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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 18 "MAT331 - Project 3" }}
{PARA 257 "" 0 "" {TEXT 256 8 "Fractals" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1065 "This project is concerned \+
with ``fractals'', that is, objects whose Hausdorff dimension is great
er than their topological dimension (or, roughly, ``things which are s
o wiggly we have to say their dimension is not an integer''). Notice t
hat I am purposfully omitting a precise definition here; first, that's
 your job, and second, there isn't a single definition that everyone a
ccepts.   If you like, you may stick to self-similar fractals, as we h
ave in class.\n\nAs usual, this project has both an expository and a c
omputational portion. For the expository part, you should write a shor
t paper (3-5 pages or so) on fractals, dimension, etc. You will probab
ly want to refer to books and articles in the library(good possibiliti
es are books by Peitgen/J\374rgens/Saupe; Falconer; or chapter 4 of  t
he \"Chaos\" text by Alligood/Sauer/Yorke).  You might also want to se
arch the World-Wide-Web; there is quite a lot of information on fracta
ls  on the web, but be careful: as is always true, some of the informa
tion on the web is misguided, misleading, or just downright wrong." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 393 "
For the computational part, you should make some fractals listed below
.   You may use the \"turtle\" approch as we discussed in class, or wr
ite programs to make them directly, as you prefer.  I would suggest th
at you incorporate the fractals you compute in the computational porti
on into the expository portion, using them as illustrations, etc. Then
 include the maple to do them as an appendix." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 5 "1.   " }{TEXT -1 
143 "Make a vonKoch snowflake, then modify it  so that it becomes \"sp
ikier\" or \"flatter\" (see pictures below).  Compute the dimension fo
r each case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
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7$F<F(7$F<$!+++]i:F3F)-F$6$7$F;FAF)-F$6$7$7$$\"+++](o%F3F(7$$\"++++]iF
3F(F)-F$6$7$7$FLFCFKF)-F$6$7$FBFTF)-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#
%,CONSTRAINEDG" 2 610 91 91 2 0 1 0 2 9 0 1 1 1.000000 45.000000 
45.000000 10030 10061 10056 10074 0 0 0 20040 0 12010 0 0 0 0 0 0 0 1 
0 0 0 0 114 33 0 0 0 0 0 0 }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 158 "    What if you replace each segment of a square \+
by  9 segments each 1/3 as long, arranged as below?  What is the resul
ting figure, and what is its dimension?" }}{PARA 0 "" 0 "" {INLPLOT "6
.-%'CURVESG6$7$7$\"\"!F(F'-%'COLOURG6&%$RGBGF(F(F(-F$6$7$F'7$$\"+++]i:
!#5F(F)-F$6$7$F07$F1F1F)-F$6$7$F77$$\"++++DJF3F1F)-F$6$7$7$F<F(7$F<$!+
++]i:F3F)-F$6$7$F;FAF)-F$6$7$FB7$F1FCF)-F$6$7$F0FAF)-F$6$7$FKF0F)-F$6$
7$FA7$$\"+++](o%F3F(F)-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINED
G" 2 606 103 103 2 0 1 0 2 9 0 1 1 1.000000 45.000000 45.000000 10030 
10061 10056 10074 0 0 0 20040 0 12010 0 0 0 0 0 0 0 1 0 0 0 0 290 64 
0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 259 1 "3" 
}{TEXT -1 53 ". Make the following fractal.  Compute its dimension." }
}{PARA 0 "" 0 "" {INLPLOT "6h^w-%'CURVESG6$7$7$$\"+++++$)!\")$\"+,+++!
