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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

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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 OCR for page 11
~ 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 bis 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:~nthe Dew number =d Amyintermix :~ 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?~:~:~-

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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

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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 =~-

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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~~

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- 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. . ^

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~ - - 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

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- 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.

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2:e =~:~= ~ ~~y ~~:s rea~ for the Die techniques :~d ~r a~d ~~- mat:~. We shd 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~. ~6StU67 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

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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~midislben itS 101~ flute will be (1/~)(~ + 6)~2> ~biletbe ~lumeofl~es=~llp~mid ~(1/3)~2

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~ ~,~ 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 ~~: ~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 :~) ~

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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 tgurecal~ ~ 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 emergesm 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 tablethat the number of ~~es at an~v stage is t~0 ~~ 0: the :~s number of ed~s a~ ~~ Awry In, - ,,

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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

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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,

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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~ OCR for page 11
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.

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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 comem 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:

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- ~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

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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#< :~ + ~ ~)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.

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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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