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OCR for page 139
A':
~ ~ ~ ~ ~ ~ ~ of ~ ~ :~ ~ ~ ~
we encount:~:r . Ad: the time - ~~( - ~~. in ~e s~n and
n wo~, in a: ~~s =d video lmmS, in omamen~ design
am ~ =i,—Or in tTi)~C p81~' and in city
abUity to Pi inte~' and create pattems is the kale to d~g
wi~ the ~~d around us.
Shamir 8{O p3ll0~. ~~C 55396S 8= V15~ ~,r-160~! TO —6~-
50~S S~S CiOVC~ ~~, C - V$~^ S6~' p~. 0~,
Bile 0~t ~~SiO~ ~~S ~ ~ ~ Id =~:~$, ~~C
highly ab~= and a~le to w~ ~~
"~e incre=~g p~t of puz~s =d ~= b:~d on the tn
of ~~s and :~.ti~-~s lllu~s the ~:~x- ~
forms =d their relations :~ld Or maW p-~- ~~ g=~er
Branko Gmnbau.~- "~s are - -A :n the simple W~tion of: ~
mund' ~ motion' or ~ ~~nc ~:~e, as I~ the intricate asse~:~ieS of
~ ~ x ~ c~s o.: cells into higher f.~-~s of bite or
Iess other examples of o~! 6~:~087~ Mimetic pattems
can se - e as rela~y aim- =~$ O: =:~7 t:~5 05 phenomena, and
their stu~v ts possIble and de~:e at all levels
But despite their ~~! impo~' students Iea~m veW Illtie
t 568905 it - ~~) K ~0 8~> of shape tas h:~v been wL
sumed u~er geomet~ (li:~ly "~h measu:~-' Bike for a 10~g
time has We~ domi:~d by po-~.~/ Biopsy and theCrems of EucIld.
#
139
OCR for page 140
I
d4~A,~-~= ~ A~Y
~~ as Shy ts =t su~-~t ~r lit-~;~e and ~~-~s
not s~i¢ient ~r =~y, so Euclid is n~ m~i-~nt fin—m~.
U~ wholes ~n all t~s =d p~$ Emit wrom ~~t ~e concepts
of Comets ~ ~ ~~w and ~~ be c~d t~ m~ the medicos
ava~le to Lima ~s he d:d not vmm ~wt ~ ~~ of :~s,
networks, or flexible ~~, all of ~~ are of central :~e ~~+
is ~ ~3 =~, =6 ~~ t~0 lO Ice
~ n~ tO ci=~ ~.~w bm WAS ~ -I 1t tn ~~, me=
inn and m - ~+ Pmpedy devoured, ~ nudy of sham
cent~ =~:t of m~ education' ~ co~t the d~
on a~ c~= to n~ cays m~i~ but alm ~e ~~s and
maw 015~: )-~nt concep~ `45~=
we 68~t ~ prems~y w68t `486~- ~~ pa~v ~~e ne~t ~~s
of ~= are ~~s bong d~- We assume we know - ~~ sh~s
are, =070 or 1055. we know one when we s~ One Or we see ll mth
our ~ or in o~ tma.~:~.
But we ~w much more than whist We know that sh~s m~ be ~~
in some ~xYs and dissent :n others. ~ ~~ll is not ~ basketball' It
both are Oh ~~d su~' ~ man~ -is not ~ ~07 6~t 50~ 8=
~~;~sY We ~~ ~~t Is may have dot pm~. ~ treadle
~ ~ ~ Y K Yin ~ ~ ~ ~ ~ ~ ~ :0 ~
that ~ ~ ~ =~ ~ d Kit K
are ~~ - our shadows; -arm ~~= they -~ in size and c=~r
throughout the—.
:~ ~ ~ ~ ~ 50 0 - Hi I 0= ~0 0t
A"
the ancient G~k philosophers. to disbar si:mil~ties and -I
&m/~g ~~3Kj ~ =~70 th6 =~ of An; =d tO r000~26
IS in. ~~t it0~. ~~i0~7 3~57 8~6 :~
=~Mi~ =0 0~: t~ p~3i t~. 0[ =~07 these tools are dosebt
into so d:~tinct:~s among 0m are to some extent a~-~ficialY Is
~m-me~ ~ too] ~r dass~fy~g patt-~s or ~ to~ for ana~i~ them?
Tn ~~' it is both. N~nheless' it is helpful to days each of these
~~:~.~N
O~e of the =~t ach:~:~s of ancient mathematics was We d:is
e~ that them are exs=~7 ~ (~x, thwe~imensionat 5689~5 ~80
5~5 8~6 00~ 0{ t6~{ p0~57 cite the same nu0~r of
polygons meeting at each -my ¢~e shy known as Be rehear
~ ~ ~ ~ ~ ~7 ._ · ~ · ~ . 4 . ~ ~
po~ are $~V:0 in tIgu~ ~
This Is ~ 68:it06 t~6 imag-
:7~0'~: of the ancients that Plato made these shapes the co-~e
-of h:s -theory of matter (~e his dialogue [~!'y; 8~6 ~6 ~~]
OCR for page 141
mare
-A ~
Kit ~
:~
- ~
FIG:~E ~ ~e :~e =~r - ~~ mb Is =~ - of ~ sI~e ty~ of r - ~r
=~ m~h the =me n~—of ^~;~s ~~ng at =ch comer. Th~ t~on'
, and :~r<~n =e m~ ~ ~;~, ~~e cu~ :s made of -I a~ the
.-~ Is made of wn~
mu~ ~ ~k X][:~:I of kits ~~ to the~r =~^ 175~v ham
:~on none of Belt ~~:na~on IVY.
