev can Ice unaging a- ~~ pasts. Chil~n to E~ e~un16r Ins ~ Me in 1be long Is alla ~ :: c~l=~ mucb leers Hen ~1~ ~u1 t%~ne~i~no~on' (i~^ ~~, id dime~io!~- ~~ lo ~ Ian in Be ~l_e o1- ~ 120r 01 Ha, get Ma He sloe o1 ~ :~men~on~ sq~ iliac_ i1S tea by 27. ~ ~ . ~~ il m#1 Beg lo ~ ~ box in tag ~^ pangs in size ~i] Singe ~ 24. Eel Mansion, Egg Ponds lo it's ~ ~~1b exited. A surmising ~~ is 1~= 1 are ~omel~c owns Abe -by egg gals ~ ~1 abbe Age. ague Me gists, gig bag a And ~ ~< . , ^ ^ . .. Of ~~i#~ ~=en~on,= ~ ample of a Laming agues- of om~dc pawns ~~ as =~t~s.- Sing the Legion of was usually wires a Spas 1ba1 is applied an finite Labor of times, is orgy #1h He al of modem purr ~phics 1- ~i1 bus po~bIe lo =~ on Be expedients news 10 Clog 1bem ^^ tally. One of lbe Dim examples of ~ Ace gas iced log beg= Ampules ~ the Polish matbem~i~c~ian ~~ S~ie~. The ha step in Craig Sie~idski's f~= is 10 =~ ~ semi lie Abe Midas of a 1~~ one. Me Abed step is Me amp 85 ~0 list: me lbe midge of cab of lbe amending lounges. ~1 His oar and Win 10 Awn web is got ~ the Pies ~~- (~ 1 83. 1'S seem le 1 Sits Gael is 1# doubly ids size poodle a E~ tubal is composed off ~~ Pies of He od~n81 f~ u=. Tbis is ~~ slangs because our enamels gab files and cuts Boa 1b~ doggie agog me ~~ pat of 2: if ~ double ~e size of something of dimension one, ~ ~11~0 copies of He od~- nal, abets if ~ dote ~c size of sometbi~ of Edmonton 1~0, me ~1 Our conies of lhe on. Tbc Sieging ~S~1, Abe must bag ~ dimension some be Pen one and 1 Once a Actions dimension. (Specially' its dimension is the mar ~ #~ He paw easy Pa 2< = S; this n~u~b~r ~ is 1be Io~lb~m of lobe 10 able bag > namely 1.~....) F~=alscan ~ used1o m~iv~eala~enu~berof matbem~ical di~sio~s.Since~y~i~ asa~su~ofaninDnilep~xe~ can d~is~s~din ~l~tionlo ~ometric~he-sor=~a1~ingd~imals.The unusuildoub~lingprope~iesof5~c1~1s~vea ~ome1ricinte~~ta~tion

- ~ --in A - DIG ................ ~ MEADS ~ ~4 , me ~4 ~ - . ~ ~ ~ ~ BAA S.-''' ;L ~ ~ A;'' ''it ~''~ ''--'a ~ - ~ - ~ - A Pi 0~ ~ ~ . ~ ~~ ~~ An =^ I, $ me ~— ~ - J~ ~^ ~` ,^^ ~ - ^ - 2~ ~ ~ - ~ 3~^ ~ ~ 0~ ~ ~ me ~ - ~ ^~ ~~ ~~ x~ ~` ^` ^~. \} ~^ - ^ - ^ ^~ ~~ : ~ 8- This tn5~y pu~ - tnan~, known as sle~ q =~ A= ha]~e coptes of i: I: ~ ~ ( ~ dImens)on were one ~ two Hence t! has ~ ~1 dIm=~n CAM - ~~n one and ALLA r the I~thm to ~e two. Ether ~~ p-~s Iead to 6~s Il~ ~e Mandelbrot set~ incIu~g some Of the :~t Stn~.~g eXa~S of ma~emadc~ an.5 O~ of the ~~ imp; s:kil:~:s w~e =n ~~e our ~~s :s the Ails id to- i; data ~:met:~+ The geometry of area =d volume can help students undemand concepts like rams~ accumulat:~3, and avem~ value. fIere are t;:~= simple—amples that illust:~c this ~~- Denver travels at 40 m::~s per hour for ~ hour, then at 46 :miles per :~r for 2 :~- :~w or d<~s she :~ and what ~~s her a: speed?

as too 27 ha_ ~~o BIKE 19. A ~r Mob elms art ~ ~ Id SEWS ~ ~~ Ages _ ~~ - ~~ He heifer of ~ sat ~~e : Abe amp ~= a~ Me me -~ A Isis m $40000 ~ ~ far ~1 year Id gem $46~000 Me mat 2 a. -^ - ~d ant ~= bis ~~ ~~? ~~ ~ -~-~ @ ^ b~ tack is alla ~ d dupe of 40~ filled ad 1~ i~i~ am ~~ Lied lo ~ Tab off 46 Names. Sot is depth of the alar in the finks? ~ of age problems Ida 1be sag ~i-, ad ~ an awed on Me age dig_ (~ lisp. _~ios ~ be in ~- domed ~ the q~a of ace ~~- . ~ awe ~11 ~ ~e ~be~t of a and acme Me we bag ad lbc sag ~~ Ha. I1 is ~~ pond ~ gab ~ ac=^ Con in ~ ~ ~~ inches early ~~ ma mats ad =~d (or h~ Mae money In Cab ~ ~ 10~ 150 1 ~ ^ Men egged) ~ ~ lid (am 2~0). /~1# / /2. 86) i/ / /1.40) a. FlG~URE20. ^linc~r~pbdisplaystbe~<u~ul~ion~om1b~c p~blem$.indi~li~nglotal mild of or doll feed. The relation ~t~n tbc Sponging bar andlin=r Isis pour 10 occurs

~~ :~o N~y bar ~ r:~.~Mi=: of rates ( - it :~s ~ nep fun=~) Iea~ to ~ a=~:~ation ~ph ~~—m straight 15~= (~' ~ ~~: ~~). ~e pro~ of hnd~ ~e rate fro~m accumulation ]~ds ~~-~iV tO ~e d1f~ =1~s ~~nd 5n6 ing =~.~s -~m rates l~s to ~e In~! c~.~.. MthO~ ~ ~ semi n~ Feces ~r stu~;~.ts -to realize ~~ c=;~:~= -I they dewIop thelr Ding o~f ~s =~::~S cr ~~:= am -gamins'——stO~t ~ ~n—~ Ads ~ of m=~ma6 em ~~ ~ pmp~= ~ c.~= and as pry :~r~:~- ~I'um~ wu:~ *~11 chill in Fly kindergarten prying drawing. They pl~ed mth drawing -ok one level while ~~ learned on. ~~" T60\t 1~6 ~ ob~e sphered ~~-~' and c~, ult:imMel~t Hey :~:~d to draw Yet ~=v ~. ~n our =v :~re ts ~t as :~cn em~;:s on drawing ~ ~ mIss oppo~= to d~p ~e ~~ of our Stu~s to v-isu.~.~e gec:~:~cal relationships. The ~st common way of repreSennng ~ cube :n mo~ books is ~ CITY ~ my then t:~e it ~~g a:n obl:~-~ axis (~v ~ ~ 4~ :~ina tion) and then to ~:~= =~ndi-ng points (37~:m 21~. ~:~-~ this is ~ pe - Skin Iliad ;- off the stmct-~e of ~ tra~:~t cube~ no view ~ ~ cube actu~:~v looks like this image. We wYe look ~ ~ c~e' if one ~ce appears as ~ ~~' thm ~ must ~ ico~g dir. mIv t~ that ~~. in his =~e t'he ~~e ~~e w:~l be direct): behind the face we see and not o~ to the side as :t ls in the trad::iona! 1~ - This is tme - ~~r we u~ ~ at: `~m ~ · ~ promotion or I (~-e 22~7 whew t~e back ~ appears sm~r loan t~ - ~~. Another popular method of drawing uses lw=~c p~:~^ -which c:~s thr~ ed~s of ~ cube as se~s of equal ien~h , / ~ MU ~ ~ ~ ~ The typI=! mp=w n~t ~ ~ ~ as two >~-~) In: wllb -I connect~ :s qu~e unrea~ s>~e no cute ca~ 0\t(: Aim 3~$t tt^~$ ~~.

D! US ~9 / . FlGUR.s 29* Two co~= v1~ - of ~ c~ am ~~n bY ~e -~- pr~n -IBM ~~! ~~) ~ ~¢ I~ 0r ~~ ~ A ~ ~ ~ t~ A ~~) .~ ~ / r ~ "` / OURE 2~+ ~e Sv==~ne "~- v)~ ~ ~ =~ ~:b mi~ and tra:~t ~~ b~ ~~ ~ on-e comer ~ ~ 43O =~~ ¢~n the ~~e c~r :~^ dlr.~v -I -~e - m mmeG only S~ I. aw di.~.~d in- thts vIew - ~- \ - - .. ~.~ ~ ._ \. \. .~a - . - .._ _ \ \ Hi. Be. FICUR~ 24 - ~O v)~ of ~ cube ln general posli£~x :~ - gmpt:C (~ Also i05~) Ails O~-~t Is isle t5~ ~s-S~-') m0~g at ~ )~ an~S (:~C 23~. Fit m¢~ haS the d:~:~e that t~ Aims of the c~ a~ rep=~:~d bY the same Is +: a .. : - .~ ~ Tf we widths ~ more general imps of ~ cube" we mum d=wY ea~; face as ~ n`~Sua;~e -~:~=m (~:~g ~ I pupation) or as t:~20fid (if ~ us0 I ~~) (Figure 24)+ The Thai—~~n ~ Or o~h'tc) p~t'lo~S is pa~)Sa~v easY to GROW Slnce the p^Sctu-~.e 0: ~ Is :5 :~tsb=~Y I 0~= t~6 p0S;~;~ 0f thy 0~5 3t 0~0