*!\"*7$$\"+MLLL#)F*F+-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$F/$!+********
*)F-7$F(F:F1-F$6$7$7$$\"+nmmm\")F*F+7$$\"+++++\")F*F+F1-F$6$7$7$$\"+nm
mm\"*F*$\"+,+++IF-7$$\"+++++\"*F*FLF1-F$6$7$7$FA$\"+JLLL$)F-F@F1-F$6$7
$7$F/FUFTF1-F$6$7$7$FJ$\"+JLLLBF-FIF1-F$6$7$7$$\"+LLLL#*F*FinFhnF1-F$6
$7$F.FZF1-F$6$7$7$FA$\"+hmmmwF-7$FDFhoF1-F$6$7$FTFgoF1-F$6$7$7$FOFinFh
nF1-F$6$7$7$$\"+++++$*F*$\"+JLLLVF-7$$\"+nmmm$*F*FhpF1-F$6$7$7$$\"+nmm
m$)F*$!+LLLLjF-7$$\"+MLLL%)F*FcqF1-F$6$7$7$Faq$!+********pF-F`qF1-F$6$
7$7$F(F\\rF[rF1-F$6$7$7$F($!+mmmmwF-FarF1-F$6$7$7$F/$!+KLLL$)F-7$F/Ffr
F1-F$6$7$F^sFerF1-F$6$7$7$F(F\\sF[sF1-F$6$7$7$FDFUFTF1-F$6$7$FNFapF1-F
$6$7$F<FesF1-F$6$7$FCFisF1-F$6$7$Fhn7$FJ$\"+rmmm;F-F1-F$6$7$7$FfqF\\r7
$$\"+++++&)F*F\\rF1-F$6$7$7$FA$\"+,+++qF-FgoF1-F$6$7$7$FDFduFcuF1-F$6$
7$FeqF\\uF1-F$6$7$FjoFiuF1-F$6$7$Feq7$Ffq$!+mmmmcF-F1-F$6$7$7$F^uFcqFe
qF1-F$6$7$7$F($\"+JLLLjF-7$FaqF^wF1-F$6$7$7$F(FduF]wF1-F$6$7$F]uFivF1-
F$6$7$7$$\"+++++()F*Ffr7$F\\xF\\rF1-F$6$7$7$$\"+nmmm&)F*F\\rF]uF1-F$6$
7$7$FOFgt7$FO$\"**********F-F1-F$6$7$7$FcxFfrFbxF1-F$6$7$7$$\"+MLLL')F
*FfrF_yF1-F$6$7$7$FO$!+++++5F-7$FO$!+nmmm;F-F1-F$6$7$7$F/FduFdwF1-F$6$
7$7$FdyF\\rFcyF1-F$6$7$F^xFfzF1-F$6$7$F]u7$F^uFfrF1-F$6$7$FcyF[xF1-F$6
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F1-F$6$7$7$FcxF:F[]lF1-F$6$7$7$F^uF:F_]lF1-F$6$7$7$$\"+++++**F*F:Fg]lF
1-F$6$7$7$F^uF\\sFc]lF1-F$6$7$FftFhxF1-F$6$7$F[]lF]^lF1-F$6$7$FgoFd\\l
F1-F$6$7$F_yF[]lF1-F$6$7$7$$\"+nmmm*)F*$\"*LLLL$F-7$F^_lFjxF1-F$6$7$7$
$\"+MLLL!*F*FjyFiyF1-F$6$7$7$Fg_l$!*MLLL$F-Ff_lF1-F$6$7$7$F^_lF]`lF\\`
lF1-F$6$7$7$F^_lFjyFb`lF1-F$6$7$7$F($\"+,+++]F-7$F/F[alF1-F$6$7$7$Fg_l
F`_lF]_lF1-F$6$7$7$$\"+++++*)F*FjyFf`lF1-F$6$7$7$F($\"+hmmmcF-Fj`lF1-F
$6$7$7$FfalF]zFealF1-F$6$7$7$$\"+LLLL))F*F]zFablF1-F$6$7$7$Ffbl$!+LLLL
BF-FeblF1-F$6$7$7$FfalF\\clF[clF1-F$6$7$7$Ffal$!+++++IF-FaclF1-F$6$7$7