~ m<~ ~ ~~e ~~ ~ ~ Am. ~ fiche -I f th ~ d~:~: may of
the regular polybedm. In its time it was a mace ~t of mathem~
)n ~0 tm pi806 it ~Ct ~ -£~
; ~ =~ ~~ ~ ~~ ~~ ~~ ~= the Bier 2~ $~: ~ C0=i~
~~- ~~ in ~~ =~t t00~0 ~ Me ~~s have pmpen~=
that Ah them—m miter ~~ts and be ~le to ~~ze
- or dist:~shing £- -in ~ ~ambi~s ~. Seco:~-~- you must
be ~le to u~ these cntena to tnd out precisely dice o~ts ~~
them. ~ one ~~s just how the ancients made their d:~, but It
is eaSy ~r yW~ -mailmen tod~, e~eciallv :if they have ~~ar pol;~3~-S
to pl~ Ails to convince ~~:~s that the Bit of ~~: polyhedm lS
complete (Figure 24*
The k~ i- of math~] cIass:~=ion wew ~~ in use
ah- of arms ~,~. by ~
id t~ obj~=s :n that class. What ho changed thmu~ut the :~057
8~6 ~~} CO~ti~6 tO C~' 3~-0 t6C ~~ Ot C53:~:~tiO~$ t58t seem
impotent to us an-d the Ohmic that we use for enumerahon" ~~e
3 chows several cIasses of ~~=s that ~ ~ ~~ed together Am
~ mathematical point of IBM ~5 such as th=e =n st~e
student d:~n What properties chametenze each cIass~ A:re :-m
~~t wavs go- clas,~ 7~e o~;~ what mb ~ ~ ~
Bide cla~9 We mention hem ~ fen of the -~$si6~n s~:~s that
~ twe in mamt appi:~atiO~$.
-~£~g~ I Si~ ~O ~~S 3~0 -~; it t60V 8~C CX~
80117 I; Berm tO t~ i8~!;it 6~;~11 X~ {God their position :n space*
Cans of tomato mop (of the same brand) in ~ He—. store~ square tiles
on ~ 0~, and has in ~ -~;~t pattem aw all ~mi:~r e:~s
of con=~t 6~-~. Two o~-~s are similar if tt0)r Bite O~7 it pO
section and wale. Similan:ty teems to be ~ w~ ~:~dam-~! In:.
Preschoolers unde~-~d that I an:~- doll clothes' and pl~v
houses are all small versions of familiar Hi: The :~ct that even such
young -~il~:~n know ~~t t50$e fit o~s are su*~6 to mpresent
OCR for page 142
1~42
^~ ~ ~_~:
ma
/ \.
~ ~ .^ .
\ it/ \
ma:
b
~ 1
d
~ / ~ /
e
^
v
Of
.^
. A.
\ /
^
FIGUR.E 2. lb~c am only h~ _~; -~m ~~ abed ~ only ~ awns
IS of con~nl, _1~ ~1~$ gut a ~~1 tba1 ~n ~ gad up to maw ~ ~~`
~~1 ages. H~ ~ ~ the Me a~n~men1$ lair ash big completion as
ages ~~ =n ~ ~1~ up ~ ~~ me aria gyms.
saws tab they 1niu~i^1y unwind change of ~1~
Build and
wing ~~ scale gels o1 1~> an_> houses, sagas of ad kind
Me 1be chi~l~d~f a~ alma h~ Map of ibis if.
\~: A squaw is ~mme~cil: if you ~c
9~> 18~,27~>~36~~~. ma.
il bas Cur lines of mirror symmetry across Daub you ^n Hillel it onto
itself Chum 4~. It is gag ~ think of other others 1h~ bee 1be same
s~mcldes, or selLcon~en~s, as 1be Guam: Abe Rod Cross sy l,
a braced ~i1b Cur buggy speed ~ads, ~ circle of Cur dancers and
a Leaf clover (~i1bOU1 i1S stem) are a ~ examples. Summed
dodges o~ecls avowing lo 1be a~n~mcnl of 1beir con~ilucn1 pads.
OCR for page 143
~.~:=
:c
nwRE 3:. ~~= of ~~d ~~w I: tn~ u=~ c0~-
~t -I tt~ ~8 i~ =.~t 0~$ It ,.~ (~0r~:= ~
. ~ ~
~ : ~
or:' >"" ":~ ~ ^~ .w:.:~,,,^
.' ~ ~ \:
~ . ~..,~,.~,,,. .. ~...~ ,~-
_~
"em N
,,rr~x... ~ <=
rile At' ~
as.. · a.,,
REV
ICY
his ~ ~ mher subde, for example~ the ~ polybedra :n Fume S:b
:~.~ the same ~~s
- . Al
Just as commend Ieads ~ dry (—I'll ~s ~ =~:h~ n~e
~ =~7 ~ 5~. 6~8 ~~ t~ At
"~e ~~:c ~ct of ambetlc~ ex~- ac=~ to an h)~onan FIJI.
Gombnch ~ "~s th~ ~~t IleS sO~:~e - ~~:n ~~ and. con~
:- ~~s this lS o~= Of ~e reasons w~ ~s =d o~= ~F
s1:~ar ~=s a:e ~~t
8~> :8 HIS ~5 t~>r~= ~6 ~6 ~ )05.0 ~~;~. Self
si.milanb~t has r.-~y b=n reco.~i=d ~ ~ ~~d concm in namre~
The awa~ng ~ ~ Ned pnze for the formation of "~al:~ion
=~" and the cu=~t wo~e cmss~iScipllna.~- instep in ch~s
shag md:~e the profound implications Of similarity and scale Ah*
s:~e and mat:~tics~ ~e ~~ of ~~g MS ~im:~d (~ b=n
im*~ 'by) the stu~r Of - ~~s a~d Other selF5i:~:ilar ~~.
ford*
~~! pm~. ¢: and sl.mi:~:~:V =e metnc co:~*-
Or if, bC Hi b: ~3
4t
_
Am. '`
- `,, Mar 4tb
f
5_
~ ~ f
_ _ _,~ W ~ _ Ala
r ~ ~
<' ~ `%
s
:1
:;
a:
-
~-
f
.
.
