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U¢d*c Z@th¢Z@t*c@l C#nc¢pts @nd Th¢*r @ppl*c@t*#n

t# %ns#lU¢d Z@th¢Z@t*c@l Pr#Vl¢Zs:

Thr¢¢ Pr##fs #f F¢rZ@t's L@st Th¢#r¢Z

S.K. K@p##r

*nd*@n *nst*t%t¢ #f Z@h@r*sh* U¢d*c

Sc*¢nc¢ @nd T¢chn#l#gy

Pr¢f@c¢

Th¢ f#ll#w*ng thr¢¢ pr##fs, V@s¢d #n th¢ @ppl*c@t*#n #f U¢d*c Z@th¢Z@t*c@l c#n

c¢pts, @ddr¢ss @ f@Z#%s %ns#lU¢d pr#Vl¢Z #f Z@th¢Z@t*cs, F¢rZ@t's L@st Th¢#r¢Z. Th¢

f#ll#w*ng p@ss@g¢ #f J@V@l* %p@n*sh@d pr#U*d¢d th¢ str%ct%r@l ¢y f#r d¢U¢l#p*ng th¢

Z%lt*!d*Z¢ns*#n@l sp@c¢s %s¢d *n th¢ @rg%Z¢nt f#r F¢rZ@t's L@st Th¢#r¢Z"

@ddr¢ss c#rr¢sp#nd¢nc¢ t#: S.K. K@p##r, U*s*t*ng Pr#f¢ss#r

*nd*@n *nst*t%t¢ #f Z@h@r*sh*'s U¢d*c Sc*¢nc¢ @nd T¢chn#l#g

Z@h@r*sh* !@g@r, "h@*@V@d, %P $& (, *nd*@

Z#d¢rn Sc*¢nc¢ @nd U¢d*c Sc*¢nc¢,U#l%Z¢ , !%ZV¢r &, &))

+ &)) Z@h@r*sh* *nt¢rn@t*#n@l %n*U¢rs*t

$

Th¢n @*pp@l@d* @s¢d L#rd J@V@l*, &T¢ll Z¢, L#rd, th¢ s¢cr¢t, s%pr¢Z¢ r¢@l*ty. h@t *s

t@ttU@ (¢)*st¢nc¢+ h@t *s *U@ (*nd*U*d%@l l*f¢+ h@t *s p@sh% (th¢ s#%l+ h# *s *sh

(th¢ Z@st¢r+ h@t @r¢ th¢ Z¢@ns t# ¢nl*ght¢nZ¢nt+& -¢ s@*d t# h*Z, &U¢ry g##d ¢U¢ry

th*ng th@t y#% h@U¢ @s¢d, * w*ll ¢)pl@*n t# y#%, @s *t *s n#wn.& @g@*n h¢ s@*d t# h*Z,

&-#w *s *t th@t y#% n#w th*s+& @g@*n h¢ s@*d t# h*Z, &fr#Z Sh@d@n@n.& @g@*n h¢ s@*d t#

h*Z, &-#w d#¢s h¢ th¢n n#w th*s+& @g@*n h¢ s@*d t# h*Z, &fr#Z *sh@n.& @g@*n h¢ s@*d

t# h*Z, &-#w d#¢s h¢ n#w *t fr#Z h*Z+& @nd @g@*n h¢ s@*d, &fr#Z %p@s@n@ (w#rsh*p.&

*n @n ¢@rl*¢r w#r, Th¢ /@t*#n@l¢ #f 0Z1*ts F#rZ%l@t*#n, S*gn*f*c@nc¢ @nd #cc%r!

r¢nc¢ @nd @ppl*c@t*#ns *n @nc*¢nt L*t¢r@t%r¢,* c#ZZ¢nt¢d #n th¢ @V#U¢ p@ss@g¢ @s f#ll#S"

Th¢ *Zp#rt *n th¢ @V#U¢ n*n¢ Z@ntr@s #f th¢ J@V@l* %p@n*sh@d *s th@t @*pp@l@d* h@d

@s¢d J@V@l* /*sh* t# ¢nl*ght¢n h*Z @V#%t t@ttU@, *U@, p@sh% @nd *sh. J@V@l* /*sh* h@pp*ly

pr¢p@r¢d t# *nstr%ct h*Z @V#%t th¢ 2%¢st*#ns @s¢d. J%st @s J@V@l* /*sh* w@s @V#%t t# V¢

g*n h*s d*sc#%rs¢, @*pp@l@d* *n2%*r¢d h#w h¢ J@V@l* /*sh* h@d @ch*¢U¢d ¢nl*ght¢nZ¢nt.

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Z#D¢RN SC*¢NC¢ @ND U¢D*C SC*¢NC¢

#n th*s, th¢ J@V@l* R*sh* d*scl#s¢d th@t h¢ w@s ¢nl*ght¢n¢d #n th¢ p#*nts Vy Sh@d@-

n@n. #n th*s h¢ f%rth¢r @sk¢d fr#Z wh¢r¢ Sh@d@n@n h@d @ch*¢U¢d ¢nl*ght¢nZ¢nt @nd

t# th*s th¢ @nsw¢r #f J@V@l* R*sh* w@s th@t Sh@d@n@n h@d r¢c¢*U¢d ¢nl*ght¢nZ¢nt fr#Z

*sh@n.

Sh@d@n@n t# *sh@n

Th¢ *ns*ght #f d¢sc¢nd¢nc¢ #f Vr@hZ@U*dy@ f#r th¢ L#w¢r Z@th¢Z@t*c@l d#Z@*n *s th@t

th¢ pr#c¢ss*ng sh#%ld V¢g*n fr#Z w*th*n th¢ f#%rth-d*Z¢ns*#n@l d#Z@*n *n t¢rZs #f *ts

c#nst*t%¢nt, th@t *s @ pl@n¢ *s t# V¢ @l#ng *ts d*@g#n@l wh*ch w#%ld V¢ f@c*ng N#rth-¢@st

(*sh@n) @nd w#%ld V¢ l¢@d*ng t# th¢ s*xth-d*Z¢ns*#n@l d#Z@*n (Sh@d@n@n). S¢q%¢nt*@l

#rd¢r w#%ld ¢Z¢rg¢ V¢c@%s¢ #f d¢dh-d¢U@t@ Th¢ pr#c¢ss*ng wh*ch w@s

*n*t*@t¢d fr#Z w*th*n th¢ f#%rth-d*Z¢ns*#n@l d#Z@*n l¢d t# th¢ s*xth-d*Z¢ns*#n@l d#-

Z@*n. Th¢ pr#c¢ss*ng w@s @l#ng th¢ d*@g#n@l. Th¢ d*@g#n@l *s gr¢@t¢r th@n ¢*th¢r s*d¢.

Th¢ d*@g#n@l *s @ls# l¢ss th@n th¢ s%Z #f V#th th¢ s*d¢s (#f th¢ tr*@ngl¢). Th*s c#nc¢pt *s

th¢ c#nc¢pt #f #n¢ @nd @ h@lf %n*ts. Th¢ pr#c¢ss*ng l*n¢ #f #n¢ @nd @ h@lf %n*ts *s 4x3/2=6.

Th¢ %p@n*sh@d*c ¢nl*ght¢nZ¢nt #n th¢ p#*nt *s th@t th¢ d¢U@s @r¢ 1, 3/2, 2, 3, @nd s# #n

(Vr*h@d@r@ny@k@ %p@n*sh@d). Th¢ c#nc¢pt #f d¢dh-d¢U@t@ *s th¢ sp¢c*f*c

pr#c¢ss*ng c#nc¢pt. *n t¢rZs #f th*s c#nc¢pt th¢ pr#c¢ss*ng @l#ng th¢ N#rth-¢@st l*n¢

(*sh@n) wh*ch h@s t@k¢n th¢ f#%rth-d*Z¢ns*#n@l d#Z@*n t# th¢ s*xth-d*Z¢ns*#n@l d#Z@*n

s¢q%¢nt*@lly w#%ld c@rry th¢ pr#c¢ss*ng f%rth¢r, n@t%r@lly t# th¢ n*nth-d*Z¢ns*#n@l d#-

Z@*n @s 6x3/2 = 9.

Th¢ s*gn*f*c@nc¢ #f th¢ @V#U¢ w*th*n th¢ @r*thZ¢t*c d#Z@*n *s th@t *n #rd¢r t#

%nd¢rst@nd th¢ str%ct%r@l fr@Z¢s @nd syst¢Zs #f n@t%r@l n%ZV¢r 9 (*n %p@n*sh@d*c

l@ng%@g¢: Vr@hZ@U*dy@) w¢ h@U¢ t# d¢U¢l#p th¢ %nd¢rst@nd*ng *n t¢rZs #f th¢

str%ct%r@l fr@Z¢s @nd syst¢Zs #f n@t%r@l n%ZV¢r 6 (*n %p@n*sh@d*c l@ng%@g¢ Sh@d@-

n@n V¢st#w¢d ¢nl*ght¢nZ¢nt %p#n J@V@l* R*sh*). @nd %nd¢rst@nd*ng th¢ str%ct%r@l

fr@Z¢s @nd syst¢Zs #f n@t%r@l n%ZV¢r 6 Z%st V¢ *n t¢rZs #f th¢ str%ct%r@l fr@Z¢s

@nd syst¢Zs #f n@t%r@l n%ZV¢r 4 (*n th¢ %p@n*sh@d*c l@ng%@g¢ *sh@n V¢st#w¢d ¢n-

l*ght¢nZ¢nt %p#n Sh@d@n@n). Th*s r¢U¢rs¢ s¢q%¢nt*@l pr#c¢ss c#nt@*ns th¢ str%ct%-

r@l k¢y.

Th¢ V@s*c U¢d*c Z@th¢Z@t*c@l c#nc¢pts %s¢d @r¢ th@t th¢ %n*ty (s*ngl¢-syll@Vl¢

#Z) *s pr#c¢ss@Vl¢ q%@rt¢r Vy q%@rt¢r (Shr* P@d@ pr#c¢ss*ng l*n¢) @nd th¢ f#%rthq%@rt¢r *s th¢ *nt¢gr@t*#n #f th¢ f*rst thr¢¢ q%@rt¢rs (Z@h@r*sh* pr#c¢ss*ng l*n¢). @s

s%ch, * *nt¢rpr¢t th¢ @V#U¢ %p@n*sh@d*c p@ss@g¢ *n th¢ f#ll#w*ng w@y: th¢ str%ct%r@l

fr@Z¢s @nd syst¢Zs #f n@t%r@l n%ZV¢r 4 (@nd h¢nc¢ th¢ f#%rth-d*Z¢ns*#n@l d#Z@*n)

@r¢ t# V¢ h@ndl¢d @s %n*ty, @dZ*tt*ng pr#c¢ss*ng q%@rt¢r Vy q%@rt¢r, @nd th¢ f#%rth

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U¢D*C Z@TH¢Z@T*C@L C#NC¢PTS

q%@rt¢r (th¢ %nZ@n*f¢st¢d q%@rt¢r) *s t# V¢ pr#c¢ss¢d @s *nt¢gr@t*ng th¢ f*rst thr¢¢

q%@rt¢rs (Z@n*f¢st¢d q%@rt¢rs). Th¢ pr#c¢ss*ng w*th*n th¢ f#%rth q%@rt¢r @l#ng th¢

N#rth-¢@st d*@g#n@l w#%ld l¢@d t# th¢ d#Z@*n #f Sh@d@n@n (@dZ*tt*ng str%ct%r@lfr@Z¢s @nd syst¢Zs #f n@t%r@l n%ZV¢r 6 @nd h¢nc¢ @ s*x-d*Z¢ns*#n@l d#Z@*n). Th¢

%p@n*sh@d*c c#ZZ@nd *s t# r¢U¢rs¢ th¢ pr#c¢ss, @s th¢ pr#c¢ss*ng *s t# V¢ h@d *n th¢

d¢sc¢nd*ng #rd¢r fr#Z 9 t# 6 @nd 6 t# 4, V%t h¢r¢ *n th*s w#rld ¢U¢ryth*ng *s h@ng*ng

%ps*d¢ d#wn %rdhU@Z%l@Z—Vh@g@U@d-G*t@, 15.1). Th*s Z¢@ns th@t

wh*l¢ #%r @*Z *s t# pr#c¢ss w*th*n th¢ %nZ@n*f¢st¢d q%@rt¢r *n #rd¢r t# g# fr#Z th¢

f#%rth-d*Z¢ns*#n@l d#Z@*n #f n@t%r@l n%ZV¢r 4 t# th¢ s*xth-d*Z¢ns*#n@l d#Z@*n #f

n@t%r@l n%ZV¢r 6, w¢ h@U¢ t# pr#c¢ss @l#ng th¢ N#rth-¢@st d*@g#n@l V%t *n th¢ d*r¢c-

t*#n wh*ch l¢@ds fr#Z th¢ s*xth-d*Z¢ns*#n@l d#Z@*n t#w@rds th¢ f#%rth-d*Z¢ns*#n@l

d#Z@*n. F%rth¢r, th¢ pr#c¢ss*ng Z%st V¢ q%@rt¢r Vy q%@rt¢r, wh*ch Z¢@ns th@t wh¢n

n@t%r@l n%ZV¢r 4 *s t@k¢n @s @ %n*ty, *ts q%@rt¢r w#%ld V¢ #n¢, @nd s*Z*l@rly, wh¢n

n@t%r@l n%ZV¢r 6 *s t@k¢n @s @ %n*ty, *ts pr#c¢ss*ng %n*t w#%ld V¢ #n¢. H¢nc¢ th¢ pr#-

c¢ss*ng @l#ng th¢ N#rth-¢@st d*@g#n@l *s t# V¢ h@d Vy d*U*d*ng *t *nt# s*x p@rts. #%t #f

th¢s¢ s*x p@rts, #nly #n¢ p@rt w#%ld V¢ %nZ@n*f¢st¢d @nd th¢ r¢Z@*n*ng f*U¢ p@rts

w#%ld c#U¢r th¢ Z@n*f¢st¢d p@rt. W*th th*s th¢ pr#c¢d%r¢ #f str%ct%r@l pr#c¢ss*ng

st@nds @s f#ll#ws:

St¢p 1: Th¢ pr#c¢ss*ng *s t# V¢ *n th¢ f#%rth q%@rt¢r #f th¢ #Z f#rZ%l@t*#n.