3Q~ l \ \ ¢~:m:mc~s ~ 3;U~'CY - ant ~ \ / D it. \ FIGURE 95 in an o - ~~c d~ paml~ lln=. In t~ :~ arc ~~d as ~~! lln~ on t~e =~~ ~ ~e Act: comer is specified. In ~ onh~c p~' p~! edges of the c~e appear as p~:~:~;1 edges ~n the images so we ~= In: complete the p.~e once we know ~e pOstt10n All- the three e~ at a~ =~r (;F:=e 251+ God ~ ~~w how to kept ~ thme~ I. on ~ ~;wo- d:~1 pa~ or =~-r Ethics sweep? we can ~ on to ~ much more I—ercise~ that of drawing ~ ~~:Mimensio~ =~e of ~ =1~, c~ - ~ ~~ or =~t :~v students enco~r the !~ of ~ f our~} mens) Onal cube ln sc: en:~e 5=i on or ~~Y ilteratu re ~ such as Robe~ HeinIcin,~ stow ~ ~ ~and He Bu'6 ~ ~~—uSet0 or Madeleine L,509s ~ ~~e in nme:~4 or Even Abbot! Ab^~s H.~.~- A ~ :. - \ - . \ SAC \ \ \ :~E )~x BY addl~g ~ ~~h dlr=~n ~ ~e t - ~:~. th~ne c0~er that ~~ts thme~.~.~.l IS we la\. ~ ~~n ~r dr8~g ~~l o~. I~t shim Ch~ £.^ t- ' h. ~ ~ ~ ~ ~ ~ - , ccsm is, W~r~.~-

31 Usually ~ ~ is consumed ~ Ovid ~ ordinal ^ in a direction ~~i~ 10 our spa. Oboe he And awfully ~ Icicle of ^~1 sue ~ cog SECURE 27. ~c ~m~cd Bum ted by castling I vents 1~ ~~= of ~ cut. ~ ~ .~ immune 3$ ~ cut Rabin ~ cut

~2 EN be, To ~ , a C== o~.e -Gus m~ an ~~ I ~~ ~ SUi~C '~ - 3$ t50 Cow. ~ ~~y . ~ in__ I. . . ~ ~ Hi. ~ .\ ~ . I' n-~:RE 29~ S~ mw:~ng ~ ~~ o~——an ~~;~or h= ~ d~, ll )s ~ ~ =e ~m ~¢ :~ of ~ =~e ~~= ~ s~¢ t~ ~ ~~$~- . \ V CU5C—~S dl~t—= d ~~<t p~ Arts A ~~= on t5 ~~r hand al~- ~ 10~s Like ~ ~~. A=- ~~ -~e i~k at it, :~t :~-~s t56 =~:~. :: ~ =~ 3~ ~77 then -I vi~ ~~c images that are elil~ws -in -~:~nt ~~-~:~-s (~w ?~. Students also need to be e of the bas:c p~-~= of d~g the~ Imps ~~. Tt :s t that ~ c:~e aims :: I::~e an -: inciud:~g the extreme -me W~ ~~e Tne ~~:~e ls ~~! ~ cime Or wh~e ill Gus 1~o ~ do~Y covered ~~t ilne S~ ~ :-t - - ~ ~ ~ ~ i: ~ obw - ~~g this ~~t ma~s it easier to Omw cOnvInc=g ~~:~ and Co~es <~um 30~. ~[:~m compu:~s are ~~ e~= to produce ~ seque~e Of ~:~s show1n,8. di~-~:~.t V.~.~$ -of a, :~-~a,!~.~g cause or ~~, ~voi,~;,3 ~e it,, Tus:-~ of ~ ll:~:~onal o:~. ~~s pr-~ss ~s I ~~at to ~ ude:~s who :~e ~~n up -aim m* speck! 0~5 Id t-01~;~;~ 400~t i815o We In =~ ~;w -of this 0~0 to ~:~e stude~.~s new app~ia.tion for mathem~i=! Amp. As 01.~.~- live p~ms become :more ~~Iv 3:~ailable7 51*~15 0; ~i =5 :~0 ~ pp ~ ' ~ bef ore possit~!~ ~ to mama u.~t ~ and =~e gecm¢~c ~-~S in three and h~r dimensions. Q~e o) tne mo~ :~t :~-~s we can t:~sm~t t<:> students act all levels is the Ail:- of =~e descnp;ti-~S bOth tO sp¢~' 10~tionS and to g:ve inst=~- Examples o? coordl~es can be mad¢ available ~ St3~ O; ~ 37~6 ~ 633~t ~ 1~-~ :$ ~0 5~t I fit 6;~r~l0p

33 undemanding Of d~= =:~.~e d~:~:. You ~~t have t:o I=m ~e fim bitt =~:~N be~e ~~ ::~o the se=~ and 0n the chit (=d ~^ The l~on to ~~me mordlnates mom a* -dimen~ ~~t ~s arguable at ~! ~~. -~e I: h~ to m~e stu~ts ~=e of wh~—V ~ seel~- ~:~ expenences of d~:~t ~~s are ~~ pr=~:~' ~t is useful ~r ou: prewnt =~is to: sep=~e p~:~a a=~g to the ~~: of =~s nee~ to :~.~e ~ position or ~w an m~.ion~ ~ Lams ~ c~s ~~n ~ :a w~ :~ - a~ chud~ :~n u~d ~ ~~ Of a~. hate =n apprec~.~ ~ o~ ~~m used. {~ r fn~ :~2 ~ =~Ci~ ~~ t~ t6~ ~ itS ~~ -~ ~ ~ ~ ~ ~7 tt~ =~ t~0 =~t ~ ~6 hilly {: it - ~'5 ~: 50 t~0 0~0 7cO are I~u~g—, wu =e ~~. Tf nm, ~ to ~ ne~ bad aM chew :~ amber f :~t is In to the one v~ ~~~ keep ~~ >:n that ~~*=ion~ If 1t ~ ~= ~~' ~ in the -mar heady ~~ - ~n you g~ to the ~~r you Want.. DI~:n often ~~ sImple ~~m Illu~s ~ nu - ~r of Any po=~t t~- We ~~ ~ iomli~ by ~ Specific ~~' and We along ~ on—:~ p~, ~ fine Oregon or an—er' to ~t :~m one po~n to anode*- ^~r ~e, ba~c prOce~= Is, un~d One Can a~ are. w~ ~ wheth~ ~e a~s is on the ewn or o~ I- of the ~~. Es—=~;g the displace one tlas to tr~! in order tO ~t - m one ~~n to an~r is =~= w6~, Ieadw to A te~ ~ 0~; phi 0~ Each exe~= can ~e Nace on ~ :~r Idle ~ positive a~s or on ~ ~~s :~n ~ sc~r number bum. ~~r the =~e notions =n be =6d ~: SC~CS Wit ~~-\r~' flit tC=~C, whem the Vertical Onenm~iOn o~f Me ~~= emp~s ~ dimcti~iw. "~ bend—-en the tem~re ~s ~om 65 d~ to 40 de~ 2~- "~t ~ down -~v 25 .~- such ob~s c~ ta~ place tar ln adva.~-~ of fig s0~d :~. Many cities use directional addres=s :n their ~~et pl= (~-, ln N -God ~:~> ~~e its tm~ ~ ~~ and an ~~ 42nd Strew. Tn this case |:he alwn:~hm to tind ~ bulidi~—m knowledge of 115 address is s:~W di:~t but still =.~V enou~ to discuss ~ an elementa~ school i - ~~- Th~e distance ~~n t~ ~~s on the ~.~e side is add- ~ usual, wide the d:~ce betwee~ two :~-~atio-ns -on died sides is the sum of their a-~. No memon;~tion :s :~:i:~-~d for such ~ mteme n.t ~ Stoned nu:~= do not bave to ~ ~~s

3:4 be: w: ~~y :~ s~= a:: A. ~~ tn ~w a~ - ~K ~W er ~ is~ I: or: ~~, de=~g ~= - ~~: Ad: nm~ one: =n go: ell as lewd Se~ ~ watch ts ~~tW ~;~nt—m ~~ ~ ~~ 8~7~ 0~ 8~ ~6 t~ ~5 58~. ~ ~~ ~~ ~ =~[ ~' ~7 ~~ p=~= 0t i ~ ~spemfe ~ - sash :s ~~s to the pmb~m of ~~ a ~- ~u m -I a~ -Emmy am~ ~ ~r ~~- cou~ one ~~;,.~n , ~~e ~ ~~r Em, the~ mber ~ i: me pmbiem of deeding on ~ ~~ ~ Iocam ~ ~~ss on ~ hi. drive is ~ ~,=~e of ~ kmd:,~-f Chid p~s `t ~ ~3,5 =~, as in m~ mhe=' ~~= ~ no s—e ens- ~~ ~ ~~es th~ will achieve 'K ~ ~~ ~~g ~ =~ K the dices =e al then ~ ~= be ~e a~a~ ~~f bade of Hi: e~n :s ~ e~ to underdo ~~= An mimm=:~ng One ~m,6~n,81. e>;~s . ,- one ,, :,0 Iocate ~ s shou~ i~ ~ I,. a ~ ~ . ~ ~ . ~ point. DI~:~s ~~r mov!~ - m o~ position tO =~ =e ~~o one~men~- "~ thr~ hous~ to ~e n~= or"~ around coun mclockw)se 6ve spa=s or KCo h~:v~ay am~ the ~e ~ the o 5~ ~~:~= city i~ ~~ 0t in 6~$ 0~ t66 ~0 0[ ~0 ~0 =:d ma. ~~ the begI:~:~g of ~ ion of ~~= m=~:~. ~~ ~ cIock~ ~~er =~ue or d:~l, prov16= ~ =~nt ex ample of "~.~- ThIs :: ~ alm ~ v10~ on I=x: s~e, ~: example, on the sel=~or of ma~V car radios. In ma an~e d - ~~ t~ Hi In:di.:~r ~~s at the eXtmme Im or : treme n~t,—its tn the d~! w=-~ns ~e md~or simply g=s—m the mp value to the bottom. : ~ particular ra~o ~~n then pmsents mu. d1,.~t ~~S of pmblem..s Amends on the n-~= of the :~o sele=~r ~- ~~ I 0: ~ - )S 8~ i: kept t5~t 3~S - ~: and ow~ again in mat;~ema~;~ as well as ~~. As students b~e mme Hi I :n ~e kinds of numbed they use~ they =n Ire Actions or decimals into n~r lines =d nu - ~r mwIes~ ~~;g telep bo nc ~~e alO ng ~ ma ~ in ~ Aim: a=a Mali r~ 8' d: ~~t kind ~ ~ ~16~' using ~~S or real nu—e~ ~~nting a=~: distanced e numbers become more compli=~- tut the pro=~s rema:n the . L00311~g o~3~s or aOdres~ in ~ are-; wodd can be ac cOm ~ fished ~ ~ cicnt) ~ bs ~ ~ ~ ~ i Amp.;: al~t ~ ~ ~ Or the . ~n atlo ~ of 1t that divides each :~-al arc: ~ pmc~:re with almo~ Dial K ~ ~ ce t~t :s =~6 t0: 0~6 t~ t50 :~) t6~ig~6 ~6 to §~d phone n:umbe=- F:~t you ma~ ~ guess to divide your problem