OCR for page 144
1~
^~ ~ ala
o
He
x
r ~ ~
o ~
hi
v
^
i: ^^ ^^^^^^^ ~
1X
U
X
IO
. :
nGURE 5. ln =~ h^~ me ~ sides of me bow ~ i~nliBc~b~
~~ ~~a~
man ~= wand ~ ~ an i=~ ^~ ~ ma ~ ~ gas. ~ you
~1 Limb ~ gem sills are guilt in tog _~_ ~me?
propels of saw ~^ ~~= under sum Lions. Or
example, the numb of Is and Prices of a Plan ~ ~1 Plead
if ~ So or bend 1bc pomp. Thus able 1b~ b as of Fame 7
a~ al beacons' An about may are nailer content nor simian
beg is a~ cloud top of six line Seals. Bets a began
a~
Idly AL 1be combinalod~ proxies of a shape ~ 1be
tub we can gum Ad the ~1~ a~ hoed lumber. Bus ^~
be combinalod~ point of vita lbc shapes in Fiche 3a a~ e-~-
Ien1> d~ each has 6 guess 8 vertices, and 12 ED connected lo eacb
giber in lbe He way. ~61~ problems Hen involve c~bin~o~al
problems. Or ovate, if ~ ~n1 lo demo a lining system far 1be
compules in ~ Filings ~o ~ Unaged Al wilb boding 1~ poodle
a~n~e~ls of lions and nodes 1~ An outride Me conneclions we
want, and only then need we consider bog low me -1~ HI bag 10
be.
/~ ldpolo~i ~ equi`~ienceiseven more genenElban combing
narrow eq~uiv~ence.~F~om the ~andpOin1 of 10pOl~Ogy, allpolygonsare
loopsand allconvex poRbedr~are Mike. Piggy a~guedlba110~polo~cal
concepts occur pdor lo metric ones in child development; a Old gnat
~c~izz a low ~= disli~uisb.ing among kinds of loops, such as
age. ~~a~
papery of shape.
. . ~ ~
- ~ ~
Cupola in school is own described as gabber sbee1 Omen.' I1
yields maw excellent examples 1ba1 can enlarge a Tildes concept of
Be Darius of shag. In Or ~ mew ~ Ages of Fang
Sc am indi~inguisb~le because each can be defied ink lobe other.
OlS, of the boy and Id SCOUT widely am an excellent subject ~r
bandwagon s~dy.l2 Cbi~Id~n can leam lo play Dime on a 10=S and
OCR for page 145
145
an_ ~_{^
..
aim co saga_; ~1~- Casio fly
games Pied H~ Pa Vernon globe ~ user
1~. Opus ~~ _~ he ~c~of~e~n, ibid
lbe ~~= beech ~ ~1~~ ad ~ Pus Ed ~~
endue ad ale, ad abet cap of ~~: gab ^~^
adzes he JO divans ~ a -~~ ad ~ lass Anus
zeal ~ gnu oaf. ~de~di~ cab chic ~~ y
~~ ~ Menu ad ~- as ~11 as Them
~ ~ .
~~_l~g
Sb~s nag naves. 0~ of ~ mom Ames as of Anal
is ~ asp names 10 gags. Ping is a phmili~ ~n~pl ~ is
ecb~d in ~ gabs ~ ~ as in ma ~nl~ Pious panics.
aria is the 6~ lo ^~ Pier il is lo age of a
aeon or he name of ~ sable. ~ Ant 1bink Soul slopes (~
bias d~ fir tat Myers or exploit our ideas lo oafs if ~ do
-1 he naves. Baling 1e~nic~ ages is Regimes did as
ale auivily: but such optics miss 1be pall. ~boi~1 nag a~
usual no airily; 1bey envy ~o Angus Amen in Cab
odorize lbc An me ~ naming.
far eared in En~^spea~ng counldes IN nap in~~1e 1be
Dig a~ b~1~ des~se an ~i~1du~ in ~ Big. Thus
Jones is a peon a ~~ To is a member of ~ Jones fly.
He ages of shapes sea similar -=ions: s 1e1~b~~n is a memo
fir of 1be an gaily, a ~~n~1- of lhe Sicily of boa
Bleeds lb~ bag Cur aces (see ~~ 6~. When ~ use 1~bo ~~
f ~^ I:
;""''"'' ~"~ {~
/ > \. ~~ >'1~ a\
// ~ ~ ^
~ ~ 1' ~~} \,&
~ / `/ )~` If / #<
1 #/ ~ ~~
~ ~ ~~ V/~# ~ )\
2.~<`~}~\~34 ~
~) '//~ ~~ ~ i!''''
~ ~ ~ ~ \
< 1
~ . ~ ~
t ~ ~ /
~ ~ ~ 1 -I \?
7'j'! ''ail. ~ .~#
~ >at} ~
~1 ^
~2 / / ~
I
/
1 Atom ~
t j FIGURE 6. ~^ Jonas is ~ Labor of Abe Ones
a... .
-# Emily, and Abe 1e:~b~n is ~ maimer of 1bc
~lybed~n Emily.
OCR for page 146
146
~WA~ go: ~Y
Awake - ~- ~ =~e ~ sh~' we =e ~e "me dme Agog it
y t.~e a~ (^C-~ g 1.~. 1.~. a. m..~.~l ~.
Ion-fires pmci~' Beret ~ no I- "r-=
:~o cI~ she. ~ are cask ;~.~mH~ad:~
ili= ~ many ~m ~s,-Brig = ~.e prances
us Wt example,: t.~.e do. mt the o~S :~f ~ ~£~6 the
win me- e11~' and ~ c~^ ~utiontzed-~:~..su2dy~.~-astr~y'
- :m ~s ~t ~s Do. - ~s ~ ~ donut," But
oh =.~r =d elhp=s a.w chic sechon.~ am. :-. th~ lo.= a,= ~e
bow. Age pout ~ chew of ~;y'~ ~ ~stmenon benders shapes
the encIose red 1.~e b~' and sh~= ~m h=e holes In them'
I::ke bagels.' :s end; wide ~=e bmad chased ~ ~es
aide But ~ f~b~ doer ~d n~ be happy m!h ~ bas~b~] as
s~, nor ~d ~ b=~! blat ~ whim to- :make ~ - h ~
bas~' bemuse the Al kids of b~Is have c~ It
pro As =~the:r e~:~e of c-~si:~, arching kid
that it ~s implant to b-ui.~ houws th~ =e sturdy, n~ hous~ th.~ might
wIlaps-~. This mn== tmnscends o - r ~s tMt h~s :~- commonI~
d.~, such as ia~ and s~' Bile ~+ cr ~U~7 t08~'
or Ink..