St¢p 2: F#%rth q%@rt¢r (f#%rth c#Zp#n¢nt) #f th¢ D¢U@n@g@r* #Z f#rZ%l@t*#n *s l*k¢

@ tw#-d*Z¢ns*#n@l c@rt¢s*@n fr@Z¢.

St¢p 3: S# pr#c¢ss*ng w*th*n th¢ f#%rth q%@rt¢r (f#%rth c#Zp#n¢nt) @Z#%nts t# pr#-

c¢ss*ng #n th¢ f#rZ@t #f @ pl@n¢.

St¢p 4: @s V#th th¢ d*Z¢ns*#n@l l*n¢s @r¢ syZZ¢tr*c, th¢ n@t%r@l f*g%r¢ #f th¢ pl@n¢

f#rZ@t *s @ sq%@r¢ (t# V¢ c@ll¢d th¢ pr#c¢ss*ng sq%@r¢).

St¢p 5: F#r pr#c¢ss*ng #f th¢ f*rst thr¢¢ q%@rt¢rs #f th¢ Z@n*f¢st¢d d#Z@*ns, d*U*d¢

th¢ sq%@r¢ *nt# 3x3 = 9 sq%@r¢s (t# V¢ c@ll¢d pr#c¢ss*ng %n*ts).

St¢p 6: Th¢r¢f#r¢, *f th¢ l¢ngth #f th¢ sq%@r¢ *s Z, th¢n th¢ l¢ngth #f th¢ pr#c¢ss*ng

%n*t w#%ld V¢ Z/3.

St¢p 7: @s th¢ f#%rth q%@rt¢r *s t# V¢ pr#c¢ss¢d @s th¢ *nt¢gr@t*#n #f th¢ f*rst thr¢¢

q%@rt¢rs, th¢ l¢ngth #f th¢ pr#c¢ss*ng sq%@r¢ sh#%ld V¢ *ncr¢@s¢d Vy th¢ l¢ngth #f *ts pr#-

c¢ss*ng %n*ts, th@t *s, th¢ ¢xt¢nd¢d pr#c¢ss*ng sq%@r¢, @s *t Z@y V¢ c@ll¢d, w#%ld h@U¢ @

l¢ngth ¢q%@l t# Z + Z/3 = 4Z/3.

St¢p 8: Th¢ pr#c¢ss*ng sq%@r¢ V¢*ng #f l¢ngth Z, th¢ Z@n*f¢st¢d p@rt #f th¢ N#rth-

¢@st d*@g#n@l #f th¢ ¢xt¢nd¢d pr#c¢ss*ng sq%@r¢ w#%ld V¢ 5Z/6, @nd th¢ %nZ@n*f¢st¢d

p@rt #f th¢ N#rth-¢@st d*@g#n@l #f th¢ s@*d sq%@r¢ w#%ld V¢ Z/6.

St¢p 9: Th¢ @V#U¢ f#rZ@t d¢U¢l#ps @ sq%@r¢ wh#s¢ l¢ngth *s ¢q%@l t# th¢ d*@g#n@l,

wh*ch Z¢@ns w¢ @r¢ tr@nsc¢nd*ng fr#Z th¢ g¢#Z¢tr*c@l sq%@r¢ t# th¢ U@l%¢s-sq%@r¢ (@s

*t Z@y V¢ c@ll¢d). *n th*s f#rZ@t tr@nsc¢nd¢nc¢ *s p¢rZ*ss*Vl¢ w*th*n th¢ f#rZ@t #f @ g¢#-

Z¢tr*c@l sq%@r¢ *ts¢lf *n t¢rZs #f th¢ cr#ss p#*nts #f th¢ l*n¢s p@r@ll¢l t# th¢ @x¢s #f th¢tw#-d*Z¢ns*#n@l c@rt¢s*@n fr@Z¢ #f th¢ g¢#Z¢tr*c@l sq%@r¢. *f w¢ t@k¢ *nt# @cc#%nt #nly

th¢ cr#ss p#*nts #f th¢ l*n¢s p@r@ll¢l t# th¢ @x¢s (wh*ch l*n¢s Z@y V¢ c@ll¢d p@r@ll¢l @x¢s)

th¢n th¢ cr#ss p#*nts #n th¢ l¢ngth #r Vr¢@dth #f th¢ sq%@r¢ w#%ld V¢ ¢q%@l t# th¢ cr#ss

p#*nts #n @ny #f th¢ tw# d*@g#n@ls #f th¢ sq%@r¢.

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St¢p 10: Th¢ ¢ff¢ct #f ¢xt¢nd*ng th¢ pr#c¢ss*ng sq%@r¢ *nt# th¢ ¢xt¢nd¢d pr#c¢ss*ng

sq%@r¢ %p#n th¢ U@l%¢s-sq%@r¢ #f th¢ pr#c¢ss*ng sq%@r¢ q%@ th¢ U@l%¢s-sq%@r¢ #f th¢ ¢x-

t¢nd¢d pr#c¢ss*ng sq%@r¢ *s th¢ str%ct%r@l k¢y f#r r¢s#lU*ng why th¢ s%Z #f tw# r¢g%l@r

V#d*¢s #f Nth-d*Z¢ns*#n@l sp@c¢ d#¢s n#t c#nst*t%t¢ @n#th¢r r¢g%l@r V#dy #f th¢ Nth-

d*Z¢ns*#n@l sp@c¢.

*n th¢ pr¢s¢nt st%d*¢s * h@U¢ @ppl*¢d th¢ @V#U¢ str%ct%r@l k¢y *n thr¢¢ d*ff¢r¢nt w@ys

t# pr#U¢ F¢rZ@t's L@st Th¢#r¢Z. Th¢s¢ pr##fs @r¢ s%VZ*tt¢d n#t #nly f#r th¢ p%rp#s¢

#f s%pply*ng @ pr##f #f th*s %ns#lU¢d th¢#r¢Z, V%t @ls# w*th th¢ @*Z #f @pp¢@l*ng t#

#th¢r sch#l@rs t# @ppr#@ch th¢ Z@*n ch@ll¢ng¢s w*th*n th¢*r d*sc*pl*n¢s thr#%gh th¢ U¢d*c

w*sd#Z.

Th¢ U¢d*c p¢rsp¢ct*U¢ #n Z¢th#d#l#gy *nt¢gr@t¢s #Vj¢ct*U*ty w*th s%Vj¢ct*U¢ ¢xp¢r*-

¢nc¢ @nd @s s%ch th#s¢ wh# @r¢ tr@*n¢d *n #Vj¢ct*U¢ Z¢th#d#l#gy @r¢ r¢q%*r¢d #nly t#

l¢@rn h#w t# s%ppl¢Z¢nt th*s @ppr#@ch w*th th¢*r #wn s%Vj¢ct*U¢ ¢xp¢r*¢nc¢ #f th¢ U¢d*c

R¢@l*ty. F#r th*s * f¢¢l h*ghly pr*U*l¢g¢d t# V¢ @t th¢ f¢¢t #f H*s H#l*n¢ss Shr* P@d@ V@-V@j* wh# *n*t*@t¢d Z¢ f#r th¢ Shr* P@d@ pr#c¢ss*ng l*n¢ t# pr#c¢ss th*s q%@rt¢r Vy q%@rt¢r,

@nd @t th¢ f¢¢t #f H*s H#l*n¢ss Z@h@r*sh* Z@h¢sh Y#g* wh# *n*t*@t¢d Z¢ f#r th¢ Z@h@r*-

sh* pr#c¢ss*ng l*n¢ t# pr#c¢ss th¢ f#%rth q%@rt¢r @s th¢ *nt¢gr@t*#n #f th¢ f*rst thr¢¢ q%@r-

t¢rs. * @Z @ls# h*ghly #Vl*g¢d t# Pr#f¢ss#r Kr*shn@j*, Ch@*rZ@n, *nd*@n *nst*t%t¢ #f Z@-

h@r*sh* U¢d*c Sc*¢nc¢ @nd T¢chn#l#gy, f#r ¢xp¢r*¢nc¢d g%*d@nc¢ @nd p¢rs#n@l *nt¢r¢st

*n th¢s¢ @nd #th¢r st%d*¢s @t th¢ *nst*t%t¢.

78

Pr##f Vy D*r¢ct C#Zp@r*s#n

#U¢rU*¢w

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79

8/17/2019 Azzura

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80

8/17/2019 Azzura

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81

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82

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83

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84

St@t¢Z¢nt #f th¢ Th¢#r¢Z

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85

Th¢r¢f#r¢, T *s n#t @ r@t*#n@l n%ZV¢r.

St¢p 5: H¢nc¢, *n th¢ ¢q%@t*#n %N

= UN

+ TN

, U @nd T @r¢ *rr@t*#n@l n%ZV¢rs. Th¢r¢-

f#r¢, w¢ h@U¢ t# l##k f#r s#l%t*#ns #f th*s ¢q%@t*#n *n th¢ f*¢ld #f r¢@l n%ZV¢rs. H¢r¢ *t

Z@y V¢ r¢l¢U@nt t# n#t¢ th@t %N

, UN

, @nd TN

@r¢ r@t*#n@l n%ZV¢rs, s# % 1 = %N

, U 1 , = UN

@nd T 1 = TN

g*U¢ r*s¢ t# th¢ ¢q%@t*#n % 1 = U 1 + T 1 wh¢r¢ % 1, U 1 @nd T 1 @r¢ r@t*#n@l n%Z-

V¢rs. Th*s h@s th¢ #VU*#%s s#l%t*#n *n th¢ f*¢ld #f r@t*#n@l n%ZV¢rs. H#w¢U¢r, wh¢n th¢

s#l%t*#n *s r¢q%*r¢d f#r %N

= UN

+ TN

, wh¢r¢ U @nd T @r¢ *rr@t*#n@l n%ZV¢rs, w¢ h@U¢ t#

sh*ft t# th¢ f*¢ld #f r¢@l n%ZV¢rs.

St¢p 6: Th¢ f*¢ld #f r¢@l n%ZV¢rs c#nst*t%t¢s th¢ @r*thZ¢t*c@l c#nt*n%%Z wh*ch *s

¢q%*U@l¢nt t# th¢ l*n¢@r g¢#Z¢tr*c@l c#nt*n%%Z #f @ str@*ght l*n¢. N#w th¢ s#l%t*#ns #f

th¢ ¢q%@t*#ns %N

= UN

+ TN

@nd ZN.

%N

= ZN.

UN

+ ZN.

TN

@r¢ d*r¢ctly l*nk¢d %p. Th¢r¢-

f#r¢, th¢ s#l%t*#n #f th¢ ¢q%@t*#n (Z%)N

= (ZU)N

+ (ZT)N

, wh*ch *s n#th*ng V%t SN

=

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p#*nt r¢pr¢s¢nt*ng th¢ n%ZV¢r z¢r# (0). @s @ s¢c#nd st¢p, f#r @ g*U¢n r@t*#n@l n%Z-

V¢r Z, w¢ Z@y c%t @ cl#s¢d *nt¢rU@l [#Z] #f l¢ngth #Z = Z fr#Z th¢ l*n¢ #R. @s @

th*rd st¢p, f#r @ny r@t*#n@l n%ZV¢r X < Z, w¢ c@n c%t @ cl#s¢d *nt¢rU@l [#X] #fl¢ngth #X = X fr#Z th¢ *nt¢rU@l [#Z] #f l¢ngth #Z = Z.

4. Th¢ *nt¢rU@l [#Z] g¢ts d*U*d¢d *nt# tw# p@rts, n@Z¢ly cl#s¢d *nt¢rU@l [#X] @nd

th¢ #n¢-s*d¢d #p¢n *nt¢rU@l (XZ]. V#th th¢ p@rts, n@Z¢ly [#X] @nd (XZ], d# n#t

h@U¢ @ny c#ZZ#n p#*nt. @ll n%ZV¢rs r@t*#n@l @nd *rr@t*#n@l l¢ss th@n #r ¢q%@l t# X

@r¢ *n [#X], @nd @ll n%ZV¢rs r@t*#n@l #r *rr@t*#n@l gr¢@t¢r th@n X @r¢ *n (XZ]. @s

s%ch, th¢ *nt¢rU@l (XZ] *s #f @n *rr@t*#n@l l¢ngth.

F¢rZ@t's L@st Th¢#r¢Z

5. St@t¢Z¢nt: "*t *s *Zp#ss*Vl¢ t# s¢p@r@t¢ @ c%V¢ *nt# tw# c%V¢s, #r @ V*q%@dr@t¢

*nt# tw# V*q%@dr@t¢s, #r *n g¢n¢r@l @ny p#w¢r h*gh¢r th@n th¢ s¢c#nd *nt# p#w¢rs #f

l*k¢ d¢gr¢¢."— F¢rZ@t

6. *n Z#d¢rn Z@th¢Z@t*c@l l@ng%@g¢ th*s *s r¢st@t¢d @s: xn

+ yn

= zn

@dZ*ts @ s#l%t*#n

*n n@t%r@l n%ZV¢rs #nly f#r n = 2. Th@t Z¢@ns f#r g*U¢n n@t%r@l n%ZV¢rs z, n, @nd x,

w*th x < z, w¢ c@nn#t f*nd @ n@t%r@l n%ZV¢r wh#s¢ nth d¢gr¢¢ *s ¢q%@l t# zn

-xn

.