Day 35 into two- p=~ o=~ the p:~e book or ~ picking ~ number. In wu =~W Or ~~s wish ^~t ~u want and ma~ ~ :~ ~~ W3t :$ in whichevet p8~ (~ 0r b~ow -~ur firm gue~) that contains wh~ It ls that Mu are :~g ~~* s:~r s - :~e can be used to 5~d ~e "added- of the Ie~= of ~e hymnal of ~ sq=re mthOm mpu1nng ~ ~~or w11h ~ squ~ roo: key. binding ~e dec~! eq~t of ~ £~=i~ can ~ we~d as ~ ~ ~ vein 0: th0 0~0 6 im If we ~~t w: 6~d 3/ ~ 7, we =n multiply did dec~s by ~ ~ ~o ~e if the pmduct is bl~ Cr s~r shad. S-,. All ~~.~.s ~t put 1.~o One catted or ~e at;: :t ~r c~es out; ~~. ~ S-~16 on the other :~ 60= (~ 0~t ¢~7 50 ~~ is ~ ~ l~ 0n ~ 60 ~~} ~ :~.0 ~ ~ t5~ 50~ ~ t~ A ti Th~ ~ndame~!—met:~ pmblem ~r One~*r>~al phenomena is the dete:~:~.~n ~ dls~e ~~ ~ w~^ ~ :~S l*~*~ calcu~on or tampion of =~ of =~s =d pol~ ~ is One gentle tnc =:~ - ~ ~t all ~~.~s sho~ ~~ to un~W ~10 :~15 ~~) 51~-~007 =~St =~0 ~ ~Ot ~~ how to 8~ Id. 0~ BYTE 35~ - At ~ i5. ~~ i~' ~5 [~6 bitt :3 ** ice ~~:~' 3~:16 or 22~7, walnut :~:ng in either case whether ,, K ~ too ~ 0t t00 5~ ~5 0~t dig ~ ~ ~ ~ fi ~ ~s o ~ ~0 p~> ~~\ 50~Og Dim tt A ~ ~ =~t ~ t~6 bit ~ ~e area of ~ ~= ~ the Ida of ~ ~~*~:re m~ s1de ~~! tO the radius-" 750 ~~t t6= t5:~7~ - ~ =~$ 3~C All- =~C :87 0: 000~, ~ =~: t500~ mm of ma~i~~ C>ne can* ~ ~ t~m=~us amount of—e~e out of ~ co nt i nul.~g dI~ s~o ~ of ~ he e~n of ~ ~ ~ ~ the f =t ~ tme ~ hi:: student [*~0S th~ ~e belt around ~ ~ wacky= a* 11111e =~= t~ t~= t:~8 =~5 t50 t0~' ~ ~70~: 0~5 ~~¢ # temIs ~r arc Ien~. Flnd)~ the c~wumfere~ce Of ~ cIrcie :s ~ one dimensional p!*ob so :~s answer shoutd 3b=e ~ =~:~ o~ ~~e summer line ~~.~e ls *if! ~~w can we cete~e wnetner or not ~ given number is I0Ss tha~ thlS le~th or greater? Com~=s ~th the =~=ce of circumwnbed and inscribed ~1~s :s an e:fl~;i~ ~~:~v Ad* dealing w)~b these question S. Alt'hou~ Such compansOns cannot d~e ~ e::~, they can =~#~ci~>Y Em - ~~r 22/ I: is slightIv ~~e or s1.~.~\ b-~w

3:6 3~ w:~:Y ~ ~ muntI~ ~= aw :~ - impost ~~r d~g Wren few :in the anthmmc of ~~c ~~:~- Stude~s =n e tnst=~on Scare says "~ ~~ ~ m~" or " - (= or B3~, and th~ ~ 0~w ~ Is m~ mun~. en ~ev c~ ~ a~d to ^~ ~ off ~ a: ~t accom ph~ the ~~ :~t By ~~m dOub~ or tnp~ # th * e~= with ~ idea: of m~ - g ~ s:~ nu~ ~~ ~ Gove r ~ va~s ~ ~e ~me ~ m=~^ ~~e =~n g ~ the B~ Is ~om ~~:~d ~~s ~e ~ as~ ~~ ~~p ~~e Bl2' him ~~ ~~ :84 ~s :~s ~e ~ ~ t~ng ~~ o~ F:12. One m~t tntmd~ ~ ~~+ P3~ _ "pm ~~ ~~e B~ = - ~~h is ~e ~e ~ T3~4 ~ "-~e up three ~ Amp' S':.~.~, P85 TF: and PF2 ~ TB2' ~~g ~ c~e ~~= of I-* The p~! ~~e am. ~,:~d ~.~s ~ that ~ use them tO=~-OS ~ :f61* o~raticns. ~e nde th= ~e pr~ct of t~ n~e numbe~ ~~s ms~- ts one of the ead:~t ~~g bl=~ teat mm:~ce many ~~ms t~ math~cs :means IBM not Appl*~e ex~e—h c~g ~~s -em =~e its tuido*n to ~e ~= of n~ -*~ · i .qt ~ ~ ~ ~ . · . ~ * ax ~ Boa~ ~es help Students appr=~..~e m~e ~~e ~ In* t.m mtt simple addition (~ - Wh:~ ~~\r~:~; dCpCD~ OD th~ throw Of ~ bait ~ (LlC0) 8~d 13107 in mom =~pli=~ed ~es whem ~e mere can ~ In: ~ Vega;. ~ -~g =~ are ~~y on~imen-~. m~s =d.~S Children should be=me ~~d :n both following and among direct tio~*.~.. Ally -I should Ie~ how to d:~*m ~ =~n ~*m, one pm Of th<¢ Ah- to another =d p—aps *to describe the n~ of 50600lY ~~-~*~ the ~~= ~r mnng - m one ~~t ~~s to another in an ~~ w~n m.~t ~ gulte compl:~- an ides to\~* h~ ~ simpler *I We can imagine ~ sequin*= of imp towns mth dimwit dimensions properties—a-~tier tO~ ~! ~~d out alo~ ~ single st~m or a -villa I~d =t on ~ ~~ar plot. ~ mode~! I. - =~6 Ilk ~ ~ 66~ 0: tt~ I ~6 ~ ~0 0~ ~#~5 6:~10~ could Chip their and to~ would ~~w Or mom van~ at1oyn,** No matter what t6~ Amp are named' we can stIll ~e Oly~-~s all ~ mid by - I: "~o n~t two t\:~87 t00*~* *rum Ieh and ~ three blocks.- W: =~*~*s with ~ -clear one-~:n, the inspections =n ~ vaned~ "(ic east two blocks, thin nonh them blocked ~e hrst tnst~*~n depends on the di.~ti.~n that the pe~n is racing' and the second does not.

D= If the Am: of ~ vllI~ is h=~g on ~ a, we :~n up the natum! cOor~e ~~* "~o ~ two bwks and by: three-" ~~n 0= of i:~=i=s =n then be =~eci. "Go :~D—o' then up th~" =-d "~o ieh th=e and. ~~ W' c.~'5i~e tO give "~o Ie~ ~ =d two.- B)Y pl~g this ~~e wnh c=~.~, we can easily tnt - ~~e ~e m. ~ o; a6~ng 0~6 p~5 806—~~O of m: nu - ~6 pa~m by positive integers. :~: ~ :~e "~- and "taker opem~, we =.n extend the I ~~a Of signed numbe~ to an geb:ra of two~emional quant:~= Notice ~~t th~s ~~m of its does At req:~e the use of coor~ I.~. the plank ~e exercise CameS ad~.ional v~w - ~n ~~ dressy are ~~= in te~s of amp n-um~ or mmp=s ;di:~ctiOm. W: one ~~.~& th~ avolds the =~.~s ~~d m' n~ nu~. ESNd ~ E7~2 retires ~ move ~ E4~82. ¢e co= deuce ~ his commomense appr=ch a~ the a - ~:C ~~nt (~' 9) ~ (3~* 4) ~ (~, ~~) ~s some~:~g ~~ c~ c~e much ~= :n ;~ am de~s me d~ ~ me a. great ~~ pecple who are mOftsed by n=~ Ins ~A' "Taxicab ~~- -es an; emotive Paris ~ the -~e of radios ln~-~" Em. pl~ ~e Me of ~~, telling bins how to ~t - m one locator to anmi3~. "~ ~ three ~~ts no~ =d ~ ~~= west ~~d be such ~ -* ~e Lacier of ~~e i; the pro6t ~ the Cob Iota on many f~= su~ as -I ~~' accidents, am -giraffe band One =n ~~v Mom ~ bo=d ~e thy would mo~l maniac city tra~c an~ ~ ~~s used Do ~~ :~= of ~ tw - ~~:~ instm=~n set" ~e su~ce cf the =dh is -a~er ~~ example of ~—md~ a; o~^ -in ~ - :~t eM~s in -~ space' we need oni: two num~, ~~e a~ I~: ~ ~w - ~ I~ti=. divider of Chips =n ~w :~ns ~ ~ iD mars ~c east =d then ~ my= -~e north. On ~ Ire of ~e em :~t on ~ 0~t plane— It O:~{ Otl60$C OpC=~S =~ ~ 6~^ ~~ ~ ~~= 6~C ~~ - and then 10 -~es due =~ =n p~ ~ ~~p at ~ Laurent ~sitiont ~~e t of th:s d~e is an i- :~r of cu - ~~. In teaching ~~ we sh~d not ~~re the m:~we v:~ ~me Tod~s students take ~r =~d Me ~~t that we can manipulate ink a~ 0n ~ -I 5~n 57 push~ng by;* taming If 0 twi st i ng ~ ov5~ ~ ~ amp ~ tlke LoOO o~r ~:~t ~ ~ ~~:e nce in giving simple geOmetnc indentions to m~e Potts and o~ts around on ~ scwen. This g:~s mathe:~m teachers ~ cha*~;e ~o ~~e a number of impotent :~$ ]~6i~ t~ 0907~i0~5 t : ve pm0~5~5 (~: drawing—=~ 0 -I space-fi:~ling cu - ~~.