CI=~tio-n ~.s dwel~ ~^ w~Y ywng chiffon team to
wco~ize ~ =~ mam-~ s:h~ without be~g f¢~atly ~. emit
wodd :s Am: ma~ of ~^ sh~s thet ho-Id ~' such as ~s
a~ bags and ba~ ~, shapes to A- ~, such as balls =d p~s and
5100~' 55~= tO- use' such ~ Axis and s}~ns am b~. ~Ms
r 5~]r~ children team
FI:~:S ~ ~ Th^~e hex~S th8t am
t~ IN ~ it~N :t 0~ i8~ A ~,~0 p}3 ~(
I. ~ j),) 0:~-~,= ~ ~ ~
7 ~ ~0 ~) leg) tN0~$
In (b I: and (~) a:= tnt~n
ran. ~t ~ 3;~0 -~;33~t N0~:5 At
~,.~N;N - - ~Ntr~$ I; 0~t ~
008. <~t,¢ ~f\,[;85~N~$j (3,33 0~],) ;3N$~;0 CONS
o~f thew 3~0NNp0$
\/ / ~
ARC
.~
/
OCR for page 147
tat?
nG:~S ~ FOur natum! ~ (~) ~ ~ the =~ palm (~) homs Cf ~ :
~) Act: ~n Aft: ~ ~ colonng and 1~' :~< the ~ :~* ¢e
¢~OD 553' S~=S ~ (~CO Ct=~¢ =~$~'-p:~¢ ~¢ ~ J:~-
A ate~ =~¢ 306 ~ ~ ~#. ¢=
nam~ ~r some of them, such as I: In: pol~#~-~, a~d some
e po~h-~-
s, in our ^~Is :~fi~on and cIass:fi~atio~n of shapes usu~:v
~ just at ~e point - ~= they =n ~i.n to be reaLv mtem:~tin~
where they ~ to e>:~re ~es In thr=-~onal ~^ How
maw ~e T~iz:e th~ even ~s that are not h~ =n ~ ~ng
and I:? :~ molecul~ have =~! shapes' but o~n these
po - ~s am- ~:~ =d their Con~tio~s am t~ :~~ to -I
chem:~ pmpmms (~m 7~. Bes:~s fiche ~ns and poly~ns
~ 60~ ~ (~-7 ~= 8~: I- ~ ~ ~ 5~
By broadening ~e defi~;~n of polygon to :~ any cIcsed Ioop, we
may also ~dy knots" :In addition to ~r-`~us pectin importance
~r tying, things' knms enter into the design of reworks Cub as cIos~-
1~ and are helptu! in understanding the ~:re of: some biological
molecules So~ bubbles soap §~;, ajar f~:~s am also endless sources
of ~i:~g ~:! pnncip-~.
The ~dv o~f pol-~ra can be cxte:~¢d from s::~le shapes that are
easy ~o =~: ~o othe~- such as star polybedra' that are more corms
9~- ~V I'm 870 p8~55 5~05 35 ails 0 t~0 pi8~C, t~8t
are beautify.! as well as use~. The helix and the spiral am ~.~!
~.,;- - 4;,~
OCR for page 148
1:48
~ Air ~O,4~ TO N~4=
to t:~ an~ a~ AL web as~ to~ ma~em=cs. ~: 8m wen tm
day' ~ "~.~.e,1.iX~ h=~ ~e ~ ~ houseboat phrases. ~w
p - ~e reach; ~¢~ Isa ~ 6_~'
hit ~= =~ =~ :~: M::~a con~ dis~:~: ~' ~d ~ ~
(~m if*: :~:~s~d :~m ma: ~ day I.: - - ~s
Atlas ~+ Awake-~ ~ we Ace: ~ 50; ~ if ~ Our~ ANAL ~ -~
t - f id, ~s ~ whM~ ~e unl~ ~d :.: h~ ~ ~ axis
wew ~ 5,¢~mest : ~ :~ ~ ~ i: I'll ~ ~ i: 'I ~:
ANTIS
: In; aver to ~;~=t and awe pa=:ms >n to-ts :mage-pae3~d
I :~t IS: n~ end ~ ~w =~= s:~= and ~,
w~e ~o ne~ to an~:;~e that ~s ~ us to He ~ As ~
I~ sha~s are butit -A s~-ones and ~ Awe =~ms =d thmr
pmpe~s . ~ ~
Wan chi.~n m~e sh~= out ~ bloc~ or ~' th~ oDen imitate
the d:~e com:~sit:~s ~ ~ see ~ar~:~em (Firm 91. Nature
too creates patterns. Like man-made In n~T ~s a~=
,~,t, Wagner I. art ~ 0~.;~*i706 into =~ ~0 ~::~;0 ~
0~i~ iOt0 0~5 i3~:6 =~57 ~ti~ oily t~= 8= 00~n the $~U~itS
of ~ h:i~.~ :=re: o~i=ti~.^
Warm =0 -~=i~0 p-~5 =~97 ~¢ {in:] i;~t t50 5~6 fats
.
and a~s Bear ~r and over - ~7 - ~ when the Elects
_:
n:~:~E 9. Many ~^ ar~ bu:~t :~m ~r ones~ Th:e
~:~g beams a* ~ bn~ :* :~w wpeated p:~:~:~s a:m
USES }~ ¢~g 3~6 8-~-~' 3$ t~ ~.~' t~ ~ 3# ~0
out Ale p3#-~S#
OCR for page 149
E t0~ Young: ch~= =n :~e the ways :n - ~~h
6:~ t~r ~ t11e a. p~e '~0X
— . ^_
invoked are wo did ThIs is not Just ~ mincide~.