Pr##f

7. L¢t U = zn

-xn

.

8. z, n, @nd x @r¢ n@t%r@l n%ZV¢rs, th¢r¢f#r¢ zn

@nd xn

@r¢ @ls# n@t%r@l n%ZV¢rs.

L¢t Z = zn

@nd X = xn

. Th¢r¢f#r¢, U = Z-X.

9. @s Z @nd X @r¢ n@t%r@l n%ZV¢rs, s# th¢ cl#s¢d *nt¢rU@l [#Z] #f l¢ngth Z c%ts @

r@t*#n@l l¢ngth #n th¢ r¢@l l*n¢ #R. S*Z*l@rly, th¢ cl#s¢d *nt¢rU@l [#X] #f l¢ngth X

c%ts @ r@t*#n@l l¢ngth fr#Z th¢ *nt¢rU@l [#Z].

10. U = Z-X = [#Z]-[#X] = (XZ] = *rr@t*#n@l l¢ngth.

11. H¢nc¢,U N" f#r ¢U¢ry n@t%r@l n%ZV¢r N, @s Nn

*s @ n@t%r@l n%ZV¢r s#

w#%ld ¢q%@l @ r@t*#n@l l¢ngth @nd n#t @n *rr@t*#n@l l¢ngth.

12. Th¢ @V#U¢ pr#U¢s th¢ th¢#r¢Z V%t f#r th¢ r¢str*ct*#n f#r th¢ n t# V¢ 3.

R¢str*ct*#ns f#r n t# V¢ 3

R@t*#n@l¢ f#r th¢ r¢str*ct*#ns:

13. T# @rr*U¢ @t th¢ r@t*#n@l¢ f#r th¢ r¢str*ct*#ns f#r n t# V¢ 3, w¢ h@U¢ t# g#

V@ck t# #%r d¢f*n*t*#n #f th¢ d*Z¢ns*#n *n t¢rZs #f *ts g¢#Z¢tr*c@l r¢pr¢s¢nt@t*#n.

14. Th¢ d¢f*n*t*#n #f d*Z¢ns*#n c@nn#t V¢ ¢xpr¢ss¢d ¢xc¢pt w*th*n th¢ Z@n*f¢st-

¢d w#rld #f thr¢¢-d*Z¢ns*#n@l #Vj¢cts, th¢r¢Vy p¢rZ*tt*ng *ts @rr@ng¢Z¢nt @s @ l*n¢-

@r g¢#Z¢tr*c@l c#nt*n%%Z #f str@*ght l*n¢s, ¢@ch @ Z@th¢Z@t*c@l Z#d¢l #f th¢ @r*th-

Z¢t*c@l c#nt*n%%Z #f r¢@l n%ZV¢rs. *n c#ncr¢t¢ t¢rZs, g¢#Z¢try @cc¢pts d*Z¢ns*#n

@s th@t wh*ch c#Zpl¢t¢ly p¢rZ*ts r¢pr¢s¢nt@t*#n @s @ str@*ght l*n¢, @nd th¢ g¢#Z¢tr*-

c@l %n*U¢rs¢ f#ll#w*ng *s th¢ thr¢¢-d*Z¢ns*#n@l sp@c¢ w*th th¢ thr¢¢ d*Z¢ns*#ns

n#th*ng V%t str@*ght l*n¢s. *n f@ct, h¢r¢ l*¢s th¢ r@t*#n@l¢ @nd th¢ @nsw¢r t# th¢ q%¢s-

t*#n why r¢str*ct*#ns @r¢ pl@c¢d %p#n n t# V¢ 3. H#w¢U¢r, c#Zpr¢h¢ns*#n #f th¢Z@n*f¢st¢d w#rld c@nn#t V¢ @cc¢pt¢d @s @ pr##f @s s%ch #f th¢ ¢x*st¢nc¢ #r #th¢r-

w*s¢ #f th¢ h*gh¢r- d*Z¢ns*#n@l sp@c¢s. F#r th*s w¢ h@U¢ t# h@U¢ @ p%r¢ly Z@th¢Z@t-

*c@l @ppr#@ch.

15. F#r th*s, w¢ Z@y h@U¢ *ns*ght *nt# th¢ *nt¢rn@l str%ct%r¢ #f Nn

Vy @cc¢pt*ng *t

87

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14/30

Z#3¢/4 S5*¢45¢ @43 U¢3*5 S5*¢45¢

t# V¢ @ r¢g%l@r V#dy #f l¢ngth 4 #f nth!d*Z¢ns*#n@l sp@c¢ 6f#r th¢ pr¢s¢nt t@¢ @s @

d¢f*n*t*#n th@t d¢gr¢¢ n r¢pr¢s¢nts th¢ n%ZV¢r #f d*Z¢ns*#ns7.

Th¢ F*rst *Z¢ns*#n@l -¢g%l@r V#d

89. T# h@U¢ pr#p¢r *ns*ght *nt# th¢ *nt¢rn@l str%ct%r¢ #f h*gh¢r!d*Z¢ns*#n@l r¢g%

l@r V#d*¢s, w¢ w#%ld V¢ r¢2%*r¢d t# p*n!p#*nt th¢ r@t*#n@l¢ @s t# why th¢ U¢ry f*rst!

d*Z¢ns*#n@l r¢g%l@r V#dy 6ll7 d#¢s n#t p¢rZ*t d*U*s*#n *nt# sZ@ll¢r r¢g%l@r V#d*¢s #f

th¢ f*rst!d*Z¢ns*#n@l sp@c¢.

8. Th¢ @nsw¢r *s V¢c@%s¢ #f th¢ U¢ry d¢f*n*t*#n #f th¢ n@t%r@l n%ZV¢r 8. Th¢ n@t

%r@l n%ZV¢r 8 *s th¢ sZ@ll¢st n@t%r@l n%ZV¢r, s# th¢ 2%¢st*#n #f th¢ ¢)*st¢nc¢ #f st*ll

sZ@ll¢r n@t%r@l n%ZV¢rs d#¢s n#t @r*s¢. -¢nc¢ l8 :

8 ; <

8, wh¢r¢ : @nd < @r¢ n@t%

r@l n%ZV¢rs.

8=. Th¢ @V#U¢ s*t%@t*#n d¢s¢rU¢s pr#p¢r Z@th¢Z@t*>@t*#n. F*rstly, *t t@¢s #n¢ @s

%nd¢f*n¢d. S*Z%lt@n¢#%sly *t g*U¢s %s th¢ fr¢¢d#Z t# @cc¢pt @ny l*n¢@r l¢ngth, Z@y *t

V¢ ¢2%@l t# r@t*#n@l #r *rr@t*#n@l %n*ts?n%ZV¢rs, @s @ l*n¢@r %n*t @nd h¢nc¢ n¢.&

8. %s*ng th¢ @V#U¢ fr¢¢d#Z #f ch#*c¢ t# @cc¢pt @ny l*n¢@r l¢ngth @s th¢ d*Z¢n

s*#n@l %n*t h¢lps %s t# r¢d%c¢ @ny r¢g%l@r V#dy 4& t# th¢ f*rst r¢g%l@r V#dy #f nth!

d*Z¢ns*#n@l sp@c¢ 6A7, wh¢r¢ 4 B #n¢ %n*t. Th*s @ls# c@n V¢ t@¢n @s th¢ f*rst r¢g%

l@r V#dy #f th¢ f*rst!d*Z¢ns*#n@l sp@c¢ 6l87 s*nc¢ 8

n B 8 B l

8.

CD. Th¢r¢f#r¢, *n #rd¢r t# %nd¢rst@nd th¢ *nt¢rn@l str%ct%r@l @rr@ng¢Z¢nts #f d*Z¢n

s*#n@l r¢g%l@r V#d*¢s, w¢ h@U¢ n# #pt*#n V%t t# c#ns*d¢r th¢ str%ct%r@l n#t r¢sp#ns*Vl¢

f#r tr@nsf#rZ*ng %n*t l*n¢@r l¢ngth *nt# @ r@t*#n@l fr@ct*#n pl%s @n *rr@t*#n@l fr@ct*#n.

C8. 3¢s*gn@t¢ th¢ p#*nt #f st%dy @s @ str%ct%r@l n#t #f d*Z¢ns*#n@l r¢g%l@r

V#d*¢s. ¢ f#c%s #n th¢ st@t¢Z¢nt th@t #n¢ l*n¢@r %n*t *s ¢2%@l t# @ h@lf!%n*t r@t*#n@l

l¢ngth pl%s @ h@lf!%n*t *rr@t*#n@l l¢ngth.

CC. V¢f#r¢ w¢ t@¢ %p th¢ @V#U¢ st@t¢Z¢nt f#r ¢l@V#r@t*#n, * w#%ld l*¢ t# @dd

h¢r¢ th@t th¢ r@t*#n@l¢ f#r th¢ @V#U¢ l*¢s *n th¢ f*rst pr*nc*pl¢ 6#f Z@n*f¢st¢d w#rld

@dZ*tt*ng str%ct%r¢s7 th@t th¢ str%ct%r@l %n*t *s h@lf @s c#Zp@r¢d t# #n¢ @s @ d*Z¢n

s*#n@l %n*t.

¢f*n*t*#n #f @ Str@*ght L*n¢

CE. #n¢ @ttr*V%t¢ #f th¢ str@*ght l*n¢ *s th@t *t h@s @ Z*ddl¢ p#*nt. @ s¢c#nd @ttr*V%t¢ #f th¢ str@*ght l*n¢ *s th@t *t h@s @ Z*n*Z%Z #f thr¢¢ p#*nts. *t Z@y n#t V¢ p#ss*

Vl¢ t# g*U¢ @ pr¢c*s¢ d¢f*n*t*#n #f @ str@*ght l*n¢ *n t¢rZs #f s#Z¢ @ttr*V%t¢s. -#w

¢U¢r, th¢ V@s*c @ttr*V%t¢s #f th¢ str@*ght l*n¢ w#%ld h¢lp %s s¢ttl¢ s#Z¢ pr@ct*c@l d¢f*n*

t*#ns. Th¢ pr@ct*c@l d¢f*n*t*#ns Z@y h@U¢ w#rth f#r @ny d#Z@*n #f pr@ct*c@l *Zp#r

t@nc¢ V%t th¢ s@Z¢ Z@y n#t V¢ @cc¢pt@Vl¢ t# Z@th¢Z@t*cs. @ pr¢c*s¢ Z@th¢Z@t*c@l

d¢f*n*t*#n #f @ str@*ght l*n¢ c@n V¢ pr#U*d¢d Vy th¢ @r*thZ¢t*c@l c#nt*n%%Z #f th¢ r¢@l

n%ZV¢rs. Th¢ f#rZ@t V¢n¢@th th¢ #rd¢r¢d d*spl@y #f th¢ s¢t #f r¢@l n%ZV¢rs w#%ld

Z@th¢Z@t*c@lly 2%@l*fy t# V¢ d¢s*gn@t¢d @s @ str@*ght l*n¢. @s s%ch th*s *s @cc¢pt¢d @s

th¢ d¢f*n*t*#n #f th¢ str@*ght l*n¢ f#r th¢ p%rp#s¢ #f ¢)pr¢ss*#ns #f d*Z¢ns*#n@l

fr@Z¢s, @s w¢ll @s f#r th¢ p%rp#s¢ #f c%tt*ng l¢ngths fr#Z @ r¢g%l@r V#dy.

Str%ct%r@l %n*t U¢rs%s *Z¢ns*#n@l %n*t

C. 4#w l¢t %s t@¢ %p th¢ ch@ll¢ng¢ #f th¢ str%ct%r@l n#t 6p@r@gr@ph C87. -¢r¢

th¢ s*t%@t*#n pr#c¢¢ds *n tw# st¢ps" f*rstly, *n t¢rZs #f th¢ @nsw¢r t# th¢ p#s¢r, *f w¢

==

8/17/2019 Azzura

15/30


c@n tr@nsc¢nd th¢ thr¢¢ d*Z¢ns*#n@l g¢#Z¢tr*c@l %n*U¢rs¢ @nd h@U¢ @ g¢#Z¢tr*c@l

c#nt*n%%Z, @nd s¢c#ndly, *n t¢rZs #f th¢ f*n¢ c#nn¢ct*#n V¢tw¢¢n th¢ str%ct%r@l %n*t

@nd th¢ d*Z¢ns*#n@l %n*t Z@*nt@*n*ng th¢ r@t*# #f 1/2 : 1.

Sq%@r¢ @nd C%V¢

25. F#r V#th th¢ st¢ps, *n*t*@lly w¢ Z@y pr#c¢¢d w*th th¢ str%ct%r@l c#ZZ#nn¢ss

#f th¢ sq%@r¢ @s @ r¢g%l@r V#dy #f @ tw#-d*Z¢ns*#n@l sp@c¢ @nd th¢ c%V¢ @s @ r¢g%l@r

V#dy #f @ thr¢¢-d*Z¢ns*#n@l sp@c¢.

26. Th¢ s*gn*f*c@nc¢ #f th¢ str%ct%r@l c#ZZ#nn¢ss #f @ sq%@r¢ @nd @ c%V¢ *s th@t

th¢ r@t*# #f @r¢@ @nd p¢r*Z¢t¢r #f th¢ sq%@r¢, @nd U#l%Z¢ @nd s%rf@c¢ @r¢@ #f th¢

c%V¢, @dZ*t s¢q%¢nt*@l #rd¢r @s @ f%nct*#n #f th¢ d¢gr¢¢ #f th¢ d*Z¢ns*#n@l sp@c¢.