:: a: I: i: ~ em~Y ~' wh~ Int - ~:s =~m t~ ~= :~t ~ ~ ~ when a:~t is ~d o~ ~ ie~ stde ~ ~ mmou~ s=~" It ~ - ~ the: ~::~ helm on me ~t ~* Thls Is: i: lo Ale::: ~ Me ratio dI~ w~ :~ ~ - ! ~ ~:= o=~g on ~ cI=~ ~ same vim Its endpows ~ ~ ~ ~ = =~* A~ ~, ~ f ~ t~k of t~ po~ on ~e IeD ~ :~f = s~en ~ wm:6ed with t~ m~ng points ~on the~ n~t ~' then we ~ de~g ~t m~ ~ A= mc~e ~m ~ - cynic ~ - ~ :~ ~ 5~. it iS O~ ~C =$0 t~t - = ~ ~t :~s o~ ~ tad of ~e s~^ 1t r=~ - ~ Am. ~p nn ~ := ~0 ~.~ ~ ~ ~ ~ c/~e ~ t~ A , ~ ~ ~:~- his Eves ~ Film Slim ~ item tube' - Ash :~ i ~ w~ ~e geome~ of ~ tom~s is :n mm~e w~s :~ t~ of the pl=~, bm tn - ~r Wry ~ It is w~ ~- Tn the pl=e am p- ol~ ~ ~es- not into :~: divides the pl=e mm t~ ple~" 13~: lf ~ ~6 ~ ~:~.6 ~i~ ~ ~5 3= ~6 ~ top o: t~ ~ A ~ ~ sepame he toms ~o ~ pieces. its :~ is the ~e as in outs:~^ R~ed w ~:s phenOmenon is the ~ ~t on ~ t~S ~ ~ fig c10: - Cu - = th~ cro~ ~ .~y :~e warm ~ ~= ~ ~ ): - ~s If t~ 0~6 I. i ~0 pi&~0 ~5 (~: jam t0~5 ~t ~RS :! ~ toms, t~ m^~1~! name ~r ~ I 56~6 5~('0) \5 ~ :~ - 'A 5~, 8~0 )~ ~ 5~^ t~ ~6 CU~S =n Int~: In 3~t one pout and: ln whit: a- c1050d' cu - e ~ ecd n~t wpak ate ~ ~ ~ ns ldc ~ro ~as oulS Ide A

DMIE~ an - ~:n number of ~~. An I: ~~t is idea ~r Bins track. of pairs of num~= - m C;~. fig Fly wars the tows It is- ~ sho~ sew mom two to three d:~. From the twin 6:~$~ Vi i~ i~O Ot ~ We C~ =~0 tO t~ =~i Ot ~ (~` W~ CtC ~ ~ he~t ~r e~ portion as ~~} as ~ Wstnon on to ~d We can au~ent :~0 ~~+ Both elewamr ^~:~. We sp~' ~ Minim t~ t57= ~~, ~t =~07 ~9 =:~g to ~e n: nth door of ~ bulldIng at location ES ~ an aii:~bm ~r Attic Am this io=:~on to ~57~-. ~te t:~: in this pamcuiar Imp ~t ~s ~ big d:~e in what directions O~C I The OS=1 ~~:~= WO~ bC D-9 Emus. B~:~:ing with DA ms wu to: ~ Be, I - ~! but :n the w=~g buildl~ The sI:~n ~-~16 50 dissent ~: ~ ~~e p:~6 0n ~ j—6 gyms ~~ t~5 to- m - c ~ One position to ano - r ~ ~-~ ~ Akin di~e te~ ~~ ~ ~ ~ ~ ~~ 7 i:; 1n ~ o^~- Ano~r th=~-dimen~! geometry' anws if ~ ~t to - Fry the ~~ 05 ~ 8~07 ~~g it~ i0~ l8tit~, =6 ~~. ~6 a~n, :~t m~es ~ ~:~ce :n which o~: we give the num~n that lndI~e ~ ~~n lo=~n or ~ ~~=ions Or =~ti:~:g from one point to =~.~. ~ . T~ rntu:it10ns th~ ~~:ts a=~e in de~ with Ordinate patm In the plane a~ I. tnples :in thr~) sp~ iead :~ura~- :o ~ina;te geometry in higher dim=~ions ~ thorou~ up a ding of ~ =:d thwe ~~s ~~= ~ i.: Datsun ~r the ~~l ~~:~ions o~f ~~r and ~~x aigebm in scions and engmeen~ in economics and socm} sewnce' and e - =~IV in com=~r science and ~~.~. ~ illumine thIs p~si~ with two- eXamples. . ~~C VC~i0~ 0 ~ 54~6 080 tC ~~O 57 t0~: points (~), ~ ~ ,0), ~ ~ 7 ~ ), an~ (Q ~~. ~ o5~n t~C VC:~i:~S O:t ~ =~7 ~6 =~ t~0 tt~ ~0i~5 0t ~ sguare w:th zem in the th1~ coord~te and ~en m~e the squar0 one unit :~ the third d:~n ~ obtain ~~r more ve~' with ~ :1 :i:n the iast cOo~.: (~^ (~7 i'0~, (~':~- ('0: ~ i'N (~O i)' I; Be,

^~ ~~ And :~^~ ^ ~1~:^ =~ ~~ ~ ~~ digests Pa ~ ~ go Pant ~~= gad fit ~ ~~~ an. = ~ _ ~cd~ ebber Me Guam or He Age as ~~ emit eider 0 or 1 is. em Count. The p~edu~ ~ne~i~ aulomid~y: ~ obtain ~ ~^ of , ~ ~q wit 1be em emits of ~ ~ and pal O in 1~ and chine and 1hen Me 1be ~ in ~ Numb dictions ~ ~n em mod Dials Cab 1 the last dominate: ' ~ ~1 e _ j~ ~ . ~ . ~ :iS )~< (I <~^ ~^ (0~0> 1~0)7 (1~0, 1~0), (1> 1, 1,0), (0: 1> 1~0)> (0~0~0> 1)> (1~0~0> 1), {1: 1~0, 1), (0, 1~0, 1)> (070,1~1)$ (1~0~1~1)> (l, bI: l)> (O>l, al). Bus ordain 1be sigh Bates of a ~=u~, gab O or 1 in eat of Cur Shingles. I1 is his ~~ of -~= n lb~ is it far , . , commu~lcal~g ~1 ~ ~ computer. ~ sound topic 1b~ Kneels= in a gem nice ~ is 1be Ion am. if ~ think of 1bis tb~eo~=m as a way of ~lcul~ing lbe lamb of be diurnal ~ a =~e bulb Men sides, then the extension 10 age dim~n~ons is im~mediale: Even a Slid bound by Granular sides, ~ f~ Bay the teem ~ one sue arid 1bon aDDlv il 10 ~ =~an~e ~~ ~ ~ ~ a. ^ ^ ~ ~ gulp oar lee ~1 dla~ne1 (Fiat 32). ~ easily g~ ~2 = c2 + ~2 = ~ ~ + ~ ~ ~ ~ ~ ~2 _ ~ sides e~ 6, and ~ is 7~2 + 62 + ~2 The pained is est~lisbed, and 1be distance Paula in ~ur~imension~I space Slaloms almost immed~i~ely. Students En 1ben calculate 1be ~lenglbs of diagonals of 1be ~~=u~ be ~1 Finales. 11 Was out 1b~ 1be Fob of Be m~cr

dirty of a ~~men~on~ ~~ Ham (0~0~0~0) ~ (1~1~1~1)- is at= 2, Ibid is Mice labs lest of a side. CO\O~GL~O~ SOWS 4~1 Me cawing Inceptions tab a~ ~ used in giving lotions and d.i~=ion in ~Hiar spats of one: 1~0> and 1~c dimensions got Tautly ~1 ~r p~ben~ome~a cow silica Quits mom 1ban abbe flume. Explosion ~~ antis, a Solicit 1ecb~ique far dealing bulb 1bese represen1~1ion~s~is one oflbe mod i~mpo~an1 applies capons of di~:ensio~sin curTcn1 research. Tbe ability to visualize and . . . . ... . . . 1~=e~r~ mu111Ul~mensl~=al gala Is ma, ~ one of-lbe Cast Has ~ can . , patent our stunts in ibis gem Ha. Some of He mad used Id impeding cx~^ off b~e~men~on~ ~ pbe~om~ena occur ~ Non Loos of ~omethc oidecls nepresent.ing = ~ in ~ cactuses or m o11onsin~tbe gal ~odd. lobe clog familiar spaces are The onc~di~endonal oolle~1ion of points on ~ line>1be go din~ension~1 coBk<$ion of points in a T4ane, and lbe lb ee~dim~en~on~ chlleclion ofpoin~s1~ space. Ski Ace can also consider the coJleclion oflincsin fb~e~plane~tbe 0011~1ion ofTJenesi~n space, the collection of at possible dries in ~ plane, or the collection of spheres in space. EVE iIluslrale this process by paling sever ex as off Nomad gal lag lo bi~e~m~sion.~ ~nb~#ion ma. Consider He allowing (slimy un~i~ic) Nun: Be lima dilator of oar 1 tar teas 10 bang ~ ~1 off alias aver Me ash so AS 10 aluminate Ion pads of 1be Boor al akin lima. Sometimes the size of a sag is supped lo ~~ pang 1be Aura of ~ Ins So metimes one colored Bade is supposed 10 be con ~ ned in an - or. Ilo~v can she keep1rack ofell1~e circles ofl~ghland tb~en desig~ligbting di~ec1~0ns so tbe1 en asista~1 can carry Abe out? In this particular fb~eaer1ho~lighls a] b _ 1be same fornn. A Anne bulb is suspended fto m a sire banging dozen from the ceilings and conical abate direclS 1be~Iigb1 oul in ~ tba1 meek the fk>or in ~ disc of~Egb1. lDhe Odes ofibe shade co me doing at a 45~ allele, go lag [8OluS 01 IDe DISC IS egu21 10 TON ~Clg~T OT ago Oulo ~OoVC age noor (Figure SS). Ibis makesi1 easy fbrtbe director 10 specify the 10cation ofanylighl, Lace she can indicaletbe position ofibc center oftbe disc using 1be same coordinates 1be1 The director of the play uses 10 give her ~ins1~ions Thai uses loo Ingress bus 1be Wing director needs a~nolber nu mber 10 represent lobe radius of tb~e all=. She could, as an al1e~nal~ive, specify the beighl oflbe buIb above 1be Coor, since fig 1bisidk~]ized situation abase 1~0 flu abed are 1be same. Lance any particular disc can be represented by Bee coondinales; 1~b~ ffs1 1~0