~~ -of mo~ oat=:~s :s Herded by ~ By few baste ~~s of
fo~: ~~' and d~t ~ ~
:~k ^~m In ~~w ,0: Peter St~= :~s s~ wa~ ln which
n81~] ~5 8~6 ~~ 5~6 35 ~~$ t=~& =~'
p8~tiO~' CiO$C ~~, 3*~6 C=~^
`' ~ The results ~ thew m~s
Of ~~=ion .~.m rem~y If.- - ~~e \,~.e vane~~ of mate~s on
Bail th~ - o~e (~ee FI~= 8)
]~t a~s of pattem -I can be grasped ~ -~-~g
the ways in which cop~s of o~s -me be pa~d toge~-~* Stu~s
q=~V hisser ~at ~~re are onIv ~ I- ~ ways to -do Whist delis
ndamen~ pro=~y of Am can be studied ~ ma~ I~^ For ex
~~ it can be smdIed i: and "~ when the o~ts
BAN ac~d · ~ ~ ~= ~ ~ tA
quadniateMs' and hexagons Cadre iO)*
Ol~r ch)~:n can expen
ment with ie~ re~r -amps and discover some sumnsing th-~- such
as the ~~t that am~ quadnIate~, Been one that 15 :~t co:~' mi! t:~e
t:~-e p:~ne (:~re ~ ~ ). (~ is ~ Su~nSing but ve~ simple cO-~egue:~ce
of the fact that the sum ~ the i? of the an~s of ~ -~:~:!
is 360 -3 Him ^~1 i?? 030 51~ deeper pmpe~ics of sphere
P30Li~8 AXES Ii1:~' =~5 8$ :~Ci: a?) ~.~ - 3~6 t0~ rhea can be gener
ate~- (~:nxbaum and Shephard,~ ~~ and Patl~~0 is the definitive
resource for *~rial on tilings. 3
OCR for page 172
of
172
^~ ~~ ~ \_~c'/
PI ~~ ~=d#~ ~ gait ~i~$,~~=,
~ It's
paw ~1~ ~ pit ~ ~ ~ ^-
~~ ~ ~ ~ SO
Ha; fit
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Same ~ ~ Nazis
Cadge; =~n
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e ~ bite
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ma; Amp
tb~
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DO Ian a~ using
simply mans
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Clef ~= and 1~1 cut
I
atoms ~ 3~0 --
~u~
^o ~~ ~1~ of 1~ abut;
potions; mat
land iamb
ala; ~im~~ibl: h
admit tiv~d~ ~cb~i=1 doing; ^
~
1c ~~ Talent a~ micmsco~ Has mat; tbc ~~m
Bang I ED card
Tuna ~~1~ Gaping On ~~ 1bc Cog Spurn Aphid
Spur
P1GLRE 30. An a~n~mcn1 of lOpiCS Flared lo $b~ abed
pates I Ad cam ~ -~ ma apse
as an milt collection of quite Isle loci=.
OCR for page 173
Ad:
I:
-:v
nGU~ hi- Me s~= ~ ~~e
~ ~ lt X ~ ~ ~ ~ ~ ~
. Th~ 15 ~ tM ~~ Of diamond
asks ~= ~~s ~ - = ~~s x
of ~~:7x
- :~5 0t :~t 563~5 - ~= TV ~~s come eas~ ~ ~~t
The SiE=n -my Which. bas trans~:~d Me indu:~li.~d wOdd lD
iust ~ few dec~s, is bash on ~ ~~re that is ~ camer Of tucre~ly
mad circuits.
^~ ~e circuits shed are complex,
~ ~ *~x=~ that hou=s them his -a s1:~e mo~r
stm-~.
~r -~ample' bins silicon Is built of I~d z~g ;
noes (~ 31)' ~~h am =~ to make end ~~ ~~ ~~" :
the silicon ~~= the nn~ aw i~:~ed to fo~ ~~-~:ke pol"~+
~:~ schoo! ~~n c~ ieam to build and identify these su
at middIe whoo:l children =n Ieam to p~ them to~' a~
heir school ~.~dren can stu~ the ~~:ion b~n the all-icon st-~=
=d the prope~s that make ~t so use~.
The ~T scan =d other ~~s of =~ed 1~ recon~
n h~e revolut:~:ized m-~: dig:- in recent "ma. Whim
hi 8~ 05: ~ 6> ~~> i~ 80 exer~ A
seen is an ~~e in wc=~:~ting images - m t:~cil* cmss se~.
t::~e the c:~uit~+ cn ~ s:~-~n chid the Ames r=~n used in this
t-Cchnolo~x 1s ~ com~ex pr~ss ~t the simple~ ge~-~al ~ -
that underlie it are easily unde*~.
Hew ~~n ~ 6~d that the same ~~c pnn-ciple$ are =~M to
manly §~. W: example the con~:ion of shapes ~om sections =d
~~s teas been the task of architects and builders ~r --. Wh:ile
it is not be to bnng ~ C^¢ scan :~ch~:~e or ~ co~tion sne
into the classmom, many p~ts suitable ~r school can help students
under~d the relation betwec:n shadow -of cross section and shape.
5~ ~~pg¢~:,g,:~r tt~ {gg¢~;~h0~3, 0~57 8~0 0~:~=i~" ~6il6~ 0~
ten ieam to make paper snowpakes in s-~^ an cX0'rClse that =n easllv
OCR for page 174
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Representative terms from entire chapter:
coherent mathematics
A£~ ~:~.~Y
:~UR:E ::~w Bmnc~d sn-~es BYE ~e ~r
s-~ of 70e c - ~$ w=~d ~ I:~e at ~ - sc~e
bee =:~=i to ~ ~dv of t~:r symm~+ The hex~! Mew of
the snowshoe decides an intm~tion ~ the Mm - of polygons' :t
:s an :deal submit :fior the elementa-.Y cIa:~m~
:13~ut the s=~ke has mu~ more ~ teach us. T~ the hm: place ~
snowflake :~:ks {~e ~ patmm ~ m~ht see :n ~ kdicidoscope, =d so ~:
ls~ This schists ~ ~y of the :~. Quiche. as we hew seen,
~s an Pain of the ponc~;~es of mirror ~* These same
refl~:n p:~:~$ un~:~d mntem~:~ te¢h;~* one need only
think of the :~fle=:~on beams of bu~= ala~s and I~ or of m~r
and sonar. M:~e schoo! Ohm em; eas;~:~<^ understand and appr=~t
su~ ap~i~. At the h~ - ~! i - ~: An- emer~e of hexagonal
symmetry fmm aggm~es cf wamr mol=~es can be explo~ and so
c'an the C - ~' dendnti'( ~th^A' or branching.