W¢ Z@y d¢f*n¢ th¢ d¢gr¢¢ #f n-d*Z¢ns*#n@l sp@c¢ @s n. Th¢s¢ r@t*#s f#r l¢ngth '@' #f

th¢ r¢g%l@r V#d*¢s *s:

@2

: 4@ @nd @3

: 6@2

27. Th¢ @V#U¢ r@t*#s h@U¢ th¢ f#rZ%l@t*#n @ n

/2n@n-1

: 1 f#r n = 2 @nd 3.

28. Th¢ @V#U¢ s¢q%¢nt*@l f#rZ%l@t*#n *ZZ¢d*@t¢ly s%ppl*¢s th¢ k¢y t# %nl#ck th¢

str%ct%r@l kn#t #f th¢ d*Z¢ns*#n@l r¢g%l@r V#dy. *t *s th@t f#r @ny d*Z¢ns*#n@l r¢g%l@r

V#dy th¢ d#Z@*n p@rt @n

(wh*ch f#r th¢ sq%@r¢ *s th¢ @r¢@ #f th¢ sq%@r¢ @nd f#r th¢ c%V¢ *s

*ts U#l%Z¢) @nd th¢ fr@Z¢ p@rt 2n@n-1

(wh*ch f#r th¢ sq%@r¢ *s *ts p¢r*Z¢t¢r @nd f#r th¢

c%V¢ *s *ts s%rf@c¢ @r¢@) Z@*nt@*n @ r@t*# d¢p¢nd¢nt %p#n 1/2,th¢ l¢ngth (@) @nd th¢ d*-

Z¢ns*#n@l d¢gr¢¢ (n). Th*s r@t*# c#Z¢s t# V¢ @ /2n.Th¢ str%ct%r@l s*gn*f*c@nc¢ *s th@t f#r

@ny @ @nd @ny n, th¢r¢ c#Z¢s *nt# pl@y th¢ f@ct#r 1 /2.Th*s f@ct#r, wh*ch Z@y V¢ @cc¢pt¢d

@s @ h@lU*ng-f@ct#r, *s r¢sp#ns*Vl¢ f#r pr#U*d*ng th¢ r¢q%*r¢d str%ct%r@l kn#t t# V*nd th¢

str%ct%r@l @rr@ng¢Z¢nts t# c#nst*t%t¢ d*Z¢ns*#n@l r¢g%l@r V#d*¢s. Th*s *s th¢ str%ct%r@l

k¢y t# %nl#ck th¢ str%ct%r@l kn#ts #f d*Z¢ns*#n@l r¢g%l@r V#d*¢s #f @ll l¢ngths @nd @ll d¢-

gr¢¢s. Th*s f@ct#r V¢*ng fr¢¢ #f th¢ l¢ngth @nd d¢gr¢¢ #f d*Z¢ns*#n@l r¢g%l@r V#d*¢s @c-

q%*r¢s %n*U¢rs@l @ppl*c@t*#n. H¢nc¢, ¢U¢n th¢ r¢g%l@r V#d*¢s #f %n*t l¢ngth #f @ll th¢

d*Z¢ns*#n@l sp@c¢s #f @ny d¢gr¢¢ @ls# @dZ*t th¢ f@ct#r (1 /2).Vy th*s Z¢@ns w¢ g@*n *n-

s*ght *nt# th¢ *nt¢rn@l str%ct%r@l @rr@ng¢Z¢nts #f l1

, l2

,13

...1n

.

Fr@Z¢d D#Z@*ns S¢q%¢nc¢29. V¢f#r¢ w¢ *nU¢st*g@t¢ th¢ *nt¢rn@l str%ct%r¢ #f r¢g%l@r V#d*¢s #f %n*t l¢ngth #f

d*ff¢r¢nt d*Z¢ns*#n@l sp@c¢s, l¢t %s %nd¢rst@nd th¢ s¢q%¢nc¢ [@ n

/2n@n-1

f#r n =

1,2,3...]. Th*s s¢q%¢nc¢ Z@y V¢ d¢f*n¢d @s @ fr@Z¢d d#Z@*ns s¢q%¢nc¢ @nd *ts *nd*-

U*d%@l t¢rZs @s *nd*U*d%@l fr@Z¢d d#Z@*ns #r s*Zply fr@Z¢d d#Z@*ns. Th¢ r¢@s#n

f#r th¢ ch#*c¢ #f t¢rZ*n#l#gy *s th@t th¢ r¢g%l@r V#d*¢s h@U¢ th¢*r d#Z@*n p@rt c#n-

t@*n¢d w*th*n th¢ fr@Z¢ p@rt @nd th%s c@n V¢ d¢s*gn@t¢d @s fr@Z¢d d#Z@*ns.

F*rst Fr@Z¢d D#Z@*n

30. Th¢ f*rst Z¢ZV¢r #f th¢ fr@Z¢d d#Z@*ns s¢q%¢nc¢ [@ n

/2n@n-l

f#r n = 1 ,2,3...],

th@t *s @n

/2n@n-1 f#r n = 1, th@t *s @

1

/2xl@1-1

, th@t *s @/2, *s th¢ f*rst fr@Z¢d d#Z@*n.

Th*s *nd*c@t¢s th@t #%t #f @ny cl#s¢d l*n¢@r *nt¢rU@l w¢ c@n r¢Z#U¢ #n¢ cl#s¢d *nt¢r-

U@l #f h@lf l¢ngth wh*l¢ th¢ s¢c#nd h@lf w#%ld n#t V¢ @ cl#s¢d *nt¢rU@l. H¢nc¢ th¢

str%ct%r@l %n*t *s h@lf #f th¢ d*Z¢ns*#n@l %n*t, wh*ch ¢ss¢nt*@lly *s th¢ l*n¢@r %n*t.

#n¢ Z@y r¢U¢rt V@ck t# th*s fr@Z¢d d#Z@*n @ft¢r h@U*ng V¢¢n thr#%gh th¢ *nt¢rn@l

89

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Z#3¢/4 S5*¢45¢ @43 U¢3*5 S5*¢45¢

str%ct%r¢ #f s¢c#nd @nd th*rd fr@Z¢d d#Z@*ns, wh*ch @r¢ th¢ w¢ll!n#wn g¢#Z¢tr*

c@l f*g%r¢s #f @ s2%@r¢ @nd @ c%V¢, t# f%lly r¢@l*>¢ th*s *Zpl*c@t*#n.

*nt¢rn@l Str%ct%r¢ #f @ S/%@r¢

E8. Th¢ s2%@r¢ *s @ tw#!d*Z¢ns*#n@l r¢g%l@r V#dy w*th th¢ s@Z¢ l¢ngth #n V#th

d*Z¢ns*#n@l l*n¢s. T@*ng l¢ngth t# V¢ @ %n*ts, th¢ s2%@r¢ w#%ld h@U¢ @n @r¢@@$ @nd

@ p¢r*Z¢t¢r 0@. Vy s%*t@Vly ch##s*ng @ l*n¢@r %n*t, w¢ c@n h@U¢ th¢ l¢ngth #f @

s2%@r¢ ¢2%@l 8 6l*n¢@r %n*t7. Th¢ l¢ngth 8 6l*n¢@r %n*t7 w#%ld Z¢@n @ cl#s¢d *nt¢rU@l

#f l¢ngth 8 6l*n¢@r %n*t7. Str%ct%r@lly th*s w#%ld Z¢@n th@t w¢ c@n d*U*d¢ s@*d cl#s¢d

*nt¢rU@l *nt# tw# *nt¢rU@ls, #n¢ #f wh*ch *s @ cl#s¢d *nt¢rU@l #f l¢ngth &1$.

EC. @lg¢Vr@*c@lly w¢ n#w 6@;V7C B @

C ; @V ; V@ ; G

C B @

C ; C@V ; V

C. H¢#

Z¢tr*c@lly, t@*ng @ B (#:, @ cl#s¢d *nt¢rU@l, @nd G B (#:7, @ #n¢!s*d¢d #p¢n *nt¢r

U@l 6t# V¢ r¢f¢rr¢d t# @s @n #p¢n *nt¢rU@l7, th¢ @V#U¢ @lg¢Vr@*c ¢2%@l*ty w#%ld @c

c#%nt f#r th¢ *nt¢rn@l str%ct%r¢ #f th¢ s2%@r¢ #f l¢ngth (0I.

Str%ct%r@l Vr¢@2%p #f @ S/%@r¢

EE. H¢#Z¢tr*c@lly th¢ cl#s¢d *nt¢rU@l (##' c@n V¢ d¢c#Zp#s¢d @s th¢ cl#s¢d *n

t¢rU@l (##' B (#: ; 6:#' B (#: ; (#:7, wh¢r¢ : *s th¢ Z*ddl¢ p#*nt #f th¢

cl#s¢d *nt¢rU@l (##'. Th¢ str%ct%r@l Vr¢@!%p #f @ s2%@r¢ #f l¢ngth 8 6l*n¢@r %n*t7

c@n V¢ *n t¢rZs #f th¢ str%ct%r@l Vr¢@!%p p¢rZ*ss*Vl¢ Vy th¢ l*n¢@r %n*t @s @ cl#s¢d

*nt¢rU@l (#:7 #f &1$ %n*t l¢ngth @nd #p¢n *nt¢rU@l (#:7 #f &1$ %n*t l¢ngth. Th¢

@V#U¢ f*g%r¢ sh#ws th¢ str%ct%r@l Vr¢@!%p #f @ s2%@r¢ #f l¢ngth (##' w*th : @s *ts

Z*ddl¢ p#*nt. Th¢ @lg¢Vr@*c ¢2%@l*ty (##'C B (#:

C ; C(#:(#:7 ; (#:7

C wh¢n

tr@nsl@t¢d *nt# g¢#Z¢tr*c ¢2%@l*ty, @s h@s V¢¢n sh#wn *n th¢ @V#U¢ f*g%r¢, w#%ld d*

U*d¢ th¢ g¢#Z¢tr*c@l s2%@r¢ *nt# f#%r g¢#Z¢tr*c@l s2%@r¢s. #%t #f th¢s¢ f#%r s2%@r¢s,

th¢ f*rst s2%@r¢ r¢pr¢s¢nt¢d Vy (#:C *s @ c#Zpl¢t¢ s2%@r¢ w*th @n @r¢@ ¢2%@l t# th¢

s2%@r¢ #f th¢ l¢ngth (#:, @ p¢r*Z¢t¢r ¢2%@l t# t*Z¢s th¢ l¢ngth #f (#:, @nd th¢

f#%r V#%nd@ry l*n¢s @nd f#%r c#rn¢r p#*nts *nt@ct. Th¢ s¢c#nd @nd th*rd s2%@r¢s r¢p

r¢s¢nt¢d Vy C(#:(#:7 @r¢ n#t c#Zpl¢t¢ s2%@r¢s, s*nc¢ #n¢ V#%nd@ry l*n¢ @nd tw#

c#rn¢r p#*nts @r¢ Z*ss*ng. Th¢ f#%rth s2%@r¢ @s w¢ll *s n#t c#Zpl¢t¢, s*nc¢ *ts tw#

V#%nd@ry l*n¢s @nd thr¢¢ c#rn¢r p#*nts @r¢ Z*ss*ng.

D

8/17/2019 Azzura

17/30


34. Th¢ #r*g*n@l sq%@r¢ Z@y V¢ r¢c#nst*t%t¢d *n th¢ f#ll#w*ng f#%r st¢ps @s d¢p*ct-

¢d @V#U¢. @s @ f*rst st¢p, #%t #f th¢ f#%r sq%@r¢s t@k¢ th¢ sq%@r¢ w*th f#%r V#%nd@ry

l*n¢s @nd f#%r c#rn¢r p#*nts *nt@ct. @s @ s¢c#nd st¢p t@k¢ #n¢ Z#r¢ sq%@r¢ #%t #f th¢

r¢Z@*n*ng thr¢¢ sq%@r¢s w*th #nly #n¢ V#%nd@ry l*n¢ @nd tw# c#rn¢r p#*nts Z*ss*ng.

Wh¢n th*s sq%@r¢ *s @dd¢d t# th¢ sq%@r¢ #f th¢ f*rst st¢p, *t c#nst*t%t¢s @ r¢ct@ngl¢ #f

@r¢@ ¢q%@l t# h@lf th¢ @r¢@ #f th¢ #r*g*n@l sq%@r¢.

35. H¢r¢ *t Z@y V¢ r¢l¢U@nt t# n#t¢ th@t g¢#Z¢tr*c@lly th¢r¢ r¢Z@*ns n# U@c%%Z @t

@ll wh¢n th¢ tw# sq%@r¢s #f ¢q%@l l¢ngth, #n¢ #f th¢Z w*th *ts #n¢ V#%nd@ry l*n¢

Z*ss*ng, @r¢ j#*n¢d. Th*s *s V¢c@%s¢ [#X] + [#X) = [#X] + (X#'] = [##], th%s pr#-

U*d*ng @ c#nt*n%%Z thr#%gh#%t th¢ V#%nd@ry l*n¢ @l#ng wh*ch th¢ tw# sq%@r¢s @r¢

j#*n¢d t# c#nst*t%t¢ @ r¢ct@ngl¢.