- - ~ - - :~ ~~E 33* ~ ~~t w,~ ~ Ode ~ ~ ~ 45--~- ~ ~1 t0~.m ~ ~t ~ ~e h~ Of ~ ~ - ~ =~s eq~s ~e b~ ~ ~¢ I~ AYE t~ ~- beim t~ 10~ of t~ ~~r and ~ ~4 m:~ Be: - ins (~r O~ Skip ~, ~e :~- In ~~s ~ ~ ~ ~~t ~ ~~.~n of ~~ in ~ pla~ is I. Id ttiS =~ ~ 3D—3~ 0[ ~ 00~D ~0 =~h A ne eie=~t ~ t50 ~1~: ~ =~ty ~ ~ bookkeeping ~~-lce' ~ ~~=o m:~ ~~ the po~n of each light by Dim ~~ c~inat:~. for exe - ~, (6, 8-' >) re~s to ~e I~ht w~:h center at eke (~, 8;) position on the 6~or and ~ radius (or height) of 5* To =11 ship ~ ~ indicates something more ma c<~-~e of recombs It is ~ signal ~~.t ~e Ah.. ~ th es -of ~ ~~:~ of I—so For =~e, ~ spotlight m~ coon dina;~s (~, 8' 5) stays on the stage' while t~ I:~t (6s 4' S) shines o~ the front of the stem It is easy to deadline ~ mIc ~ tell when ~ I:~t s=~,~s ~~;y from the ~~t n~ of the stem namely ~~t ~e =~d m=di:~e be I~r ~~n Me third:. . . ... .. .. Mare co-~ex problems :~ci~ the lighting dimcto-r mn also be no: - d by ~~ to tne c~:~in~+ ~r eXam~, - ~n will one ~~t ~ en t~~y wpame fmm a.: {~ wo^, ~s he—~:n the ~~e Between the po:~ts :n ~ plane Even ~ tM fi st t~ coordinates is r the ~e sum of the third =~- Tn symbols, the =-~ t10n is e~d bY —' Tn this mn6~ration ~~ce the th=e -~inates do not; ploy the sam~e $~s of mIes, so even tho~ the ~~.~-- of the mnf~:~;~n sp=e ls thre - Ill it tre=s the i~t c~ni~e d:~y ~~m the firm two~ T! :s not identity] ~ the usua:! geometry, off o:~- three~e, where the JP~ha~n theorem treats 611 =~s the ~~e way. An IBM aspen. of conD~n spa=s are the special ~~:~s they : .~.

~sw.x ~3~= or Iater eve~e hears thM time is the :~h d;rme~. That idea, however, Il.mitS the Am' of I ~ Alr=~v in the Iast centu~, ~~= realize that there a~re ma~ s:~ns in wh~ time can be viewed as ~ mash ~i=~$iO~' 6~t ~V ~O =~S 600-S it 60~6 anv special rOle as ~e ~~h dimension. when p~sic~s, e~y rel:~:~:y p~, ~~V an event by m~ ~~e ~~e coordinates and one dime coord:~e, Why are us:~ ~ ~u—imensional con6 t:~n ~~. his - ~e has ills o~ ge~:~' th~ :~s not th~e same as the ~~ of ~~sio~ Eucl~n s~' who distance is oven by the Iced twang theowm~ {n the t:heo~* of relat~' the ~~e t=~:n two cows :~s given ~ the exprm~n · ~ ~ ~ ·! ~ ~ — · ~ re t) me is mea~d ~ ~~t unns wI~d to ~ speed Of bile ¢e three dimensional mnfi~ra:t:on sp~ of sp~li~ts pmv:~es use ~] ~ nalogy f or ~ f o u:~dimen~o nal sp=e u:~d' :n mo ~~.~= modeling. ~e a:~s :~:: Moe up ~ :~e ~ be reo~ t~ ~~) 5~5 of d~t radii" Ibe ~~:~n o! ~ p=~.~r m~:~:~:~- I~e the dewnption :~f st~ i~ Assists of ~ I.i~ of ~~S of ~~t s1.~ in ~~nt ~~tions~ Each sphere require= the coordir~ to Specie its c~r and o~ coOrdi:~e ~r the - ~~. ~.~S the con - mion —~e of atoms is ~dimensional, and ~ mo:~:~e is ~ collection of su~ atoms a~^ in ~ particular :~ti~- Using ~e I=~ of ~e con6~n Sp~, we can - ~~ - Moliere ~ ~ =~r and a~ it tO display dI~t v1~. if ~ ask the =~er to check ~at ~ =~ms do n~ :~=, this :~es gum cond~-~. in ~~r coo~, namely ~ ~ ~ ~ ~ ~ wok The ~~t of this Fin space is much cio~ to that of rela I: thecm than it :s to ordinal :~idean ~~sion~ :. lnterest~in~y it is this sO~ of question:—avoiding i:~=ions—that a~ p08~5 in ~0 50i0~= 0 —Oti037 U5i~g ~~ ~~ 05 =~i~08 tO keep track of Jew m - ~~g throu~ fin ~~s of him din mens:~:~- Suppose ea~ light on our =~le sta~ possesses ~ rheostat that can metro! the cu:~t—h== the bn=~f the spot. if we a bars to the coordinates of the spo~li~, the~ the conduction

~4 ^~= do. ~ = mitt ~ :~.~.~.~. If we ~t tO enc~e ~e mi~ of ~h as—I, ~en the d~:~ality u:mp-s acme -~e sin n: cn:~r r.~-],1.~. ~.~e I- c~.~s r=~2 el~r :~- s~ . ;~;, .,~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ · ,'; ~ ~ , , ,, ~~.,s ~ on =d V~e or ~ 1~1ve a~ts 0? re~—~~' a~ blue ~~r ~~) 0t - ' ~7 ~6 5~ ( - ~~ 30 tt~ ~~g ~~t `,11 -I ~ ~~ =~s ~: ,~ ~~. Wo~ ~r ~~:;~i~ ~ ~{t=~-7 a~t6~t(~= TM5 ~ mike Aide c~ Iead to ~ =~*~= ~~e ~:~ ~^al1.~# Relai~y Isis bed ~ =~g ~~ ~~ mth ~~ ~~mm Or spa~ am ~~ne ~~ t~. ~~3i alum push ~s bell dad ire =~ca£¢~^ ~~e cu~ m~s ~~p ~ -of seven <~-~= ~~ ~ ~ ~ dme to ~ve ~ ~ t~ensmnal con~:= sp=e ~~r )~nt model uS~ ~ co.~6gu~= spew with 2:6 d~.~.s I:~. I,. c=e the ~~ of ~e ~~ depends to mme bee on the ~~ of ails t53t ~V i~ t5680 61~:~$7 88 8~ 816 t~ Wiping t~ 0[ t6~ =:~X :~ among ~~s -in ~~w h~men~ ~~. ~~s ~ another t~e of c=~n 89800~7 50t ~~ 6~ ~ 5~6 FINK ~: ~6 86600: 800~6 -she ~ ~~s w=t to dec~e the back =~! O~f ~e h~1 mth ~ patw~ of p~:c—~~. Thev—ide to stretch-~m f~ the Cowhand Me tn~ and ~~r t~ week U¢~e ~e show, they mme up Ruth a; 605~ us~ =~-~0 ths~ twen~ 5~ ~ey cants i~e t50m up um:i the show ~ they twe to ~d ~ ~ of ~~1:~g the posit:~:~s so the"v =n pet them up ~n iater. W~ many numbers ~ tb~ ~ to specify the position of =~h - ~-~ ~ lS ~ dimen~-~? Of the m~n It is ea~ to see that the -~si-~:ity of tbis =nfigurat:~n s~e is two:. lt takes ]~ - o :~S to io=~e ~ men ~~' one ~:~g the 0~r and one up the left ed~ ~ the wall' an-d each of th~e if: can be ~i6~d by~ ~ si:~e n:~. The pair ~ num—rs (4' 33), ~: 0~? 00~ t~t the sting that ~~s - m the point ~~r ~~t over on the fleer to the point three ~~t up on the wall edge (:~ure age. The =~:~ction of pairs, one spur ~r each st~' t-611s the positions of ~! $~~ It is even possible to :~-~d thew ordered pairs 'n ~ specific sequ~e so the is ~~! knotty which order to follo~ when they . In ~ wav t:~::s coding :s die the -old ~me of "~:~ct the dots~ chew ~ pol~n is d: by ~ wpuence of ordemd pait$7 50 t~ fig

- - ~ (of:) ) A7W e fi) ,~ S.} nGURE :~. ~ 'in s~e of |~ . ca~ r~t Lh~ ~s flops ~ at m:n f mm the f =r to lm tl:~S on the Ieh cd~ o~ Lbe wall. the dots :n—er we draw Chic wIy~. Tn -ant sculpture sm~ the bas:c element~s are not poi:~s but ~=sC~'tS. ~ ~:~.~g the s-~e -of Is ~me~.~' we re~e the wall wuISpture. f ~ i~sC C3~ the dimensionality -of the COOs6~8tiO~ ~~, W~ =~ allo~t the -I ~ the stnng tO be pl=~d a~e ohs the - with the t~ bill somewhere on the :~h e~ of the walI' w. ~ st111 sand one n`~r ~r the he~, but :~w th`e record will have to inci:~tle two num- be~ ~r the fleer COO~ii~s810-~. USA (~tiO~ O; ~~S WOU]-6 tt~ 5~ t~S~m0~S5i0~S, jiOis6~g =~: I O; mare i. $:. B~ allowing ~e nnn`~ to ~~ ~~:~e o~ USA VC~iC~7s W3~: 3~6 ~ _ is ~ d. ~ I.. on. ~;~\ n<)ior we w0~ haw a. I- ~ ;: farm ]:mens:~` sy~. si=*sp~se ~~bm woul6 ~en e~le one to predict, ~r -~, w:~-~r or not two Id-. are aim to .inte~. ~~n -~-e are I~ng stnn~ ~~ng ~ wall, it is commonplace for them to Isis+ ~SU-~ ~s~iODS 8~-0 -~0 ::f we am Ash ~ thee - ~~sional colie~ion 8~ =~: 5ti:~} ~t t50 0.~6im6~tS0~S31 ~~= 0: ~5 )0 5900~~ it is elm :~ting to :hok for confsgumt:~ns cf s~:~ts that co amp to fam:~:r con6~:ions in -exit ~~. What col:~-~n O: S~0 ~ tS iO ~ Is C~ ~~O ~ Sp 8~C CO =~ ~ line joining two winstS, Who S~tf~S :~ ~ t~C—i~`C~SiO~3 CO~io`~s~ tts'0~^s -~s] ~ ~ c~e p~e !n ~9 -~ions Ashes as thesc Ins ~~d st~k:~g and ~sDptOdi1C~C ViSU~ O~S id. th~ St~g sculptu:~- ~0 ~i~s-~s~s3]s:~, O: ~ ~st8tiO~ Sp8C't becomes cs=~:ia:~:~ ::m~ p~` ~~ ~~ :~s~id~: ~<~=i: I. ~7~ ~ print i~ maxim o~` ~ line, we can d=0dsbe its state at am ~~` time ~ biding two num~ and 0~ ~: its p05iti0~ 3.~6 ~ IS ~t itS VC)sOCit>. ltC St8t() Sp8~6 iti; I: t~i=S0~S5;~a}~ 8~6 ~ paint =~:~g 80 S0~i~: t~ ~ 'Is physi(~7` i=~- iil(0 ~ 53~} tt055i~g ~~p Char I WAS ~ ~~:~r~ ~~ii describe cu - e in th~ state space Similarly ~ point mos ::~g Ash ~ ci=3sC~ 11~-C