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lFcdlF1-F$6$7$7$$\"+nmmm()F*F\\clF[clF1-F$6$7$7$F[elFfclFjdlF1-F$6$7$F
`wF[dlF1-F$6$7$7$F\\xFfclF`elF1-F$6$7$7$Fdy$!+mmmmOF-7$F\\xF\\flF1-F$6
$7$F^flFgelF1-F$6$7$F\\z7$FJF]zF1-F$6$7$7$Fdy$!+LLLLVF-F[flF1-F$6$7$7$
F\\xFjflFiflF1-F$6$7$7$F\\x$!+++++]F-F_glF1-F$6$7$7$FdyFdglFcglF1-F$6$
7$FiflFiglF1-F$6$7$7$FcxFjflFiflF1-F$6$7$7$FcxFdglF`hlF1-F$6$7$7$F^uFd
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7$7$FAF[alF[jlF1-F$6$7$7$FAFhpFdjlF1-F$6$7$7$F/FhpFhjlF1-F$6$7$Fb_lF`j
lF1-F$6$7$F]alF\\[mF1-F$6$7$7$FA$\"+hmmmOF-7$FDFg[mF1-F$6$7$F[sF9F1-F$
6$7$7$FfalFin7$FfalFLF1-F$6$7$FhjlFf[mF1-F$6$7$7$FfblFinF`\\mF1-F$6$7$
F\\jlFhjlF1-F$6$7$FgilFh\\mF1-F$6$7$7$FfblFLFh\\mF1-F$6$7$Fefl7$FJF\\c
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FL7$F(FLF1-F$6$7$7$F\\xFL7$F\\xFg[mF1-F$6$7$7$F/Fg[mFe^mF1-F$6$7$Fjp7$
F[qFg[mF1-F$6$7$7$F[elFLFj^mF1-F$6$7$Ff[mF__mF1-F$6$7$7$F[elFinFg_mF1-
F$6$7$Fh\\mF^`mF1-F$6$7$F[^mFf[mF1-F$6$7$F[_m7$FdyFg[mF1-F$6$7$F[dl7$F
fqF\\blF1-F$6$7$7$FaqF[alF[dlF1-F$6$7$Fj`lF`amF1-F$6$7$7$F\\xFhp7$F\\x
F[alF1-F$6$7$7$F(FhpFj`lF1-F$6$7$F\\[mF\\bmF1-F$6$7$7$FdyFhpFgamF1-F$6
$7$F__mF\\[mF1-F$6$7$7$F(Fg[mF__mF1-F$6$7$Ff^mFjbmF1-F$6$7$Fh`mFcbmF1-
F$6$7$Fc_m7$FfpFg[mF1-F$6$7$Fcbm7$FcxFhpF1-F$6$7$7$F^uFhpFhcmF1-F$6$7$
7$FdyF[alFcbmF1-F$6$7$7$F^uF[alF\\dmF1-F$6$7$7$FfqF[alFddmF1-F$6$7$Ff]
m7$FOF\\clF1-F$6$7$FhamF`dmF1-F$6$7$F\\amFhdmF1-F$6$7$7$FcxFg[mFh`mF1-
F$6$7$7$F^uF\\blF\\amF1-F$6$7$7$$\"+LLLL)*F*F\\s7$F_fmF:F1-F$6$7$7$Fcx
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6$7$7$F^uF^w7$F^uFduF1-F$6$7$7$FfqF^wF^imF1-F$6$7$7$FdyFLFj^mF1-F$6$7$
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7$7$FcxFho7$FdyFhoF1-F$6$7$7$FAF:FdjmF1-F$6$7$7$FcxFduFhjmF1-F$6$7$F^`
m7$F[elFgtF1-F$6$7$F_imFa[nF1-F$6$7$FjhmF^`mF1-F$6$7$Fe[nFgilF1-F$6$7$
7$F[elFjxFe[nF1-F$6$7$7$F\\xFjxFb\\nF1-F$6$7$Fe[n7$F\\xFgtF1-F$6$7$F\\
em7$FOFfclF1-F$6$7$Fj\\nFf\\nF1-F$6$7$7$FfalF`_lF]_lF1-F$6$7$F^]n7$FJF