ache b:~hing of the snowflake :~s as cha~tic as its sv:~:~
t~ and is equally si~iCca:~:t in the ~dv of shape. :: =~= of
the sno~Dal
175
~ mile of geometry Is ~ pemnn~ Issue In m.athe~.~cs edum:~n
at ~T levers - m By sch~ to ~ - ~~* For Day veam
Geoff :~as been the problem c~d~of the~ - attics c~rr~lum.
glance ~^ the N~ - Junk of Cached If Demo
1987 Yak By: and Teaching ~:~e~s smug of
the ~~ quesi~ns invoked. ~e Aced w1ttL Good: :s due in pm
to Iack~ of See on - ~t Comets ill Land ~ ~ amid ~~ :~.
Do we saw it m~ discmI~ned ~9 To cream ~~s ~r
.,, . = x ~ .*,
mber so ~89 Or ~ ~ because ~ew :s implant c~nt tn Me s~=t
~~:~
= ~~~
~ him saw =~ - texts ~ m~t om exam~p~s of Comae
~:s i:n :~, scmn=' ~~o m'
con~ns ~s ever e>;~Iored in a~ depth. ~e Ethos of method
=d cat its usually unmeces^. Or Abet - ~~ =~s ~ unea~
=~m,9~mi505 It.. =~7 t=~t t~t mat jig ~~. teaching
—Dive re=oni:ng, pawing thecrems of acrid :~ problem
solving, tea~:i~ visualization, and preparing t~ ~~ ~r ~~*
The contin:ui:~g debate indicates ~~t :~e of ~~e goals is pameulad
wed ~d tn the ~~em situation.
In ~~t ~! of these Chad ~e Aching of Comet - ~~ is defended
on ~e =~ the ~t ~~s Be. purposes' r~r ~~n on the ::~-
po~ce of the su:>j:=t in its own n~. For =~ ~ wcent micle
on si.~:~? Justi50s the I of si:milanty ~h ~e ~~:~g r~
t:iona.~.
S:m~lanty :~as am :~d :n ma~ ~~s of tM s~ cumenlum-+ Some
Amens ~r rat:~ numter con~ ~~ ~~ ~~ MA t6~ p8~ 0{ tte
.
~7 d~S W1~ =~:~S =~Y - = ~ == p=~$ ':~5 SI=~ -
6~sK Rat~o =d pro~;~on ~ ~ ~
g—~ ~~ Age p~:~ =~y God ~o the ~~ Sta~d
$ 3:~ AWRY p=~ Wor~ pr~.—r~ an~= God
pam of m=y inwIl~:~e tes~ stm118r gec~C :~S would s=m to prOv16e
~ helpf~ men~! tm~ '$:~r other t~ of ~~n a - ~~. slWatIOnS.
All of thaw tC880OS are ~ ~id. ones, but there is ~ Go omission. ~e
pnncipal reamn for tea~:~g rim ou~t surely to be that ~ is of
pro~nd ~mpo-~ce in un~:~ng sh~.
Mean - ~~, Outside the halls of education the co:~r revolution is
Lipidic chan~ng the wodd in which we I:~. These changes aw placing
:~ew demands on the cumculu:~7 dem=~s that are just ~~:~ni:~g to be
heard in the whools. The r~ion :n ~e stud-v of shape and form
=860 p05~6 TV t56 :~et =~ts that ~ ~ ~
~~r compromise for Be but ~ new and coherent mathematics
cumcu:~um that :~m shape tnto the ent~= cou~ of stunts
0~$ ~ ~~ ~ {-?~ `~7
IS ~~d to ~ o': gm ~~s , As: ~ ~.~t ~ use~ que~n ~ need
to am ~~:~ ~t we -want our ~~ to Icnow and w~^ Euclid
red She cafe ~~g ~~ shops I cede 5~?5
of de~s ~d a~mph=s and: veer carafe jar. In o~r
to a~ze shapes, ~ud¢~ mu~ key bow Ibex lengths, ~~'
a~ - ~~= ~ w~ a~ Shear and ~~ - As* Icy need t0~ ~~w
pr~erti~ o~pall~ an~ Ed* l~;~c aid ~¢~' =6
~6amen~ property 0t figures" M0recyer.? t~ need to ~~ 60~7
web Mer and ~~~ to =~= su~ flare :ilgures as: It:
~es, far art?: 3~6 -~uar=-
~ - ~ they've Adds ~ t~iti?~ ~~tW cu~
n~:~` ~ it can=t ~ =~d u~r it ~ a 6~:?~{ =~0 07
ex:~r active- The shapes ~~ nudges need to un~d
tok, am the I: fat ~ need to be ~le ~ do m~ t56~7 8=
too Yast -a s~= ~r ~~. ^~ - thy is mns:~=We daft: ence
of As arid purposes
Tra~it~ ~~ ~~ms mow ~n i.? Id? {~?5 mt6
s1~ C-~-~+ Tt.s Mae In ^~! 1n s~e re~S :~S an~s to Me
cum~ar :~ of- ~~? =~= l~?~?~:?~- ~ ~~nt who
~?5 ~~ 0t ~~?~t i0~5 ~?~,5? ~0~t =6 mme imp his
t0~ 8~6 ?~50 309Uit05 t~0 68~8 ~: -I =~= i=~, )~g
0~: 0~ - ~~, ~~ i8.~, 0~ t~ 0~: t8~, 8~? - i05.5 =~
0~?8 vet =~6 0~. ~~v 8~ t60 avid 06~10 Ian~ages t~t =~le
&~ ~0 :~ - ~~ i:~. I~' As Ad I=m both c1~
an-d modem i=~, If:: ~w h^e ~e time ~ oppo~;~v ~r
50~. ~ Wart. - 6~ Alit 00~? - 0~:~6 Aims ?~5 0[ ~?~0
~~ ~ ~7 598~iSh~? 0t ~-~ is O0t t~ 5~D tO ttC pt05~
^~ 0; t~0 ~~ ~ Fit 8~6 ~~> 8~0 55?~ ~ 0~?~81 6~li60=
. F6t 0~: 20~ v=~ EucI:6 ?5 ~3 ~?~5 5?~6 ~ct 0~17 85
the mmer?~? - of gecmet~ but ~~o as ~e ~ amok of mmthemat:~
08~=g ~~ 7~Og ~ 3Ki0~5 ~5 5~O ~——Itt~.? not
?~ ~ =?~th0~ti0'5 5~ ~0 ~ ~0 8~6 I* ~: 0~$
it W35 ~*~5 ~i?~06 t Arid ~ schisms =~[ my? ~~?~ 0t
the r=! ~~d that ie:d to the ~i50~? of non~n ~'0~
—:ich ?subseque:~iV became ~~e ~1 too] :in ~~ng the 18~^Y~IC
?: 0[ t~6 I. ~. ~~?~] ~~ ~5 ~~t 108t it?5 ~~$ 50t
o~r r~s :~uire that ~ Do i: tt~ =~th6~?~l ?~
of :~m 18.~ cOu~s into our curncu1~:~.