36. N#w @s @ th*rd st¢p, w¢ Z@y t@k¢ #n¢ Z#r¢ sq%@r¢ wh#s¢ #n¢ V#%nd@ry l*n¢ *s

Z*ss*ng fr#Z th¢ r¢Z@*n*ng tw# sq%@r¢s. Wh¢n th*s sq%@r¢ *s j#*n¢d w*th th¢ r¢ct@n-

gl¢ c#Zp#s¢d *n th¢ f*rst tw# st¢ps, th¢ Z*ss*ng V#%nd@ry l*n¢, @s *s sh#wn *n th¢

@V#U¢ f*g%r¢, V¢c#Z¢s @ c#nt*n%%Z *n t¢rZs #f th¢ h@lf V#%nd@ry l*n¢ #f th¢ *nn¢r

l¢ngth #f th¢ r¢ct@ngl¢.

37. F#r @ f#%rth st¢p, j#*n th¢ r¢Z@*n*ng f#%rth sq%@r¢, wh#s¢ tw# V#%nd@ry l*n¢s

@r¢ Z*ss*ng, w*th th¢ g¢#Z¢tr*c@l f*g%r¢ f#rZ¢d @s @ r¢s%lt #f th¢ f*rst thr¢¢ st¢ps. @s

*s ¢U*d¢nt fr#Z th¢ @V#U¢ f*g%r¢, #n¢ #f th¢ Z*ss*ng V#%nd@ry l*n¢s w#%ld V¢c#Z¢ @

c#nt*n%%Z *n t¢rZs #f th¢ %pp¢r h@lf #f th¢ V#%nd@ry l*n¢ #f th¢ *nn¢r l¢ngth #f th¢

r¢ct@ngl¢ l¢ft %nc#U¢r¢d %nt*l St¢p 3, wh*l¢ th¢ s¢c#nd Z*ss*ng V#%nd@ry l*n¢ #f th*s

f#%rth sq%@r¢ (@s d¢p*ct¢d @s St¢p 4 *n th¢ @V#U¢ f*g%r¢) w#%ld V¢c#Z¢ @ c#nt*n%%Z

*n t¢rZs #f th¢ *nn¢r V#%nd@ry l*n¢ #f th¢ th*rd sq%@r¢.

38. Th*s *nt¢rn@l str%ct%r@l @rr@ng¢Z¢nt #f th¢ sq%@r¢ *s s*gn*f*c@nt *n s¢U¢r@l w@ys.

Tw# #f th¢s¢ wh*ch h@U¢ U*t@l V¢@r*ng f#r th¢ pr¢s¢nt @r¢ th@t th¢ sq%@r¢ h@s n*n¢ str%c-

t%r@l p#*nts #%t #f wh*ch ¢*ght @r¢ syZZ¢tr*c@lly l#c@t¢d @r#%nd th¢ c¢ntr@l n*nth p#*nt

wh¢r¢ @ll th¢ f#%r sq%@r¢s @r¢ j#*n¢d, @nd th@t wh¢n #%t #f th¢ sq%@r¢ #f r@t*#n@l l¢ngth,

@ sq%@r¢ #f r@t*#n@l l¢ngth *s c%t #%t, th¢ r¢Z@*n*ng p#rt*#n #f th¢ #r*g*n@l sq%@r¢ c#n-

s*sts #f thr¢¢ p@rts, n#n¢ #f wh*ch h@s @ll th¢ f#%r V#%nd@ry l*n¢s *nt@ct.

*nt¢rn@l Str%ct%r¢ #f @ C%V¢

39. Th¢ c%V¢ *s @ thr¢¢-d*Z¢ns*#n@l r¢g%l@r V#dy w*th th¢ s@Z¢ l¢ngth #n @ll th¢

thr¢¢-d*Z¢ns*#n@l l*n¢s. T@k*ng th¢ l¢ngth t# V¢ @ %n*ts, th¢ c%V¢ h@s U#l%Z¢ @3

@nd

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Z#3¢/4 S5*¢45¢ @43 U¢3*5 S5*¢45¢

s%rf@c¢ @r¢@3@$. Vy s%*t@Vly ch##s*ng @ l*n¢@r %n*t, w¢ c@n h@U¢ th¢ l¢ngth #f th¢

c%V¢ B 8 6l*n¢@r %n*t7. Th¢ l¢ngth 8 6l*n¢@r %n*t7 w#%ld Z¢@n @ cl#s¢d *nt¢rU@l #f

l¢ngth 8 6l*n¢@r %n*t7. Str%ct%r@lly th*s w#%ld Z¢@n th@t w¢ c@n d*U*d¢ th¢ s@*d cl#s¢d*nt¢rU@l *nt# tw# *nt¢rU@ls, #n¢ #f wh*ch *s @ cl#s¢d *nt¢rU@l #f l¢ngth &1$.

D. @lg¢Vr@*c@lly w¢ n#w th@t 6@;V7E B @

E ; E@

CV ; G

C ; G

E. H¢#Z¢tr*c@lly

t@*ng @ B (#:, @ cl#s¢d *nt¢rU@l, @nd G B (#:7, @ #n¢!s*d¢d #p¢n *nt¢rU@l 6t# V¢

r¢f¢rr¢d t# @s @n #p¢n *nt¢rU@l7, th¢ @V#U¢ @lg¢Vr@*c ¢2%@l*ty w#%ld @cc#%nt f#r th¢

*nt¢rn@l str%ct%r¢ #f th¢ c%V¢ #f l¢ngth C(#:.

Str%ct%r@l Vr¢@2%p #f @ C%V¢

8. H¢#Z¢tr*c@lly w¢ h@U¢ (##' B (#: ; 6:#' B (#: ; (#:7, wh¢r¢ : *s th¢

Z*ddl¢ p#*nt #f th¢ cl#s¢d *nt¢rU@l (##'. Th¢ str%ct%r@l Vr¢@!%p #f @ s2%@r¢ #f

l¢ngth 8 6l*n¢@r %n*t7 c@n V¢ h@d *n t¢rZs #f th¢ str%ct%r@l Vr¢@!%p p¢rZ*ss*Vl¢ Vy

th¢ l*n¢@r %n*t @s @ cl#s¢d *nt¢rU@l (#: #f &1$ %n*t l¢ngth @nd @n #p¢n *nt¢rU@l

l¢ngth (#:7 #f &1$ %n*t l¢ngth. Th¢ @V#U¢ f*g%r¢ sh#ws th¢ str%ct%r@l Vr¢@!%p #f @

c%V¢ #f l¢ngth (##' w*th : @s *ts Z*ddl¢ p#*nt. Th¢ @lg¢Vr@*c ¢2%@l*ty (##'E B

(#:E ; E(#:

C (#:7 ; E(#: (#:7

C ; (#:7

E, wh¢n tr@nsl@t¢d *nt# @ g¢#Z¢tr*c

¢2%@l*ty, @s h@s V¢¢n sh#wn *n th¢ @V#U¢ f*g%r¢, d*U*d¢s th¢ g¢#Z¢tr*c@l c%V¢ *nt#

¢*ght g¢#Z¢tr*c@l c%V¢s.

C. #%t #f th¢ @V#U¢ ¢*ght c%V¢s, th¢ f*rst c%V¢, r¢pr¢s¢nt¢d Vy (#:E, *s @ c#Z

pl¢t¢ c%V¢ h@U*ng @ U#l%Z¢ ¢2%@l t# @ c%V¢ #f th¢ l¢ngth (#:, s%rf@c¢ @r¢@ ¢2%@l t#

s*) t*Z¢s th¢ @r¢@ (#:C, @nd @ll th¢ s*) s%rf@c¢s @nd ¢*ght c#rn¢r p#*nts *nt@ct.

C

8/17/2019 Azzura

19/30


46. Th¢ d*U*s*#n #f th¢ c%V¢ *nt# ¢*ght c%V¢s #f @V#U¢ d¢scr*pt*#n c@n V¢ %s¢d *n r¢-

U¢rs¢ t# r¢c#nst*t%t¢ th¢ #r*g*n@l c%V¢ *n t¢rZs #f th¢ s@*d ¢*ght c%V¢s. Th*s *s s*Z*l@r

t# th¢ c@s¢ #f th¢ sq%@r¢ tr@c¢d @V#U¢ *n p@r@gr@phs 34 t# 37. Th¢ s@*d ¢*ght c%V¢s

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h@U¢ *n @ll 36 s%rf@c¢s. Wh¢n th¢s¢ @r¢ c#ZV*n¢d @s @V#U¢, tw¢lU¢ s%rf@c¢s @r¢ %s¢d t#

f#rZ @ c#nt*n%%Z w*th th¢ Z*ss*ng s%rf@c¢s #f s@*d c%V¢s. Th¢ r¢Z@*n*ng 24 s%rf@c¢s

@cc#%nt f#r th¢ s%rf@c¢ @r¢@ #f th¢ r¢-c#Zp#s¢d #r*g*n@l c%V¢ @s *s ¢U*d¢nt fr#Z th¢

@V#U¢ f*g%r¢.

S%ZZ@ry

47. V¢f#r¢ w¢ t@k¢ %p th¢ g¢n¢r@l Nth t¢rZ #f th¢ fr@Z¢d d#Z@*ns s¢q%¢nc¢, * w#%ld

l*k¢ t# s%ZZ@r*z¢ th¢ p#s*t*#n ¢Z¢rg*ng fr#Z th¢ @V#U¢ @n@lys*s #f th¢ f*rst thr¢¢

fr@Z¢d d#Z@*ns. Th¢s¢ thr¢¢ c@s¢s *nU#lU¢ c#ZZ#n g¢#Z¢tr*c@l f*g%r¢s, n@Z¢ly @

cl#s¢d *nt¢rU@l #f @ str@*ght l*n¢, @ sq%@r¢, @nd @ c%V¢ #f r@t*#n@l l¢ngths.

48. W*th r¢sp¢ct t# th¢ f*rst fr@Z¢d d#Z@*n, *t r¢q%*r¢s @ d¢f*n*t*#n #f @ str%ct%r@l

fr@Z¢ f#r @ cl#s¢d *nt¢rU@l. W*th r¢sp¢ct t# th¢ sq%@r¢ @nd th¢ c%V¢ *t *s #VU*#%s th@t

th¢ @r¢@ #f th¢ sq%@r¢ *s c#nt@*n¢d w*th*n th¢ p¢r*Z¢t¢r #f th¢ sq%@r¢ @nd th¢ U#l%Z¢ #f

th¢ c%V¢ *s s*Z*l@rly c#nt@*n¢d w*th*n th¢ s%rf@c¢ @r¢@ #f th¢ c%V¢. @s s%ch th¢

p¢r*Z¢t¢r @nd s%rf@c¢ @r¢@ r¢sp¢ct*U¢ly @r¢ th¢ fr@Z¢d p@rts #f th¢ sq%@r¢ @nd th¢ c%V¢.

Str%ct%r@lly th¢ l¢ngth #f th¢ cl#s¢d *nt¢rU@l r¢q%*r¢s @ Z*ddl¢ p#*nt @nd @s s%ch w¢

Z@y d¢f*n¢ @ p#*nt @s @ z¢r# sp@c¢ f*g%r¢ r¢sp#ns*Vl¢ f#r pr#U*d*ng @ fr@Z¢ f#r th¢

(l*n¢@r) l¢ngth Vy h@U*ng pl@c¢Z¢nt f#r th¢ fr@Z¢ p#*nt @nywh¢r¢ *n V¢tw¢¢n th¢

*nt¢rU@l. (T# V¢ sp¢c*f*c, th¢ pl@c¢Z¢nt f#r th¢ fr@Z¢ p#*nt *s n#t @t th¢ V¢g*nn*ng #r

th¢ ¢nd p#*nt #f th¢ cl#s¢d *nt¢rU@l.) Th@t w@y #nly #n¢ p#*nt (t# V¢ c@ll¢d th¢ Z*ddl¢

p#*nt s*nc¢ *t f@lls Z*dw@y V¢tw¢¢n th¢ tw# ¢nd p#*nts) c#nst*t%t¢s th¢ fr@Z¢ #f th¢

f*rst fr@Z¢d d#Z@*n.

49. Fr#Z th¢ @V#U¢, w¢ Z@y c#ncl%d¢ (@nd t@V%l@t¢ @s th¢ f*g%r¢ V¢l#w *nd*c@t¢s)

th@t wh¢n fr#Z th¢ f*rst fr@Z¢d d#Z@*n, th@t *s, @ cl#s¢d *nt¢rU@l #f @ str@*ght l*n¢, @

cl#s¢d *nt¢rU@l #f sZ@ll¢r r@t*#n@l l¢ngth *s c%t #%t, th¢ r¢Z@*n*ng p#rt*#n c#nst*t%t¢s

@n *rr@t*#n@l l¢ngth @s *t *s Z*ss*ng #n¢ ¢nd p#*nt. Wh¢n fr#Z th¢ s¢c#nd fr@Z¢d

d#Z@*n, th@t *s, @ sq%@r¢ #f r@t*#n@l l¢ngth, @ sq%@r¢ #f sZ@ll¢r r@t*#n@l l¢ngth *s c%t

#%t, th¢ r¢Z@*n*ng p#rt*#n c#nst*t%t¢s thr¢¢ sq%@r¢s #%t #f wh*ch tw# sq%@r¢s @r¢

Z*ss*ng #n¢ V#%nd@ry l*n¢, @nd #n¢ sq%@r¢ *s Z*ss*ng tw# V#%nd@ry l*n¢s. S*Z*l@rly,

wh¢n fr#Z th¢ th*rd fr@Z¢d d#Z@*n, th@t *s, @ c%V¢ #f r@t*#n@l l¢ngth, @ c%V¢ #f sZ@ll¢r

r@t*#n@l l¢ngth *s c%t #%t, th¢ r¢Z@*n*ng p#rt*#n c#nst*t%t¢s s¢U¢n c%V¢s #%t #f wh*chthr¢¢ c%V¢s @r¢ Z*ss*ng #n¢ s%rf@c¢, @n#th¢r thr¢¢ c%V¢s @r¢ Z*ss*ng tw# s%rf@c¢s, @nd

th¢ l@st, th@t *s, th¢ s¢U¢nth c%V¢, *s Z*ss*ng thr¢¢ s%rf@c¢s.