46 ^~ ~ \^ By scions pendulum> ~11 bow ~ Dimensions me space Ding i1S~ angular position and amour calmly. He He ~~ of a point moving in ~ plow am ~ ~u~ime~io~l# flits ~, lotion and Faber ~ ~: Block. Scic~i~ ~ .. analyzing Be Ghan of a ~l~llile hoe lo Con in ~ s~imens~1 slate ~ace: Aid 1b~ Snares far portion Ed flee far Loci. He 1~ of physics ^11 resldc1 ~ Adds Us of ~ Them ~ Us l~e~im~~on~ ~~. In~> sciatic ~ ~ Had dew of beak ~ =~1~ =c sates of leer spaces. Pa ~ On OTTO Hauls co~poLds ~ ~ cur on a Ads ~ Unwed ~ace. He Judy of sucb b#~di~m~sion~ Amid Areas is an Away impound subject in mown Split mathematics. ~= Fidel. pied his 30~01dC USE be did nag ~n1 them 10 appear Chic. One of the frsl Ads was ~ di~spl~ of ~~= basso Its su~cnd~ ~ s1hn~ in vadous Is (aim 35). As the offs ~1~1~> ~~ i i.~.,.,.~.! F1OURE 35. F^bc~s kindest included basic cabals shot could ~ bung Ham Lyle at dim ~iliOD.S~ 1ben view ^ Siren per$~ctive$ to see vadous c~~ctio~1 ~apes.

DO :~:~s :~4 The =~! dIa~ cams beXa~ whose: st~ ed~ :~t o~= on children co~ld obse—e them—~m d::~:~nt v10ws and ultlmatel:V come to an ~~tion of their wmmetn~ ad ~~. In the mo~! I- b~ F=~' the ~:~.~- the - ~~- and the cube all bad e'\~ts a;~ed so that they =~Id be suspend~ lo d:~m ~ se c: :~s ~~: -a - ~ the Chew h~ o~v one ~~?~t l5~ - - Ilnder he'd th~. oh's in the center of an end :~c one In t~e cen~r of ~ side, and one on we'd nm. The cube ~~o ha t:~. one in the center :~f ~ ~~, :~e in the c=~: of an ed~, and one at ~ ve~- The wnous views of the~ m=~;~g objects Iead- to one of the most 1~- CXCtClSC$ i~ I: to~s :n I* ~~ the ~~ m:~n of Cros~tional silces~ One w~ to vi~e ~~s wi~:~t a:~v ap~g ~ kn:i~ to ~ mal: model: {s to l.~:e w~t wou~ hap 90n :f ~'e Ah =bme~ed the 51~cy :n waw. H~ ~fYiii the ~~ of the water Iced cha~9 l.~e ex.~.~e 1,~t, is A.. ~~.t ~r stuctent.s ls to v:~:ize the sha~ of the "~= Of ~ cube su~ed -I ~ ve:~. ~ st~Ot—O 68$ i00~ :~> 8t ~ :~ CUT ~?~) b~r~ ~ mace ~~: ~h=~ ~: ~ K ~ ttat t5e ~ 5~: i~ ~ t0~ ~ :( ~ g~= ~ ~ ~ A This ~~ can Hi ,, ~ 6~g ~ =~: 5~6 870~6 ~ 0~:0 ~ a~ model ~r the p; eces <: ~ th K ~ =~ t~ =~:i~s ~~= ~~= :~ ~~ ~~c A ~~= on Me sides of ~ :~ :~n (~ 37) ~:~: plastic =~e hay hIled web ~ colored Duly- can be ma~ i;~6 to 5~w th$ ~4~.~5 5ti005 t570~= t60 :~. If the cu~ :s ~ ~ inlaid 5? $~:~:~ ~ li) 8~?~:5 5~ ~ 000t st1ce—that 18, a silce ttmu~ the cenm—~:~ss of th0 cubes cn I it i - ? ~ ~ :~ t~ t~n 88?~ 5~-~$ t~ 6:~= 0~t ~ Bill ,d,~sition of the cube produ~ the cent~! slice with the matest area. art' ls no! the bexa~} sI100~) ,? )~ tt~ :~ :~) ~~ I. 8786~ - ~ I: 0{ ~-?5 I: =~5 it tt~ ~6 5~' 56

48 Mar ~.~O,~= ~ N~Y . Id. ~s wm~e 1~O ~ ~d 58~=S -~e =D 50 ~$~r~6 ~ ~ ~¢ 0~ dyer If, I; t~ ~= ~d ~n one of his sets another h~e-~ cone. ¢-e con:c s - ~$ am p~a that =n be seen and ~d Iong be:~-e st~s ~ Riced tO hi ~ ~ partial Fiats w:th :~;id ~ idle Be chan~g conic se~s as 0~,t,~,3) ~6 05 ~ ~.~3,606 Army It, ;3, =~:~ar $~:x stded wI~on, SO lht Ct~) SiKtCC O; ~ ~:~ t¢~ ~ ~ t - .~t ~6 IKE t5' ~ ~=- )~0 t0~0 OAK tt ~ -apt p K. 0~ Kay t6~ =.~-5 ~: (~g ~ ~ 070~- t~0 skit p)~$ =~0 3~ 0~t =~0 I've

- ~< ~5 39 ^~£~8 BY ~ try \50 ~~— Beg any / ~Ye=~n of the a=~s ~~ I'm ap~.~t i=~h ~ ~ the: tI~= wI~:~t ch~ngl~g ~~r a=~: :~h ~-e inv~tiw~on -of Wick O pOl750~ O~-~tS l=6S tO ~ l~g uuzZIe. If wee ~.~e ~ manned.= did ~ ~ '~e o~! to -~e of its ~5, we ~t ~ wn~ of tnan~* If we sI~ ~ planes parole! to one of the 0~e57 ~ ~t [~=, =6 lD the =n—p05iti0~7 ~ 59~= (~:~ 3-~. Stud=~s =n m~e =~d po]~.l mo~s of the es of ~~s—ompos:~on bY cunI~ and ~~ an appmpnat-e fit A y 6~ t~ p~t t5056 t~0 :~ti=l pleces m~= ~ ~~ ~ ~~r pymm~- ~e ~~1W ~~s two three~:~$~ anal~ue of the apt: icon that makes two :~s of ~ =] length se~ d~t :f we put a=~s on the ends ~ Fiche 39~:. Visits ~ H~r 13imens~on$ Gem one~;~ed VearS ~ chin Abbott A~tt uSed silting to iL iOS~C ~C dim 3~' i~ hit Ci85$iC him ~^ ~~t his =3t 6~: ice tO t~ tO t~0 0~ ~0 ~ri0~prOi~t 0[ ~ ~7 livid in ~ Nondimensional enliven, ~~ci~v when he -is view ty ~ at: hoe mom higher dimen~. The frustums attempts of the sphere ~ t=ch A Amp ~~t the third dimension g:w ~~ insights into the -chalk Ien~s of =~uni=tion and visitation ::n ~~. (Early pam of HatIan-d m~ be -I ~r some students' and some of the social satire dot ~ 3~6 - ~: 3t 6= t686i~- ^~ ~5 ~ 3~ emit , - ~ ~ BY , , ,,, . ~ ~ x. .. ~ . ...... , , , i. .. .. re:~r anO wo~r tOr equatIty wno was Amp tne na.~-~:~O =~-~s -of Victona;n Endued wnh ~ to dass shiv and panics ula~y with best to women~ On1~ at the end ~es ~ Square begin to i ~ eni~d dried of his society) Wade ~~ 5~ ifs ~~ ~~= I - bar 3~ 5~:~0 - ~ ~ dimension h:~r than Or~t OW~9 ~~ad of gm~ri:~;g and chan~ circles in p}~0 Ate Arch At, ~~ - bring 8~6 ~a~gi.~g 5~5 i~ ~ ::~:tin~ to :~t such an ~m :~s the :~n and deflation of ~~:~' tut the Bins; of the ex-~se is that sru<~6 ~ I@hen<~-~on could be interpreted equally well aS the sticks of ~ ~~-e pi our th~:~Sion~ underset t: ~ ~~:= were ~Pisi:~Cd by ~ cube ~om the Acid dime:~:~' he would see ~ ~Parietv of polv~ depending ok. the Amp:. of the cube