~6 id ~ ?~t tt8t 0~$ ~?~?8 =?~> p?~$ 0: I ?~d 50i
0~" {t 0~ ~ =~5 ~~ti¢~ 05 p058~6iiiti0?5 ~: i=~?~ti~$ ~-~.
:
177
inst-~io-n—~ building mod~ m~ u~ Spiders Am ~~bse~-
ti~ t~ If - = =~ip~i~ ~~ ~~* 85~ ~~ Belt
tractmna~ po~Im for enta~:ng me foamy ~ m~emau= In
tI~ :n s~ ~~t At:.
~~ ~ ~ 5~ my: ~~$
m~ ~~=s in whim she pies ~a mie aw n~ used to of
as m~hem=~s in ~ n~w or ~~ -~* ~;~, pr - limps
of ~ze and sca~ ~ not belong ebb to ma~. :~:3:=d
] s~ of queries that send us to ~e ~~ or to mileages :-in
other ~~rt~- Could there aver have be~ ~9 Could ~~;~e
]e ~ ~aL as mice? our Ohs sh~ that Mess qum~s =e
-I man =r wco~d hi ¢e =~s -~e n~ so
applic~:~s of ~mila~. ~ go t cw~ -I be suppon~ by his ie~
f they were exa~Iv similar ~ our :~ - , intro, ~e bone mass ha% to
~ :;: di~-~ionateK~. This -pi malces ~ sm~ of
bob sod more fig ~= -it wo~d be ~f the =~$ ~~e
.
sim~p ~^
PO:~ ~ is ~~ I~ ~ CI8~, ~~ Optics IS ~ Id- Of
p~' ~m~ and - ~r t=-~s are -I cone - s of bail
0~7 bibs 6~10 ~5 ad bore tO =~l the M=~*:O Of
m~- Even within ma:~tics? she :s inte~i-~ina~* it re~
Wires vista! and m*~:~! s~s, :~1 ah- and many Aver
tools. Teach:~g shed :n ~ -~:~' meaningful ~ can stimulate cIose
coopemt:~n among teachers Of many sub*
~Skt.6 Dyer 0f Ache 8~8 p~18 Cutting 80~5i;5 ~~l—~~;~.
W:r example, the sturdy of sim*~lanty can be nicely Implemented by ~
~~ of ienses, requiring an -~*~rsion into Chow :~en the :~es
aSmelated mth many' -of ~e :1.~ of 0~c oW:~s ~~e g ~~s
pnnciple) stand as te~*mo~v to the It that Wave discipline boMem
h^e *not owes been* so hide It is -~iv be~se we 6nd the same
Ago Or; t6~ Fig ~~ t~ 5~i th0=—35 p3~ 0f ~ma.~ema:ticti
in ala? 6~ ~ ~ 00 ~ tt~t ~ ~
~e stu~ of -' tS ~ Ia~ Sumac.
All of us' Children and
adu:~' ieam ~~t shapes by? mating them an~ ~~r~g m.~.~5 Chat
ure 33~- As an ancient pmve~ s~' "l hear and ~ ~~, ~ see and i:
m:~em:~, ~ ~o =~ ~ ~~'
if Arc wis]h to build ~ ~~pe—~ cubes ~ 5~} 50~e, 0;[ ~ chip>
~~r polyhedron—we :h~ to be able to cut out ante assemble pieces
of the ---I sizes. Th:s is once of the reasons that Chic ~~Y
(~0 =~' p8~) It 306 50 forth) remains i:
~W /~:~S TO ~~y
:
~ 0~ ~~ ~ ~ A ¢¢ 0~: ~~
t~ H^~:^ ~~r $~ ~ :m~d.
Co.~` h~ devoted h1s , to, Age,
en:~g patw=S 1n ~~x
Bu~ mo~' in shiv ve~ =~te Sen~' is one of the be~ w~s to
undo t~ =d prayed=.
:~n expenmentatio~n is essent:~+ F6r - ampler when we ma
Abe m:th our own bands~ ~ ~:n much more Tonsil I- :ts metric,
=~bi:~nal, =d ~abi:litv properties than if we jug :~k ~ one. {if
::~d of ca - card ~~s we :~ t~e Me few plastic I, ~~k
in b~Is of putty ~ in ma:~:mal:~ the c~e will wobble. ~~ Is
' ~e wand cube :s not ~ "~- model. ~ the c-~, ~ is
use~l one becau~ it teacheS so-~hing ~~t Oddity and n=~.litv.