50. Th¢ f#ll#w*ng t@Vl¢ ¢U*d¢ntly Z@k¢s *t cl¢@r th@t wh¢n th¢ r@t*#n@l l¢ngth #f th¢

r¢g%l@r V#dy #f th¢ Nth-d*Z¢ns*#n (Nth fr@Z¢d d#Z@*n) *s d*U*d¢d *nt# r@t*#n@l p@rt (Q)

@nd th¢ r¢Z@*n*ng *rr@t*#n@l p@rt (R), th¢ r¢g%l@r V#dy g¢ts d*U*d¢d *nt# 2N

N-

d*Z¢ns*#n@l V#d*¢s #f th¢ f#rZ Qn

RN-n

, wh¢r¢ n = N, N-l, N-2, ... , 2, 1, 0. Th¢s¢ N-

d*Z¢ns*#n@l V#d*¢s, n%ZV¢r*ng N+l, Z@y V¢ c@ll¢d r¢sp¢ct*U¢ly, f*rst, s¢c#nd, ...

(N+l)th-d*Z¢ns*#n@l Vl#cks #f d¢gr¢¢ N. Th¢r¢ *s #nly #n¢ N-d*Z¢ns*#n@l V#dy #f th¢

f*rst-d*Z¢ns*#n@l Vl#ck. Th¢r¢ @r¢ N N-d*Z¢ns*#n@l V#d*¢s #f th¢ s¢c#nd-d*Z¢ns*#n@l

Vl#ck. Th¢ R-d*Z¢ns*#n@l Vl#ck h@s (N.N-l.N-2... N-R/l.2.3...R) N-d*Z¢ns*#n@l

V#d*¢s. Th¢ l@st Vl#ck h@s #n¢ N-d*Z¢ns*#n@l V#dy.

51. @s @ n¢t r¢s%lt, th¢ f#ll#w*ng *nt¢rn@l @rr@ng¢Z¢nts #f r¢g%l@r V#d*¢s j%st*fy #%r

l#g*c @nd c#ncl%s*#n #f p@r@gr@phs 7 t# 12 th@t wh¢n fr#Z th¢ r@t*#n@l l¢ngth, @ r@t*#n@l

p#rt*#n *s c%t #%t, *t l¢@U¢s V¢h*nd @n *rr@t*#n@l l¢ngth.

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D¢gr¢¢ #f Fr¢¢d#Z

52. N#w w¢ Z@y t@k¢ %p th¢ q%¢st*#n why r¢str*ct*#ns @r¢ n¢c¢ss@ry f#r n t# V¢ 3.

*n #th¢r w#rds, th¢ q%¢st*#n *s: why f#r th¢ f*rst @nd s¢c#nd d¢gr¢¢ #f n@t%r@l n%ZV¢rs,

th@t *s, f#r n = 1 @nd 2, th¢ ¢q%@t*#n Zn

= Xn

+ Yn

h@s s#l%t*#ns. H¢r¢ *t Z@y V¢ r¢l¢U@nt

t# n#t¢ th@t th¢ d*U*s*#n *nt# tw# p@rts #f l*k¢ d¢gr¢¢s *s n#t p#ss*Vl¢ f#r @ll Zn

, n = 1, 2.

Wh¢n Z @nd n V#th @r¢ ¢q%@l t# #n¢, th¢ d*U*s*#n f#r Zn

= 11

*s n#t p#ss*Vl¢, th#%gh f#r

@ll Z 2 @nd n = 1, th¢ d¢s*r¢d d*U*s*#n *nt# tw# p@rts #VU*#%sly h#lds Vy th¢ U¢ry d¢f*-

n*t*#n #f th¢ n@t%r@l n%ZV¢rs. F#r ¢x@Zpl¢, c#ns*d¢r 1, 1+1,1+2, 1+3 ...,1+s,....

53. F#r n = 2, ¢U¢ry Zn

*s n#t d¢c#Zp#s@Vl¢ @s Xn

+ Yn

. *ll%str@t*#ns #f th¢ p#*nt @r¢

l2

, 22

, 32

, 42

. Th¢r¢ @r¢ *nf*n*t¢ly Z@ny U@l%¢s #f Z f#r wh*ch th¢ d¢s*r¢d d¢c#Zp#s*t*#n

*s n#t @U@*l@Vl¢. #n th¢ c#ntr@ry, th¢r¢ @r¢ *nf*n*t¢ly Z@ny U@l%¢s #f Z f#r wh*ch th¢ d¢-

s*r¢d d¢c#Zp#s*t*#n h#lds. #n¢ cl@ss #f s%ch U@l%¢s *s Z = 5. Th¢ d¢c#Zp#s*t*#n f#r

th*s U@l%¢ *s 52

= 32

+ 42

. N#w f#r @ny U@l%¢ #f th¢ f#rZ 5Z, @ s*Z*l@r r¢l@t*#nsh*p @ls#

h#lds, n@Z¢ly (5Z)2

= (3Z)2

+ (4Z)2

.

54. S# f#r n = 1 @nd 2, th¢ d¢s*r¢d d¢c#Zp#s*t*#n *s n#t %n*U¢rs@lly p¢rZ*ss*Vl¢

th#%gh th¢ s@Z¢ h#lds f#r *nf*n*t¢ly Z@ny U@l%¢s #f Z. Th*s r¢str*ct¢d p¢rZ*ss*V*l*ty #f

th¢ d¢c#Zp#s*t*#ns f#r n = 1 h#lds f#r @ll U@l%¢s ¢xc¢pt Z = 1. @nd f#r n = 2 f#r @ l@rg¢

n%ZV¢r (@ c#%nt@Vl¢ n%ZV¢r #f U@l%¢s #f Z) th¢ d¢c#Zp#s*t*#n h#lds @nd f#r @n ¢q%@llyl@rg¢ n%ZV¢r (@ c#%nt@Vl¢ n%ZV¢r #f U@l%¢s #f Z), th¢ s@Z¢ d#¢s n#t h#ld.

55. Th¢ r¢@s#n f#r th¢ r¢str*ct¢d p¢rZ*ss*V*l*ty f#r n = 1 *s th@t th¢ @Vs#l%t¢ U@l%¢ #f

n@t%r@l n%ZV¢r 1 @nd th¢ l*n¢@r Z¢@s%r¢ U@l%¢ #f %n*t l¢ngth @r¢ tw# d*st*nct c#nc¢pts.

Th¢ n@t%r@l n%ZV¢r 1 *s th¢ @Vs#l%t¢ U@l%¢ *n th¢ s¢ns¢ th@t *t *s n#t d¢p¢nd¢nt %p#n @ny

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d*Z¢ns*#n@l str%ct%r¢s wh*l¢ th¢ l*n¢@r Z¢@s%r¢ U@l%¢ #f %n*t l¢ngth *s c#Zpl¢t¢ly d¢-

p¢nd¢nt %p#n th¢ str%ct%r@l @rr@ng¢Z¢nt #f @ #n¢-d*Z¢ns*#n@l sp@c¢, th@t *s @ str@*ght

l*n¢. Th¢ l*n¢@r %n*t w#%ld r¢pr¢s¢nt @ cl#s¢d *nt¢rU@l #f %n*t l¢ngth. Th¢ *nt¢rn@l str%c-

t%r@l r¢str*ct*#ns #f @ cl#s¢d *nt¢rU@l Z@k¢ *t *Zp#ss*Vl¢ f#r *ts d*U*s*#n *nt# tw# cl#s¢d

*nt¢rU@ls. @s s%ch th¢ n@t%r@l n%ZV¢r 1, @s th¢ f*rst d¢gr¢¢ U@l%¢ #f #n¢, d#¢s n#t @dZ*t

th¢ d¢s*r¢d d¢c#Zp#s*t*#n. H#w¢U¢r, th¢ str@*ght l*n¢ @nd h¢nc¢ @ cl#s¢d *nt¢rU@l, V¢*ng

@ f*rst-d*Z¢ns*#n@l V#dy (f*g%r¢) h@s tw# d¢gr¢¢s #f fr¢¢d#Z w*th*n @ thr¢¢-d*Z¢ns*#n@l

sp@c¢. @s s%ch, ¢xt¢rn@lly th¢ l*n¢@r %n*t *s fr¢¢ t# c#ZV*n¢ *ts¢lf w*th @ s*Z*l@r %n*t t#

c#nst*t%t¢ @ #n¢-d*Z¢ns*#n@l V#dy (f*g%r¢) #f tw# %n*ts. Th¢ r#l¢ #f th¢ #r*g*n p#*nt #f

th¢ thr¢¢-d*Z¢ns*#n@l fr@Z¢ *s cr%c*@l s*nc¢ fr#Z *t ¢Z@n@t¢s d*Z¢ns*#n@l l*n¢s @nd f#r

¢@ch #f th¢ thr¢¢-d*Z¢ns*#n@l l*n¢s *t r¢Z@*ns @ z¢r# U@l%¢ st@rt*ng p#*nt. V¢c@%s¢ #f

th*s, *t *s p#ss*Vl¢ t# h@U¢ l*n¢@r %n*ts s*Z%lt@n¢#%sly spr#%t*ng fr#Z th¢ #r*g*n p#*nt

@l#ng th¢ d*Z¢ns*#n@l l*n¢s. H¢nc¢, th¢r¢ @r¢ tw# d*st*nct pl@n¢s @U@*l@Vl¢ f#r ¢U¢ry d*-

Z¢ns*#n@l l*n¢, wh*ch Z@y V¢ %s¢d Vy th¢ g*U¢n l*n¢@r %n*t #f @ny #f th¢ thr¢¢ d*Z¢n-

s*#n@l l*n¢s. Th*s tw#-f#ld fr¢¢d#Z f#r th¢ str@*ght l*n¢ @cc#%nts f#r th¢ d¢s*r¢d d¢c#Z-

p#s*t*#n f#r @ll n@t%r@l n%ZV¢rs ¢xc¢pt Z-l.

56. T%rn*ng t# th¢ pl@n¢, @ sq%@r¢, @ tw#-d*Z¢ns*#n@l r¢g%l@r V#dy, h@s @s w¢ll, #n¢

d¢gr¢¢ #f fr¢¢d#Z *n @ thr¢¢-d*Z¢ns*#n@l sp@c¢. *t *s V¢c@%s¢ #f th¢ r¢str*ct¢d fr¢¢d#Z

#f th¢ pl@n¢ th@t th¢ d¢c#Zp#s*t*#n *s n#t %n*U¢rs@lly p¢rZ*ss*Vl¢ *n th¢ c@s¢ #f tw#-

d*Z¢ns*#n@l r¢g%l@r V#d*¢s. R@th¢r, *n th¢*r c@s¢, w*th*n @ny f*n*t¢ r@ng¢ #f n@t%r@l

n%ZV¢rs, s@y th¢ d¢s*r¢d d¢c#Zp#s*t*#n g¢n¢r@lly w#%ld n#t V¢ p¢rZ*ss*Vl¢

@nd *n @ c#Zp@r@t*U¢ly U¢ry sZ@ll n%ZV¢r #f U@l%¢s #f Z #nly th¢ d¢s*r¢d d¢c#Zp#s*-

t*#n w#%ld h#ld. Th*s *s s# @ls# w*th*n @ thr¢¢-d*Z¢ns*#n@l sp@c¢, tw# d*Z¢ns*#ns st@nd

r¢str*ct¢d V¢c@%s¢ #f th¢ str%ct%r@l f#rZ@t #f @ pl@n¢ @nd *t *s l¢ft w*th #nly #n¢ d¢gr¢¢

#f fr¢¢d#Z. H¢r¢ w¢ Z@y c#Zp@r¢ th¢ s*t%@t*#n w*th th¢ f@t¢ #f @ #n¢-d*Z¢ns*#n@l

V#dy, th@t *s, @ str@*ght l*n¢ wh#s¢ str%ct%r@l f#rZ@t h@s #nly #n¢ d*Z¢ns*#n. *t *s l¢ft

w*th tw# d¢gr¢¢s #f fr¢¢d#Z *n t¢rZs #f %nr¢str*ct¢d tw# d*Z¢ns*#ns #f th¢ thr¢¢-


57. N#w wh¢n w¢ c#Z¢ t# @ thr¢¢-d*Z¢ns*#n@l V#dy, *ts str%ct%r@l f#rZ@t r¢str*cts @ll

th¢ thr¢¢ d*Z¢ns*#n@l l*n¢s @nd h¢nc¢ w¢ @r¢ l¢ft w*th n# d¢gr¢¢ #f fr¢¢d#Z. *t *s V¢-

c@%s¢ #f th*s th@t *t *s n#t p#ss*Vl¢ t# d%pl*c@t¢ th¢ c%V¢. *n th¢ c@s¢ #f @ pl@n¢, *t *s p#ss*-

Vl¢ t# d%pl*c@t¢ *t @s @ pl@n¢ h@U*ng #n¢ d¢gr¢¢ #f fr¢¢d#Z *n @ thr¢¢-d*Z¢ns*#n@l sp@c¢.