~ ~,~ I;, ~~ be ~ a~a1Q~S threw ~~;~ s - s of ~ ~~ h~er~e? This :~s one place who computer =~es =:n be of great Up (~s ::n ~ firm ~e ~ id:, - - 3 =~: te~ws ~~ ~~: :~ cant ~~m Em ~~: <leve~nt of =~er m~- x ~ tomography u~ computer Aphid ~n the :: of - di—-:nsi~1 ~ms few ~~= Few Wpo~ and Geoff c<: n~ Ad: amaze came ~~s Cowed ~ e - Aims of ~~t contagions above ~ below- ~¢ wed= of the ea - . S~= slic:~-~ ~ ~ ~ ~ ~ ,, ,~ ~ :~K ~ . ,. Ids ~ used ~ -~ogls~'—~e ~~en ln maten~ scl== =e beer ~~s to show ~ ~~s cf ~ th=~:n¢~sio:~ $~= tempemu~ or den~. Ex>~ry dam Arab- uses: technics- of p~¢~ns and sI:~ng to i: - e hi ~ S~s ~~m soc~ wi.~s as - ! as - m ~e p~] and bI,01. =! wmD=~. Modems of ~~:lus mI! appmm~e t~ power of sIt=ng technics=— ~r ex:~' tn relatIng: Me volume of ~ su - ~e of mvolut:~n to the chan~g amas of bits circular cm~ ~~s or in hi ~e contour lineS on the seduce Of 3~ ~h ~ ~space. ~ng bets-= ~~n s are 1.~d to t~ notions o~f cntI~ poi:~:t th~' they =n ~~V nd and apprecIate s11~ phenomena t~t relate dI~nt di- mensions~ What ha~s If Ate brie ~ ~~.~t 0: ~ band in different tions7 It ~s =~v to =~: o~t ~e actu~ expenments and see that the =e positions—~e the slice Melds ~ pair of Circles. Wss ~tous :~s ~ clip ~~t consists of two inwhocked c:~- Again, ~ ~~d w~t to see thts amid ~ to expenment with ~ :~:~: mner :~e 6 halfwav m~ colored I:~quid. C~' can be ~ su~ng ob=~ation ~ ^* ~ ~ C:~G Co:~s ~ =v =m ~n,.~1 ~ ad al~c questions ~ se in the lnve~i~t to ~ of =~-~nc ~~s, thew =n be i: at distant educational i - 40~8 =~t ~p t~ tt~ - Chid 0[ t0508~* HOw ma~ edges does ~ tnan~ar pyramid h~9 We can follow ~~'s suggestion and m~e ~ model out of toothpicks and ~~' then =~:~t the e~. or wee ca~n simplest draw ~ p:~re of the object (Figure 401- and count the six ed~s The ~~e ~r drawing such ~ di~m su~~S an a~m ~r dete~:~.ing the number o:~:~s Sta~ with ~ point' then choo~ ~ distinct po:nt a~6 hat t~6 0~6 0680 I it tO ttC 0~6 W0 3~V had~ Now choose ~ new point and con:~-~t :t to the previous two points to ge! two ~~.~' Or ~ total of t:~. (We bave t~ 50 :~) ~

f / - - \ ~ 6 40 7~6 ~07~_~0 I'd ~~: Add_ 5~5 ~~ t=~: - ~5 5~ ~$ 87~6 ~~r ~~= Vv 6~6 4: 3~7 36,6~g 0~6 .~'0~ =~t m~ ll~t (~t,:o'ns ~ =~h =_ v ous GYM—~, one can const=ct :~n $ e Compl~e g=~S on I' 2b :, 4* $~ $~ A ~ ~ plants. choOw ~e new point ~ the i1~e cOntalUi:~g ~ . e - ~) N=.t c~se ~ new po:nt not :~ng on any of the three lines d~ed b~ the edw alreadY con~, and then conn~ this n~x mint to the preV iOuS th~ ~ dais Mel~ three new c~, f o r ~ tom! ~ ~ six: ~ We =n ~t this process to d~~ the tgure—cal~ ~ co— —I b~ Dw ~~s (~ 415~- nrst choose ~ point n~ ~ any of the six lines containing :~v =~d ~~, and ~~n co~= it to the previous ~ur points ~ ~wn ~-~:r :~ edges—~r i. of IO. ~ simile: mnst~n c~ pmduce ~e cOmple~ Mach on slK points and more lf so Flax ~ Who :s the p~= that emerges—m this pro=~:~9 It becom~ .~t i? we a.~e the r - ~~ in ~ table. Nu.~: of acing:. ;E If In ~ ! 3 3 6 ~ .6 :n 63~ == t~0 n;~Ct 0 edges :5 the nu - ~: 0 p51~5 05 p01~157 ~~ct ie~s d~y to the ~:~v o~f comb-~tions~ Based on the s-~e off constmction, it is easy to ~ that the :~:mber of ed~s at stage ~ >s the sum of all numbers Ie~ than n. W: =ample, the number of ewes fo~d Alit six points is ~ ~ ~ ~2 ~ Ax. ~ ~ ~ ~^ ~ 5- some students maY know t6e ~~:~a n;n ~ iffy tor t~e sum o: the 6~t ~ integers, pertaps ~n co~on with the famous story o; :~e Amp Gauss wh~o us ~ thi COOL up all the n~ mom ~ to :~O AnOther t~e Of patte:m :~s ~~ed by the table—that the number of ~~es at an~v stage is t~0 ~~ 0: the :~s number of ed~s a~ ~~ Awry In, - ,,

52 ^~ ~ \_~7 ~ ^ . _ ^ ~ a, ~ plate fibs Ids ~= ~ Bung of 1b=~i~ dings tdan~e. Hence Sunlit :~e$ is ~ui~l~en1 10 =unling Ides ~ Brim A display of dig Is Twined by ~m- Spalial pee ion tests omen ask hums lo =1~ a simple ~= ~~= a com~pli^1ed one Candling ewes is one of lbe simple ~ such tasks. ~ex1 in difficulty Would be coupling the number of dining trig antes (aware 42~. By Barking garb thee ~ In Blend include He new ~in~ation: ~ . ~ Lumbar 0 pants: ~~ ~ ~ . Sum or eats: `y . ~ . , human o1 1~an~es: Idle 10 1 2 1 3 - 4 6 0 1 4 5 10 10 6 15 ? To 611 in 1be missing value he In Mason mom powers, maw of Ibid am just like those 1b~ relay ewes lo points. Since abed bare as maw leant ~ abed am d~i~incl 1dples of bedims, 1be 1otal Ember of trances is jug tbc combinations of a curtain neuter of omens taken 1b~ al a time. ~1e=ali~l~, as beam, he In use ~ locution rel~ionshi~p: the number of adages al am slag is Be sum of lobe Dr~ioussumberofthan~es and tbe~revious number ofed~es. Abe loller is Be easiest ~ ~lculale: il s~bows abut 1be number of Badges lba1=nbe ~~ned~om 6 Kinesis 20. [ID gonc=11benumber~r~ rointsis~-l)~-~/6.1 . . . . . . , . ,, ~ Sludenls go bye Studied some al gill be numbed lo 1be binomial ~e~cienl~ lo raffle 1bese

I+ ~~+S (~+~)2 =~2+2~+~2 (a + 6)3 = ~3 + 3~2} + 3~62 + S: #+~~+~+~+~+~ + ail ~, + ~4~ + 1o~3~2 + 1~2~3 + 5~4 + Hi (~ + 6~6 = ~6 + 6~6 + 1 5~2 + 2~s53 + 1 5~4 + ~~: + >6 i3 damming 1be liters ~1O~ leads a shined Vernon ~ Pa~al's load eats _ ^ ^ . . . ^ 1 ne tomb ~> for examples glans ln SuC~SSlOn 10{ 1 1 1 ~ 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 ~ 1 4 1bc num~= of -~1s ~ilb ~ ~i~ ~ ~m 1he ~ur ~inl~ d~s. li~, 1dan~es ~ ~e m~e. -1b 1he emp} s~ and 1~ -~e lhe ends (~he~ ~ = 0 ~d ~ = 47. Obseranl students m~ see a~ol~b~er impo~an1 ~11e=~b~ 1be su~= of a~ ~ is ~ po~er of 2. Tbc~ is a sorbislicaled ~ ~ Slalin~ 1biS ~seralion: 1~be sum of lbe ~m~ of simplices of di~n~1 di^ mensions in an ~-si~mplex-i~ncluding tbe ~bole ~e~ and 1he cmply simplex-is 2~+1 ~is ~me relationship can ~ O~e~d by s~ling ~ ~ ~ 1 and ~ ~ 1 i~ tbe ~ of binomi~ exp~sio~s or ~ =1~3 1b~ binomial coe~cients 10 1be com~bi~lions of ~ + 1 ele~menls ~en + 1 ~a1 a lime Tbe ~1~ nu=~r of ~ssible ~mbinations is tbe~ 2~+1> 1~101aI num~r of subsets chosen ^= am~ong ~ + 1 el~men~ts. Tbis basic counli~g a~umen1 can molivete many ~pics in elemenla p{Ob~ili1V. . . C-~- ~u_s ~ C Similar dbse~ations eme~e if students in~sli~1e 1be numbe~rs of ~i~s, e~s, and ~es of cubes and ~e=^s in vadous dimen- sions. Ju~ as lb~e~ is a bierarchy of s~simplices ~i1bin e~b simpla~, 1bere is ~B analogous s~quence of squa~es and cubes ~ilbin escb ~ dimensionalcubo. ~ S-cube teas ~ ve~ices>12 edges~and 6 squa~res>2s can be veiCed by an actualcoun~ A ~quare~or 2+ube~bas 4 ve~ice~ 4ed~cs>~nd lsquarz. A 1-cubeisaseg~ue~1 ~ilb ~ ve~ices end 1 cdge,

54 an/ .# ~< ~ ~ : , ~ ., __ ~ ~ ~/ FIGURE Ha. Sb~i~ beeps idiom ~ by gum ~ ~~ 1~1 flu_ in ~c ups. ~ a~ ha sly Sups ~ ~1, tab ted Tab 1be orient cup a~ its ~.~ == ~ as~i Sib tag ~~ join Each. and ~ is going Cab 1 vat. This ~a =~ D1~B~S1~: bums loud Us 3~ gums {paints) (lines) {squads) (cams) ~ Guam Point: 1 0 0 0 0: Line: 2 1 0 0 0 Squarc: 4 4 1 0 0 Cud: 8 12 6 1 0 Hy==u~: 16 ? ~ ? 1 , becomes a bi1 more difficult. WE knot bow 10 generate a bypercube - move an ordinary cube in ~ direction perpendicular 10 it As 1he cube moves, 1he ~ vertices grace our ~ parley edges. Ibis yields 12 cd@es OD lbe od@in~lcube, 12 on the displaced cube, and 8 n ~ edges 1=cedby1bemoveme~ ~rato~lof32~sonlbob~=u~<Fi~= 43~. Counting Quads present mom of a problem, but a Dawn of lbe same met. can ~ USA 10 ~1# il. nail Used 1ba1 them are 6 Quads on the owing cad a~ 6 on the di~l~d one ~ abed 12 mug Id He squads traced out ~ Be Ins of Me maid cad.