{t elm t=~es something abou~t the sha~s into which the Me can be
t=~-~.~d - :~e I. its co-~bin=~.~ :.
6~r~ $~5 t~ 3~6 t58t =:~S &~6 "manipu::~-~- are v~e
t=~s :n the ciassr~^ But too ohen one :3he~ the ::~ament that "if
~ A ~ ~ ce~ art. ~~ Ate ~~ ~ 63~6 t~ ~~
tt0= l8~t 0~- Axis ~~ 2~ =~g t~0 l~t Tut ~~ - t
i. assum~l'0~.
First, that ~~ss ~1 sh~e is t~he
main thing that ~e team - m ~~s ~~d that it can all be 1~d :n
~ 506001* )~ ~0 500~ 0~0 0~ (~7 ~~ ~0 50~- them all.
{5i5- 0: 00~' t~ ~~- tt~ ~6 ~0 mat ~ Beet t0~ ~n t~e
~~V Of SrOlUt~g3~ CO~~ Si=:~, Wild =.~: Sit:~-
iFhe ~~. assu.~:}~:~n ls tha~t the main pumice in starving ~ mode!
:s to deve}~p our p-~wYe~ of abst=~t reasoning' hem the model plays the
.~^
Me of ~~ wheels on a: b~- ~~V ~ ~t :~r ~ - tO
u~d the wn=~ in - - a:~ =~ mp~s In:
00~= 0t ~ cube. ~ ~n beet ~5 :s =~$~63 m~t 0t ~~-~ ~~li
Me ~ iot ~ I=m - m real models*
Ideal, the ~~1d be ~ :m ~ boom ~:~ At
~, ~~) schOo! ah—d have a ]a~ Were student =n expose
shape. ~ sh~ ~~Y Ode ~~ ~ ~ =d =
Motion e~, Me - gal duels of m~ ~~, m
HIS ~r bedding ~~, =d pit tO dismay ~~. Tf polite' ~ sill
:~e~ c~ m~ grapes ~~. As should ~ mp
~6 ~~t ~~, ~~t =~7 =6 ;~ 0~:
#
programs.
~e ~ ~~ [~r e~:~e It is ;~n s=d that ~~g
is :~! ~r ~~w 1~. Ce~- - it :s tme that ~~s who: h=e tm~
wi:~ a:;
^~ ~ Ala
^e ~ ~~ ~ ~ ~~ In ~ bme of ape away lbe may
of gape ~i~li1~s ~ eyed Bales far labia.
far _mpIe~
ampler Pa is ~-l~ion~ing lobe gum Shy. ~ ~~= the
supper is Oh maps En cabs so bang come
~ s ~ ~ ~
a~ Peale Yes 1~ moot of us ~~uld~n~ --e a made
.. ~
lay verb. Hey no flower ~1 ~ ~^ loam E
Ads; in_> gay me paper ~~ me Nils in Acing
me Flies of bang. abed Abed ~ ^~ Bus haul
nag ~~ ~1~bi~ Blab lbe~ USA and Me nag Pep
~~- ~ be en~u_d by it cupola gal Eva s^-
plo~lio~n of shape ~~u~u1 1be enlist Rectum.
I Abeam, OR. ~7 ~' Whelm, ^: L~i~i~ of aria am, 1969.
2. ~ . ~H C. a dam =~ an.- ~ ~~ (~- ^~ A=dc~ of
, ~ 1#, as.
3. ala, Ma. ~~ Beg #~, MY: Sp^~d~ 1987
4 Coheir, H.S a. ^~ ~ ~~ lea ad, a: Jan ~1~ ~ gas ~1969.
5. Cage H.S~. ~ art. ~ BY: her, 1973.
6 on, smug Y.. Jr. gaily, Ore 'Dingo,' And 1be 'Sag S~~-
~' ~~ man.- ~~ ~ (1984\ 225~232.
7. ^~=dor, hex ad Span, Owing. Smithy: legions al the aisle
Oar ha.- ~^ ~~ ~^ C~~^ ^7Z Redone VA: anions guns
~~_ ^~ I~1)
S. ^1~. ^~= ~~, 1610.
9. ~m~ E.H ^~ ~~ ~~ labs, a: ^~1 Universe Pa, 1979.
10. Oats. Baa ad ~~, C.S. #~ ~~ ^~ ~~ ^~, a: a.
~m=, 1987
1 1. ~ ~ ~ ^ ~ ~ 58:1 (19B5),
1 ~ 1~
~ a- ~ O.
12 golden, ^1~. ~ At. New ha. MY: Columbia Lankily Pass. 1983.
13. Mar Jobs. ~-Su_~puer pickle ~1~ Abe -= inhaler
age ha. 30, 1988), 1, 26.
14. ~~1i=~ ~u~ncil of Acts of ~~:bem~ic$. {
ADZ? Roston. a: tonal Council of Cab of ~~bom~i~, 1987.
anions Council of gabs of ~M~bema~. ^~~ ~~ r~ ^~-
^~# Jason, a: Nastier Council of Cube of ~atb~malics,
1989.
16. snag, Marjorie and ~#, At. ^~ ~~ Amber. Be: Onion
sily of ~as~chu~lls Pass ~1977.
17. anti, Redone and ask. ~ .
age, Ah: Bib. 1988.
1$. knobs, Calorie and beck, At. ^~ ~{ ~~ ~~> Olin
P=P8~l10~)
19. Nobel, adore. ~Symmcl~ fit. In Magi, lslvan (Ed.
~~ ~~ 1989.
15.
Ha.
): ~^
181
20. I, P. ~ ^~~ ma, at: Lit, Be & Id, 1974.
21. at_ ~~ We. ~ ~ ~~ ~ A man. Ibid, at:
1~.
ant am ~ Cd~ aim ~~ of ~~ vita.- ^~ 177 { 19i6),
47~?i.
~~ R. ^~ ^~ ~~ Xe~ at, as: ~1 mar, 19S5.
4_.
23
..