@s s%ch w¢ c@n Z#U¢ (#r p*l¢) @ pl@n¢ (#r *d¢nt*c@l r¢g%l@r V#d*¢s #f @ tw#-d*Z¢ns*#n@l

sp@c¢, s@y sq%@r¢) @l#ng th¢ th*rd d*Z¢ns*#n @s *s ¢U*d¢nt fr#Z th¢ f#ll#w*ng f*g%r¢:

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58. Th¢ f*rst V@s*c str%ct%r@l ch@r@ct¢r*st*c @cq%*r¢d Vy th¢ #r*g*n p#*nt V¢c@%s¢ #f th¢

p#ss*V*l*ty f#r th¢ fr¢¢ Z#t*#n #f @ pl@n¢ #f #X, #Y fr@Z¢ @l#ng #Z d*Z¢ns*#n@l l*n¢ *s

th@t th¢ #r*g*n p#*nt g¢ts ¢ZV¢dd¢d w*th @ pl@n¢ str%ct%r¢. V¢c@%s¢ #f th*s, *t h@rdlyZ@tt¢rs wh*ch p#*nt #f th¢ Z @x*s *s ch#s¢n @s th¢ #r*g*n p#*nt. Th#%gh th¢ pl@n¢ str%c-

t%r¢ @cq%*r¢d Vy th¢ #r*g*n p#*nt #f th¢ d*Z¢ns*#n@l fr@Z¢ d#¢s n#t @ff¢ct th¢ ¢xt¢rn@l

fr¢¢d#Z f#r Z#t*#n #f th¢ pl@n¢ t#w@rds th¢ #Z @x*s, th¢ *nt¢rn@l str%ct%r@l @rr@ng¢-

Z¢nt #f th¢ pl@n¢ f*g%r¢s *s g#U¢rn¢d Vy th¢ tw#-d*Z¢ns*#n@l f#rZ@t. Th¢ gl@r*ng ¢ff¢ct

#f th¢ tw#-d*Z¢ns*#n@l f#rZ@t f#r th¢ pl@n¢ f*g%r¢s *s th@t *ts ¢U¢ry p#*nt @cq%*r¢s @

tw#-d*Z¢ns*#n@l str%ct%r¢. V¢c@%s¢ #f th*s ¢U¢n th¢ d*Z¢ns*#n@l l*n¢s, @s V#%nd@ry

l*n¢s #f pl@n¢ f*g%r¢s, g¢t *ns¢p@r@Vly Z¢rg¢d w*th th¢ pl@n¢ f*g%r¢. @s s%ch, str%ct%r@l-

ly th¢y t## c@n V¢ d¢@lt w*th #nly @s pl@n¢ str*ps @s sh#wn *n th¢ f*g%r¢s V¢l#w.

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59. Th¢ *ZZ¢d*@t¢ ¢ff¢ct *s th@t wh¢n w¢ d*U*d¢ @ sq%@r¢ N2

, #r @s @ Z@tt¢r #f f@ct

¢U¢n @ r¢ct@ngl¢ (NxZ), #f l¢ngth N, th¢n l2

%n*ts #f th¢ X @x*s, @s h@s V¢¢n sh#wn

@V#U¢, w#%ld V¢ N, V%t th¢ l 2 %n*ts #f Y @x*s w#%ld r¢Z@*n #nly N-l (*n th¢ c@s¢ #f @

r¢ct@ngl¢, #nly Z-l). Th¢r¢f#r¢, th¢ r¢-@ss¢ZVl¢d p#s*t*#n f#r l2

= 1 f#r th¢ X @nd Y

@x*s w#%ld g*U¢ N x (N-l) (@nd *n c@s¢ #f @ r¢ct@ngl¢ N x [Z-l]) wh*ch *s l¢ss th@n N2

(*n th¢ c@s¢ #f @ r¢ct@ngl¢ N x (Z-l) < N x Z). Th*s *s th¢ r¢@s#n why *n @ l@rg¢ n%ZV¢r

#f c@s¢s *t *s n#t p#ss*Vl¢ t# d¢c#Zp#s¢ Z2

@s X2

+ Y2

f#r n@t%r@l n%ZV¢rs Z, X, @nd Y.

60. N#w %s*ng #n¢ d¢gr¢¢ #f fr¢¢d#Z, wh*ch Z@y V¢ U*¢w¢d @s th¢ ¢xt¢rn@l fr¢¢d#Z

f#r th¢ pl@n¢ f*g%r¢s, th¢ r¢str*ct*#ns #f th¢ tw#-d*Z¢ns*#n@l f#rZ@t #f th¢ pl@n¢ f*g%r¢s

c@n V¢ k¢pt *n @V¢y@nc¢. H¢r¢ *t Z@y V¢ r¢l¢U@nt t# n#t¢ th@t th¢ pr##f #f th¢ Pyth@g#r@s

Th¢#r¢Z, w*th th¢ h¢lp #f @ r*ght-tr*@ngl¢, Z@y @pp¢@r @s *f w¢ @r¢ %s*ng #nly #n¢ pl@n¢.

@s @ Z@tt¢r #f f@ct, w¢ @r¢ %s*ng Z#r¢ th@n #n¢ pl@n¢ Vy c#ns*d¢r*ng th¢ s*d¢s #f @ tr*@n-gl¢. S%pp#s¢ th¢ sq%@r¢ #f th¢ hyp#t¢n%s¢ #f @ r*ght-@ngl¢ tr*@ngl¢ *s spl*t *nt# tw#

sq%@r¢s wh*ch ¢q%@l th¢ sq%@r¢s #f th¢ p¢rp¢nd*c%l@r @nd V@s¢ #f th¢ s@*d tr*@ngl¢ #r, @s

@ c#nU¢rs¢, th¢ r¢c#nstr%ct*#n #f @ sq%@r¢ #f th¢ hyp#t¢n%s¢ *n t¢rZs #f th¢ sq%@r¢s #f

th¢ V@s¢ @nd p¢rp¢nd*c%l@r. Th¢n w¢ h@U¢ t# %s¢ th¢ *nt¢rn@l str%ct%r@l @rr@ngZ¢nts p¢r-

Z*tt¢d Vy th¢ sq%@r¢ @s @ r¢g%l@r V#dy #f th¢ tw#-d*Z¢ns*#n@l sp@c¢. F%rth¢r, w¢ h@U¢

t# %s¢ ¢*ght syZZ¢tr*¢s #f th¢ sq%@r¢ @nd #n¢ d¢gr¢¢ #f fr¢¢d#Z (wh*ch th¢ sq%@r¢ @s @

tw#-d*Z¢ns*#n@l V#dy w#%ld h@U¢) w*th*n @ thr¢¢-d*Z¢ns*#n@l C@rt¢s*@n fr@Z¢.

61. @s *s sh#wn @V#U¢, #n th¢ tw#-d*Z¢ns*#n@l f#rZ@t #f #X, #Y d*Z¢ns*#n@l l*n¢s

w¢ c@n h@U¢ pl@n¢ f*g%r¢ @VCD (wh*ch Z@y V¢ @ sq%@r¢ #r @ r¢ct@ngl¢). Th*s f*g%r¢

w#%ld h@U¢ @ d¢gr¢¢ #f fr¢¢d#Z #f Z#t*#n @l#ng th¢ #Z @x*s. Th*s pr#U*d¢s %s w*th th¢

p#ss*V*l*ty t# ch##s¢ tw# *d¢nt*c@l f*g%r¢s, s@y @VCD @nd @'V'C'D' @s sh#wn @V#U¢.

Th¢ d*@g#n@l, V¢*ng #n th¢ #n¢-d*Z¢ns*#n@l f#rZ@t, w#%ld h@U¢ l¢ngth ¢q%@l t# @ny r¢@l

n%ZV¢r, wh*ch *ncl%d¢s th¢ n@t%r@l n%ZV¢rs. Th¢ tw#-d*Z¢ns*#n@l f*g%r¢ @C C'@' Z@y

V¢ @ sq%@r¢ #n @ pr#p¢r ch#*c¢ #f U¢rt*c¢s.

62. *t *s V¢c@%s¢ #f th¢ @V#U¢ fr¢¢d#Z #f c#nstr%ct*#n f#r th¢ pl@n¢ f*g%r¢s *n @ thr¢¢-

d*Z¢ns*#n@l fr@Z¢ th@t *t V¢c#Z¢s p#ss*Vl¢ t# d*U*d¢ th¢ sq%@r¢ #f th¢ hyp#t¢n%s¢ (@C

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C'@') *n t¢rZs #f th¢ sq%@r¢s #n th¢ d*Z¢ns*#n@l l*n¢s #f th¢ f#rZ@t #f th¢ pl@n¢ f*g%r¢

@VCD @nd U*c¢-U¢rs@. Th#%gh th*s @ss*gnZ¢nt @s s%ch *s n#t t@k¢n %p h¢r¢, *t Z@y n#t

V¢ #%t #f c#nt¢xt t# n#t¢ th@t th¢ %n*t sq%@r¢ w#%ld V¢ th¢ V@s*c c#nst*t%¢nt wh*chw#%ld V¢ %s¢d f#r @ d*U*s*#n pr#c¢ss #f #n¢ sq%@r¢ *nt# tw# sq%@r¢s #r th¢ r¢U¢rs¢ pr#-

c¢ss #f c#Zp#s*t*#n #f tw# sq%@r¢s *nt# #n¢ sq%@r¢. Th¢ %n*t sq%@r¢ (@nd @s s%ch @ny

sq%@r¢) h@s ¢*ght syZZ¢tr*¢s, @nd th¢ thr¢¢-d*Z¢ns*#n@l r¢g%l@r V#dy @s w¢ll h@s ¢*ght

c#rn¢r p#*nts, @nd th¢ thr¢¢-d*Z¢ns*#n@l C@rt¢s*@n fr@Z¢ c%ts th¢ sp@c¢ *nt# ¢*ght #c-

t@nts, @s sh#wn V¢l#w. *t *s V¢c@%s¢ #f th*s th@t th¢ d*U*s*#n pr#c¢ss @nd th¢ r¢U¢rs¢

c#Zp#s*t*#n pr#c¢ss *s p#ss*Vl¢.

63. H¢nc¢ w¢ h@U¢ j%st*f*¢d th¢ r¢str*ct*#n #f F¢rZ@t's L@st Th¢#r¢Z t# n ^ 3.

64. N#w w¢ Z@y t@k¢ %p th¢ thr¢@d #f th¢ @V#U¢ l#g*c #f *nt¢rn@l r¢str*ct*#ns #f th¢

f#rZ@t @nd th¢ ¢xt¢rn@l fr¢¢d#Z, *f @ny *s @U@*l@Vl¢, f#r th¢ d*Z¢ns*#n@l r¢g%l@r V#d*¢s

Z#U*ng fr#Z tw#-d*Z¢ns*#n@l V#d*¢s t# thr¢¢- @nd h*gh¢r-d*Z¢ns*#n@l V#d*¢s.

C@s¢ #f @ Thr¢¢-D*Z¢ns*#n@l R¢g%l@r V#dy

65. St¢p 1: L¢t U#l%Z¢ U = Z3

c%V*c %n*ts. *n #th¢r w#rds, th¢ c%V¢ Z *s c#nst*t%t¢d

Vy Z3

%n*t c%V¢s.

66. St¢p 2: Th¢ c%V¢ Z h@s Z %n*ts @l#ng th¢ #X d*Z¢ns*#n@l l*n¢ @s w¢ll @s Z %n*ts

@l#ng V#th #f th¢ r¢Z@*n*ng tw# d*Z¢ns*#n@l l*n¢s.

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68. St¢p 4: Z %n*t c%V¢s @r¢ r¢q%*r¢d t# c#nst*t%t¢ @ l¢ngth #f Z %n*ts @l#ng th¢ Y @x*s

@nd s*Z*l@rly, Z %n*t c%V¢s @r¢ r¢q%*r¢d t# c#nst*t%t¢ @ l¢ngth #f Z %n*ts @l#ng th¢ Z @x*s.

69. St¢p 5: Th¢ #r*g*n p#*nt #f th¢ C@rt¢s*@n fr@Z¢, V¢*ng @ d*Z¢ns*#nl¢ss p#*nt, wh¢n

th¢ X @x*s @nd th¢ Y @x*s ¢Z¢rg¢ fr#Z th¢ #r*g*n p#*nt, th¢ d*Z¢ns*#n@l l¢ngth @l#ng th¢

X @x*s *s n#t @ff¢ct¢d Vy th¢ d*Z¢ns*#n@l l¢ngth @l#ng th¢ Y @x*s @nd U*c¢ U¢rs@.

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72. St¢p 8: W¢ h@d st@rt¢d w*th c%V¢ Z #f l¢ngth Z @nd U#l%Z¢ U = Z3

, V%t h@U¢

¢nd¢d w*th @ c%V¢ #f U#l%Z¢ Z(Z+1)(Z+1) wh*ch *s gr¢@t¢r th@n Z3

, h¢nc¢ th¢r¢ *s @

c#ntr@d*ct*#n.

C#ncl%s*#n

73. Th¢r¢f#r¢, th¢ @V#U¢ c#ntr@d*ct*#n pr#U¢s th@t #%r @ss%Zpt*#n r¢g@rd*ng p¢rZ*s-

s*V*l*ty #f th¢ U#l%Z¢ #f th¢ c%V¢ t# V¢ s¢p@r@t¢d @nd h@ndl¢d lxlxl = 1 @s @ n@t%r@l

%n*t *s wr#ng. H¢nc¢ X3

c%V*c %n*ts c@nn#t V¢ tr¢@t¢d @s l*n¢@r %n*ts @nd @s s%ch th¢ @d-

d*t*#n #p¢r@t*#n #f n@t%r@l n%ZV¢rs *s n#t @ Z¢@n*ngf%l #p¢r@t*#n. *f c%V*c %n*ts lxlxl

@r¢ tr¢@t¢d @s l*n¢@r %n*ts 1, th¢n *t w#%ld r¢s%lt *n @ c#ntr@d*ct*#n. H¢nc¢, th¢ th¢#r¢Z

f#r th¢ c%V¢s th@t *t *s *Zp#ss*Vl¢ t# s¢p@r@t¢ @ c%V¢ *nt# tw# c%V¢s.