to .. C; :::::::::::::: ~ : HG ~E, 4~,5 ~ ~p ~ f~t Irk, ~= \n ;~ ~ detc~.~.~ ~v the -~.~-t d.~.~03~t of ~e In; t DISK ¢~= ~5 3~6 ~t ~ ~ - ~n ~ :~= am Lemon:` as In th~ IC ~r 68~. sees In p=~l bu n~s ¢e ~ K the ~ come in ~u't gro:~s of ~ p=~! e - ~. an ~ ciass:fi~ :~n ~ur ~s ot ~ p~et squa=s one suct ~:~0 thmu~ each ve~. Two honzomal - ~s are ra~r =~ :o s~ (:~re 44~`, ~er =~up of four Venice faces- ~me cTeam: when we remow some of the =~s Ilnes ,{~e 45) student te~s can eaS11y ~G the w:~g three ~s O~f ~r squa:~. It 1s easier to do this-en ~e :~ur ~ do- n~ owr~p =d ~ - elv :~e d~ when ~e ovedap is Ia=" :: 24 ~. ~ ping ~s or ~es is- pa~l-e~= sesws ~ ~t deal of svmmet - I' as does the ~pereu~- We ~ stud~ the relat10n b~=n wmmet~ and ~ng b~ Ioo~ng at dissent ~ 5~5A S. of ~ cube' ~ square, ~ ~ s~t ~e bY pe~:~.~ t60 edges 8t 030~ W0~\ t~ It W~5 8~0 5v =~Ki0g each w~ex to ano-~r posltI~. W.e c~.~n of ~:1 --I of the cup or W~:~e Is an !~.~\ example of ~ - ~ an air structure t53t =~8 ~: pt0~=i05Y The Emmett =~p of ~ cube ~s the col'l=~n of pe~tions of 11s \~i=S that prese~ tts ~e Y The attempt to cOd1G the relation of ~p<~=tions to symmetnes of 3 YY gebra~c and ~o:~:ric ~s provided considerable impetus for the d - ~:~t of modem algebra dunng the past t~ ace::. Eden now Aim: ~-s co-~:~e to ~l theoret:~:1 work in atomic ph~s.

56 I:: ~ art The cmcm! - I: ~~ ~ - ~~ is ~M it is sO Memo that every po:~ :~ Every - er ~~: fir ~ ~~w 6~ns ~ one vmex' ~ Ken Twin harems By: ~~ ~$. ~[ eXa=~07 8t =~5 of the 16 vendees of ~ ~~:~m ~~ :¢ r ~ m~ ~ 64. BY ~s pm~s co~ e~ ~~ ED ~ ~e ~~t r of edges :s half of 64' or 32. ~ .. . At =~h ven~ mew :~e ~ cmalD numb~ In: amp: H~ ,~S m~ ~ ~~:e ~ ~s to :¢~= twO: e~s I: amp: - ~= edges ohm ~t ~ ~ wren A A" _ ~ ~ On~ ~ ham -ant Me t~mm am~ :~e tour; :~ r~n wnee pies Or ~e second, Berg thew ~ ~ ~ Firs" ~ belt' ~h ~r of -ems ~~s twine ~:~ fig ;~:~t, once tn =ch order ~ ~~se I: 0m Dyed ~6 ~~nt squares at ~ ~ {6 ve~s to~—t~n ~~d 96 Is ~t each square is c=~ed ~ur t~'—=~ ~ ~~—h- ~ lts wmc= Hence the tme I- is -~4 ~ 94 ~~ in ~ :~. ~ ~~ =~s e ~~t munt of s~ =~s ~ ~~r ~~s ~ we .~ ~ ~:~ of ~e ~~, but ~ :is ~~ed by ~ :~th~ ~ wo~ ~ ewn if applied to ~ -~ c~" S - - Ply Advanced ~~:~:s can Sprees ~~e rests an ~ gen~ ~~. t ~~ (~' no denme the num~r of idles :~n ~ n~ube. To c~te (~, n) we ~i:~' as ~e, by Amp how m~ k<~s there are at each ve~. E~ Kate is I; ined ~ ~ smut of ~ distinct eats tom among, the ~ e~ Ala- fiom each ve:~. Th=-~-re the -I -of k~s ~ each ve~ - is ~7~) ~ (~) ~ ~~ ~ ~~, t5 _ ~ ~ , ~ =~n of ~ shims ta~ ~ at ~ time. Since thew =e C(~, 3 k- =~s ~ ~~h -of the 2n ven;~-~, the I- n-~:~er of k~cubes appea~ t-o be 2~C:~' n)* BY in this count each k<~e is counted 2~ firms' so we -city ~ tt8t Dum60:r to ~ t5e 5~ ~~. ~ (~' n~) ~ 7~ ~~p n) ~membenng the panem of powers -of 2 the come—m the su~ of rows in the s:~x t~, ~ natumIly seek ~ similar pattem far cubes. Tn this ca~ the en:~s in each row add up to ~ 'w-~r of 34 O-~s ~ ha_ = ~ Lln:~. .~.~" :O :0 Hvp:~- ~ ~ )' I 4.~s sum 24 ~ 1. s:

- ~7 _ . _ . AGILE 46. Subdivision of 1bc sit ~ dealt at. Id cuts (and ~n byes info 1b:~e eq=1 pass #^ 3, 9, 27, or 81 similar small O~-~y$ ~ tar of 3

58 ^~ ~\~c, Wee am Its lo al lo this Mention. ~ ~ awe ~ al ~ of 1be tale lo Can some ~~ anion but me ~~e~u~ is aid Embed Dig. He ~ coma an. lea 1b~ cab ~610 is 1be am of -~@e~e~ MY plus the each ~ ED of ~~ one, so 1be sum of eddy ~ on is tab 1imes~@e~ am of antes in 1~ ~ous~nss~_~nl al emit ~ Tanya ~~ proof bytes Edition. ma So #~ the glade ~~a ~r the ~~ alas in > ~ sum a Mica an: oat +oDi~ ~ +~? - (< +o#< :~ + ~< 1: ~-1 + ~(2> ~)2~-2 + . + ago - 1: ~)2 + at a) _ /, 1\~ - 3~ _ /~ ~ ~ . # ago ~pm~ help B ~~ the Is ~ aver of 3. BE gasps Be mod Gail obse~1= 1b~ junior lids ~~ is tag we may divide Be sides of an -I into tab emus pans Abe Regions divide tam- cud into 3' s~1 cubes (ague 46~. He ~s~1 is ~ smog cub Mung ~= c~ gem of the of cub> one gem ebb edge one gem _ ~~-nsion~ ~~> and ~ on. We And ~11 cube is in Me center. Is the 1~1 numbs of smog Mush Mica is S~, is fug ~ 1~ sum of me number of Pubes in 1~c ~+ub~sin~ lbe~ is one small Cube ~ each point, ~> ~~> S+~) ~c. One of F~eddcb FYoebeEs kinde ~ Pen gas as a cube subdivided info 27 ~n~IIcubes.~He could haveIik@dtbisOn~ d~mon~ion. 1. fitly Edwin Abet. ~~ Radon. fiend: ~~ & Co., 1884; Humerus I, ~ ~\ (1926) ~ ~ {1952~. Bamboo Tboma~ ~6 ~~ ^~ ^~: ~~, ^~ ~ ~ ~ D^~. ~~ ma. #: amidic ~~ abed, W. H. Flagman 1 oon Brood ~~ ad Was, Is. Craig, a: Inlomati-~I Film Buggy. 197B~ 4- B~an~ Jack. ~~ ~ Bee age, MY: ~~ golf and Company, 1966. ~ Basely, ~icbacl. ~~ Egg ID D#o, CA: Academic Pats, 1988. 6. Cdlchlo~, guild. ~r /~ ~~ Dew Id. ah: Tbemos and Sudan. 1969. 7. 3 a a. 9. Davidson, Psldda and ~illcull, Rag. age/ ~~ gaff -~/ #~r c~ ^~ Nc~ Roc~lle, I: Ian Ampex of ^medca, 19$4. Deane. Alexander. ^~ ^~v ~~ ^~ Ad: PI Press. 1984. Egg, B=no. /~ #~4 /~ Id. Folk Egged: Bruin Publi~- ~i~~ 1986.

59 10 Ill, Boat. ~ . ad ~t Built a Caky gum.- in Bedims. Clihon {Ed.~: ^~ I. ~~ ma, Ad: Simon ~ ~b.u~e~ 1958. 11. Id, Fd~d~: I ~ ~~ ~~ ~~> #: D. Apron ~ Amy ~ 18~. 12. Oa^e~ ~~in. #~7 ^~< ~~ ma, am: A1~ A. #~t 197i. 13. Ace, ~m. ~ ~ ~# arm, CA: Calm Ion 1978. 14. BEAM, ~ci^. ~ ~~ /~ ~~ ~ ma, ma: arm. Stags. and Oi~ux, 1962. 15. ~~ Deny Spatter. ^~ ^~S I ~ ~^ ~~ ma, #: ~~, 1911r 16 ~> Pence and Panic, Ads. ~~ ^~^ Peg Allo, CA: Dan gout ~i~` 1978. In, alar. ~~~ /~ ^~~ ~ ~ ~~ ~ I. sambas, ala: ~11 17 ~ . ~ _ ~ ~ ~ ~ 197& 18. Person, 1~. ^~ ~~/ arm. ~- ma, #: W.~. Reams ~ Co.. 1988. 19. trucker, Rudy. ^~ ^~^ ~~: ~~ ~ ~~ ~~ ~~> SIB, ~A: moron I 1984. 20. Tuba, aged. ^~ ~1 ~~, ~~ ~~. Cbe~i~ Ha: Graphics ~ 1 983 . ^ . Ills, ~id. ~~ ~~ ~~ ~~. Lambda, Ended: Came ~~ 1968 Wing, Id. ^~ ~~# ^~ ^~ ~~. Milan Barley {ad.), including a glib of Friodd~ ^~- by Elena Blake, Sphn~eld. ~A: Dillon ~ 191) 23 fighter, Ma age, el al #~/ I. Diddle G=dos ~M~bom~i~ P#cc ~ ~ - 1986. _ ~ . 22.

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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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