OCR for page 13
¢~ 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?~:~:~-
OCR for page 14
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
OCR for page 15
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 =~-
OCR for page 16
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 ~
$~m - ~e r - en* ~ pr~: of :~e
P~:~: th=~. ~e w^:~*~e on
the hypO`ent*= =~Is the sum ~ the
~s on the Ic~ or the n=t tnan~~
OCR for page 17
-
17
~ ~ ~ as tags of unit tbic~: ~ ~1 did in
. ~~
~_~a Ala
by Its ~ a~ minip~c~ Dig Reck. 11
i~t I -h =61 Edges baa lemma Taut ~squ~ r=~ beg
y can ~o in ~ill~ion of Me Dawn Ream (Firm 6).
Can p[~ ~~ Tic buzzes mat illumed ~ompo~Jons
~11 Ed il mob easier lager on 10 Pew ~1sj
~~ ~~ ~~ ~ ~~ -at
is abet ~ ~~e awe c=~ ~ degas into Be Bead pieces
gong ~ ~~ ~- cube (ad ~S)~, Jo as a ~ is
-~0 ~ ~C0~1 Inns ~ Dine ~(~ 7).
I
~^
an.
noodle S. ~c dinged music
tion of ~ cue inn cadent pyramids
=n ~ illume by begs built ham
desponding Splat.
~ ^ ~ ~ _ . . ...
~~ 7. 1 ~~ ~~- ~1~0~
of a ~~ id I amp td-~$
sew ~ ~~ Id to ~ singe dam
Iron in tags mend.
. ^
OCR for page 18
~ - -
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
OCR for page 19
-
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.
OCR for page 20
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
OCR for page 21
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
OCR for page 50
~ ~,~ 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 :~) ~
OCR for page 51
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* $~ $~
A ~ ~ plants.
choOw ~e new point ~ the i1~e cOntalUi:~g ~ . e - ~) N=.t
c~se ~ new po:nt not :~ng on any of the three lines d~ed b~
the edw alreadY con~, and then conn~ this n~x mint to the
preV iOuS th~ ~ dais Mel~ three new c~, f o r ~ tom! ~ ~ six: ~
We =n ~t this process to d~~ the tgure—cal~ ~ co—
—I b~ Dw ~~s (~ 415~- nrst choose ~ point n~
~ any of the six lines containing :~v =~d ~~, and ~~n
co~= it to the previous ~ur points ~ ~wn ~-~:r :~ edges—~r
i. of IO. ~ simile: mnst~n c~ pmduce ~e cOmple~ Mach on
slK points and more lf so Flax ~
Who :s the p~= that emerges—m this pro=~:~9 It becom~
.~t i? we a.~e the r - ~~ in ~ table.
Nu.~: of acing:. ;E
If In ~ !
3
3
6
~ .6
:n 63~ == t~0 n;~Ct 0 edges :5 the nu - ~: 0 p51~5 05 p01~157 ~~ct
ie~s d~y to the ~:~v o~f comb-~tions~ Based on the s-~e off
constmction, it is easy to ~ that the :~:mber of ed~s at stage ~ >s
the sum of all numbers Ie~ than n. W: =ample, the number of ewes
fo~d Alit six points is ~ ~ ~ ~2 ~ Ax. ~ ~ ~ ~^ ~ 5- some students maY know
t6e ~~:~a n;n ~ iffy tor t~e sum o: the 6~t ~ integers, pertaps ~n
co~on with the famous story o; :~e Amp Gauss wh~o us ~ thi
COOL up all the n~ mom ~ to :~O
AnOther t~e Of
patte:m :~s ~~ed by the table—that the number of ~~es at an~v stage
is t~0 ~~ 0: the :~s number of ed~s a~ ~~ Awry In, - ,,
OCR for page 52
52
^~ ~ \_~7
~ ^ . _ ^
~ a, ~
plate fibs Ids ~= ~ Bung of 1b=~i~ dings
tdan~e. Hence Sunlit :~e$ 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
OCR for page 53
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,
OCR for page 54
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 55
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.
OCR for page 56
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: ~~ ~$. ~[
eXa=~07 8t =~5 of the 16 vendees of ~ ~~:~m ~~ :¢
r ~ m~ ~ 64. BY ~s pm~s co~ e~ ~~ ED ~ ~e ~~t
r of edges :s half of 64' or 32.
~ .. .
At =~h ven~ mew :~e ~ cmalD numb~ In: amp: H~
,~S m~ ~ ~~:e ~ ~s to :¢~= twO: e~s I: amp: - ~=
edges ohm ~t ~ ~ wren
A A" _ ~ ~
On~ ~ ham -ant Me t~mm
am~ :~e tour; :~ r~n wnee pies Or ~e second, Berg
thew ~ ~ ~ Firs" ~ belt' ~h ~r of -ems ~~s twine ~:~ fig
;~:~t, once tn =ch order ~ ~~se I: 0m Dyed ~6 ~~nt squares at
~ ~ {6 ve~s to~—t~n ~~d 96 Is ~t each
square is c=~ed ~ur t~'—=~ ~ ~~—h- ~ lts wmc= Hence the
tme I- is -~4 ~ 94 ~~ in ~ :~. ~ ~~ =~s
e ~~t munt of s~ =~s ~ ~~r ~~s ~ we .~ ~ ~:~
of ~e ~~, but ~ :is ~~ed by ~ :~th~ ~ wo~ ~ ewn
if applied to ~ -~ c~"
S - - Ply
Advanced ~~:~:s can Sprees ~~e rests an ~ gen~ ~~.
t ~~ (~' no denme the num~r of idles :~n ~ n~ube. To c~te
(~, n) we ~i:~' as ~e, by Amp how m~ k<~s there are at
each ve~. E~ Kate is I; ined ~ ~ smut of ~ distinct eats
tom among, the ~ e~ Ala- fiom each ve:~. Th=-~-re the
-I -of k~s ~ each ve~ - is ~7~) ~ (~) ~ ~~ ~ ~~, t5
_ ~ ~ , ~
=~n of ~ shims ta~ ~ at ~ time. Since thew =e C(~, 3 k-
=~s ~ ~~h -of the 2n ven;~-~, the I- n-~:~er of k~cubes appea~ t-o
be 2~C:~' n)* BY in this count each k<~e is counted 2~ firms' so we
-city ~ tt8t Dum60:r to ~ t5e 5~ ~~. ~ (~' n~) ~ 7~ ~~p n)
~membenng the panem of powers -of 2 the come—m the su~ of
rows in the s:~x t~, ~ natumIly seek ~ similar pattem far cubes.
Tn this ca~ the en:~s in each row add up to ~ 'w-~r of 34
O-~s
~ ha_ = ~
Lln:~.
.~.~"
:O
:0
Hvp:~- ~ ~ )'
I
4.~s sum
24 ~ 1. s:
OCR for page 57
-
~7
_ .
_ .
AGILE 46. Subdivision of 1bc sit ~ dealt at. Id
cuts (and ~n byes info 1b:~e eq=1 pass #^ 3, 9, 27,
or 81 similar small O~-~y$ ~ tar of 3
OCR for page 58
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.
OCR for page 59
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.
OCR for page 60
Representative terms from entire chapter:
shined vernon