74. Th¢ @V#U¢ l#g*c w#%ld V¢ ¢q%@lly @ppl*c@Vl¢ t# th¢ V*-q%@dr@t*c @nd h*gh¢r p#w-

¢rs @s w¢ll, @s th¢ n@t%r@l n%ZV¢rs @dd*t*#n #p¢r@t*#n *s g¢#Z¢tr*c@lly l*n¢@r *n n@t%r¢;

th¢ s@Z¢ *s Z¢@n*ngl¢ss @nd *t *s n#t @ppl*c@Vl¢ t# th¢ h*gh¢r-d*Z¢ns*#n@l %n*ts lxl,

lxlxl, lxlxlxl....

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Z#3¢/4 S5*¢45¢ @43 U¢3*5 S5*¢45¢

$. ZC 6l)l7 @nd Z

C 687 @r¢ tw# d*st*nct %n*ts. S*Z*l@rly Z

E 6l)l)l7 @nd Z

E 687 @s w¢ll

@s Z 6l)l)l)l7 @nd Z

687 @nd s# #n, @r¢ p@*rs #f d*st*nct %n*ts. Th¢ @dd*t*#n #p¢r@t*#n

wh*ch V*nds 8;8 *s Z¢@n*ngl¢ss f#r th¢ %n*ts l)l #r l)l)l, #r l)l)l)l @nd s# #n, s*nc¢6l)l7 ; 6l)l7 Z¢@ns @n #p¢r@t*#n *n t¢rZs #f wh*ch tw# s2%@r¢ %n*ts @r¢ t# V¢ c#Z

V*n¢d, wh*l¢ 8;8 B C *s th¢ n@t%r@l n%ZV¢rs @dd*t*#n #p¢r@t*#n. %nl¢ss @nd %nt*l w¢

n#w h#w th¢ s2%@r¢ %n*t *s tr@nsf#rZ@Vl¢ *nt# @ l*n¢@r %n*t, th¢ l*n¢@r %n*ts @dd*t*#n #p

¢r@t*#n w#%ld n#t V¢ @ppl*c@Vl¢. Th¢ n@t%r@l n%ZV¢rs Z%lt*pl*c@t*#n #p¢r@t*#n, wh*ch *s

*nt*Z@t¢ly c#nn¢ct¢d w*th th¢ @dd*t*#n #p¢r@t*#n, r¢Z@*ns d¢p¢nd¢nt %p#n th¢ @dd*t*#n

#p¢r@t*#n #nly w*th*n @ l*n¢@r sp@c¢. Th¢ Z#Z¢nt th¢ sp@c¢ st@nds ch@ng¢d fr#Z #n¢!

d*Z¢ns*#n@l 6l*n¢@r7 sp@c¢ t# tw# @nd h*gh¢r!d*Z¢ns*#n@l sp@c¢, th¢ *nd¢p¢nd¢nt ch@r

@ct¢r*st*cs #f th¢ Z%lt*pl*c@t*#n #p¢r@t*#n @r¢ d*spl@y¢d. @s s%ch, %nl¢ss @nd %nt*l l)l *s

s%*t@Vly d¢f*n¢d, th¢ @dd*t*#n #p¢r@t*#n w*ll r¢Z@*n #nly th¢ #p¢r@t*#n #f #n¢!


"¢#Z¢tr*c@l C#nt*n%%Z: F#%rth2*Z¢ns*#n@l Sp@c¢

9. 4#w w¢ Z@y t@¢ %p th¢ f#%rth n%ZV¢r #f th¢ fr@Z¢d d#Z@*ns s¢2%¢nc¢

4@n1$n@

n2& f#r n 5 &, $, ... . F#r th¢ pr¢s¢nt *t Z@y V¢ t@¢n Vy w@y #f d¢f*n*t*#n th@t

4@01@

6 *s th¢ f#rZ%l@t*#n f#r th¢ r¢g%l@r V#dy #f f#%rth!d*Z¢ns*#n@l sp@c¢ #f d*Z¢n

s*#n@l l¢ngth @, c#nt¢nt p@rt 6d#Z@*n p@rt7 @s @0, @nd th¢ fr@Z¢ p@rt @s @

. *n th¢ c#n

t¢)t w¢ Z@y r¢f¢r t# @ c%V¢ #f d*Z¢ns*#n@l l¢ngth @,c#nt¢nt 6d#Z@*n p@rt7 @s U#l%Z¢ @,

@nd fr@Z¢ p@rt @s s%rf@c¢ @r¢@3@$ @s th¢ thr¢¢!d*Z¢ns*#n@l r¢g%l@r V#dy. F%rth¢r, Vy

w@y #f d¢f*n*t*#n, w¢ Z@y t@¢ th¢ f#ll#w*ng f*g%r¢ @s @ g¢#Z¢tr*c@l pr¢s¢nt@t*#n #f th¢

f#%rth!d*Z¢ns*#n@l r¢g%l@r V#dy.

Str%ct%r@l @rr@ng¢Z¢nt #f F#%rth Fr@Z¢d #Z@*n

. @lg¢Vr@*c@lly w¢ n#w th@t 6@;V7 B @

; 9@

EV ; @

CV

C ; 9@V

E ; V

. H¢#Z¢t

r*c@lly, t@*ng @ B (#: @nd G B (#:7 @s @ #n¢!s*d¢d #p¢n *nt¢rU@l 6t# V¢ r¢f¢rr¢d t# @s

@n #p¢n *nt¢rU@l7, *n t¢rZs #f th¢ @V#U¢ ¢)p@ns*#n, @;V B (##' B (#: ; 6:#' B

(#: ; (#:7, wh¢n : *s @ r@t*#n@l n%ZV¢r @nd *s th¢ Z*ddl¢ p#*nt #f (##'. *f (#: B

:, th¢n (##' B C:.

=. Th¢ @lg¢Vr@*c ¢2%@l*ty (##' B (#:

; 9(#:

E (#:7 ; (#:

C (#:7

C ;

9(#: (#:7E w#rs #%t th¢ str%ct%r@l @rr@ng¢Z¢nt #f th¢ f#%rth fr@Z¢d d#Z@*n. Th*s,

wh¢n tr@nsl@t¢d *nt# g¢#Z¢try, sh@ll d*U*d¢ th¢ f#%rth!d*Z¢ns*#n@l V#dy *nt# s¢U¢n

t¢¢n f#%rth!d*Z¢ns*#n@l V#d*¢s.

. #%t #f th¢ s¢U¢nt¢¢n f#%rth!d*Z¢ns*#n@l V#d*¢s, th¢ f*rst, n@Z¢ly (#:, *s @

c#Zpl¢t¢ f#%rth!d*Z¢ns*#n@l V#dy *n th¢ s¢ns¢ th@t @ll th¢ ¢*ght c%V*c c#Zp#n¢nts #f

th¢ fr@Z¢ @r¢ *nt@ct.

8DC

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29/30


80. W¢ Z@y d¢f*n¢ @ V#dy #f s%ch fr@Z¢ w*th @ll th¢ ¢*ght c%V*c c#Zp#n¢nts *nt@ct

@s @ f*rst-d*Z¢ns*#n@l Vl#ck #f th¢ r¢g%l@r V#dy.

81. @s *s ¢U*d¢nt fr#Z th¢ @V#U¢ @lg¢Vr@*c ¢q%@l*ty, *n @ll w¢ sh@ll h@U¢ f*U¢ typ¢s #f

d*Z¢ns*#n@l Vl#cks. Th¢ s¢c#nd, th*rd, f#%rth, @nd f*fth-d*Z¢ns*#n@l Vl#cks w#%ld V¢

str%ct%r@lly th¢ f#%rth-d*Z¢ns*#n@l V#d*¢s s%ch th@t th¢*r fr@Z¢s @r¢ r¢sp¢ct*U¢ly Z*ss-

*ng 1, 2, 3, @nd 4 c%V*c c#Zp#n¢nts.

82. Th¢r¢f#r¢, th¢ *nt¢rn@l str%ct%r@l @rr@ng¢Z¢nt #f th¢ f#%rth-d*Z¢ns*#n@l r¢g%l@r

V#dy @dZ*ts @ Vr¢@k-%p @s f*U¢-d*Z¢ns*#n@l Vl#cks s%ch th@t th¢s¢ Vl#cks h@U¢ r¢sp¢c-

t*U¢ly 1, 6, 4, 6, @nd 1 f#%rth-d*Z¢ns*#n@l V#d*¢s. ¢U*d¢ntly, @ll th¢ f*U¢-d*Z¢ns*#n@l

Vl#cks @r¢ str%ct%r@lly d*st*nct.

83. N#w th¢ g¢n¢r@l V*n#Z*@l ¢xp@ns*#n #f (@+V)N

w#%ld h¢lp %s c#ncl%d¢ th@t w¢

sh@ll h@U¢ (N+l)-d*Z¢ns*#n@l Vl#cks. *n th*s f#rZ@t, 2N

N-d*Z¢ns*#n@l V#d*¢s w#%ld

c#nst*t%t¢ th¢ r¢g%l@r V#dy (@+V)N

. @s f#r th¢ N-d*Z¢ns*#n@l sp@c¢, th¢ f#rZ@t #f th¢

d*Z¢ns*#n@l Vl#cks *s t# r¢Z@*n th¢ s@Z¢, s# @s t# str%ct%r¢ @ r¢g%l@r V#dy, ¢x@ctly 2N

N-d*Z¢ns*#n@l V#d*¢s w#%ld V¢ r¢q%*r¢d.

84. Th¢r¢f#r¢, f#r N 3 @nd Z 3 (f#r Z = 1 @nd 2 th¢ th¢#r¢Z *s #VU*#%s), l¢t ZN

=

XN

+ YN

wh¢r¢ X, Y, Z, @nd N @r¢ n@t%r@l n%ZV¢rs. F%rth¢r, l¢t Z > X > Y (f#r X = Y

th¢ th¢#r¢Z h@s V¢¢n pr#U¢d @V#U¢, @nd X @nd Y V¢*ng g¢n¢r@l, *t r¢Z@*ns t# pr#U¢ th¢

th¢#r¢Z wh¢n #n¢ #f X #r Y *s gr¢@t¢r th@n th¢ #th¢r). L¢t Z = X + R. Th¢r¢f#r¢, XN

+

YN

= (X+R)N

. Th¢ r*ght s*d¢ #n ¢xp@ns*#n y*¢lds XN

+ NRXN-1

+ .... Th¢r¢f#r¢, XN

+ YN

= XN

+ NRXN-1

+ ... + RN

.

103

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Th¢r¢f#r¢, YN

= NRXN-1

+ ... + RN

.

N#w th¢ r*ght s*d¢ *s @n ¢xpr¢ss*#n #f 2N

-1 N-d*Z¢ns*#n@l Vl#cks.

Th¢ l¢ft s*d¢ c@n h@U¢ d*U*s*#ns *nt# N-d*Z¢ns*#n@l Vl#cks @s Y = @+V.

YN

= (@+V)N

y*¢lds 2N

N-d*Z¢ns*#n@l Vl#cks, wh*l¢ th¢ N-d*Z¢ns*#n@l Vl#cks #n th¢

r*ght s*d¢ @r¢ #nly 2N

-1. H¢nc¢, th¢ l¢ft s*d¢ c@nn#t V¢ ¢q%@l t# th¢ th¢ r*ght s*d¢. W*th

th*s th¢ g¢#Z¢tr*c@l pr##f #f th¢ th¢#r¢Z st@nds c#Zpl¢t¢d.

85. N#w *f *n th¢ l*ght #f th¢ @V#U¢, w¢ try t# f*nd th¢ s*gn*f*c@nc¢ #f th¢ *nt¢rn@l

str%ct%r@l @rr@ng¢Z¢nt #f th¢ U¢ry f*rst r¢g%l@r V#dy l1

, wh¢r¢ 1 *s ¢q%@l t# @n *rr@t*#n@l

l¢ngth, th¢n *t r¢@lly w#%ld V¢ ¢nch@nt*ng t# s¢¢ h#w N@t%r¢ w#rks #%t 1 = 1/2 (h@lf #f @

r@t*#n@l %n*t) + 1/2 (h@lf #f @n *rr@t*#n@l %n*t). *t pr#U*d¢s %s w*th @ g¢#Z¢tr*c@l c#nt*n%-

%Z wh*ch, @s @ l*n¢@r c#nt*n%%Z, *s ¢q%*U@l¢nt t# th¢ f*¢ld #f r¢@l n%ZV¢rs wh#s¢ s%V-f*¢ld *s th¢ f*¢ld #f r@t*#n@l n%ZV¢rs.

86. #VU*#%sly, th¢ *Zpl*c@t*#ns @r¢ Z@ny @nd f@r r¢@ch*ng, p@rt*c%l@rly wh¢n th¢y

d¢Z#l*sh Z¢nt@l Vl#cks #f th¢ phys*c@l w#rld @nd h¢lp tr@nsc¢nd t# f#%r @nd h*gh¢r-

d*Z¢ns*#n@l sp@c¢s @s @ g¢#Z¢tr*c@l c#nt*n%%Z #f sp@c¢s wh#s¢ r¢g%l@r V#d*¢s c#nst*-

t%t¢ @ fr@Z¢d d#Z@*ns s¢q%¢nc¢ (@n

/2n@n-1

, n = 1, 2, 3 ...).

azzura

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