fclF1-F$6$7$7$F_fmFdgl7$Fh]lFdglF1-F$6$7$7$FdyFdu7$F\\xFduF1-F$6$7$7$F
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F__nFdglF^_nF1-F$6$7$7$$\"+++++(*F*FdglFd_nF1-F$6$7$7$Fi_nFdvFh_nF1-F$
6$7$7$$\"+LLLL'*F*FdvF^`nF1-F$6$7$7$Fc`nFcqFb`nF1-F$6$7$7$Fi_nFcqFh`nF
1-F$6$7$FijmFb^nF1-F$6$7$7$Fi_nF\\rF\\anF1-F$6$7$7$F__nF\\rFcanF1-F$6$
7$7$F__nFfrFganF1-F$6$7$7$FDF:F][nF1-F$6$7$7$FfblFjxF`jlF1-F$6$7$7$F_f
mFfrF[bnF1-F$6$7$FgilFcbnF1-F$6$7$7$F_fmF\\rFgbnF1-F$6$7$7$Fh]lF\\rF^c
nF1-F$6$7$7$Fh]lF\\sF^fmF1-F$6$7$7$FfpF\\cl7$F_oF\\clF1-F$6$7$7$FfpFfc
lFjcnF1-F$6$7$F]_lFb`lF1-F$6$7$F^fmFgbnF1-F$6$7$Fc^n7$F\\xFhoF1-F$6$7$
7$FDF\\sF_bnF1-F$6$7$Fgbn7$Fh]lFfrF1-F$6$7$7$FfblFjyFeblF1-F$6$7$Faal7
$FOF`_lF1-F$6$7$FealFeenF1-F$6$7$F\\`lFaalF1-F$6$7$7$FOF]`lF\\`lF1-F$6
$7$FiyFcfnF1-F$6$7$Fijm7$FdyFUF1-F$6$7$7$FJFjyFiyF1-F$6$7$FeflF^gnF1-F
$6$7$7$F_oF]zFeflF1-F$6$7$7$F_oFjyFegnF1-F$6$7$7$FfalF]`lFealF1-F$6$7$
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F$6$7$Fb`lF]hnF1-F$6$7$F[dnFegnF1-F$6$7$7$F_oFgtF^oF1-F$6$7$7$FfpFgtFe
inF1-F$6$7$7$FfpFjxFiinF1-F$6$7$7$F_oFjxF]jnF1-F$6$7$7$F[elFjy7$F\\xFj
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6$7$FjfnF`\\oF1-F$6$7$Fi]nFf]mF1-F$6$7$7$F__nFhp7$F__nFg[mF1-F$6$7$F^_
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c`nF\\bl7$Fc`nF[alF1-F$6$7$7$F^gmF\\blFc_oF1-F$6$7$F`[oFjdlF1-F$6$7$7$
FagmF[al7$F^gmF[alF1-F$6$7$7$FagmFhpF_`oF1-F$6$7$7$$\"+LLLL%*F*Fg[m7$F
i`oFLF1-F$6$7$7$Fi_nFfclF__oF1-F$6$7$Fh`o7$Fi`oFhpF1-F$6$7$7$FagmFL7$F
agmFg[mF1-F$6$7$F\\\\oFjfnF1-F$6$7$7$F__nFfclF_aoF1-F$6$7$Fc^oF_boF1-F
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6$7$Fc_mFh`oF1-F$6$7$7$F[qFLFc_mF1-F$6$7$7$Fh]lFfclF^coF1-F$6$7$7$FfpF
LFecoF1-F$6$7$7$FfpFinF]doF1-F$6$7$FfjnFjboF1-F$6$7$F^oFadoF1-F$6$7$7$
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1-F$6$7$FjfnF^joF1-F$6$7$Fb`nF][pF1-F$6$7$Fg]oFijoF1-F$6$7$F[_oFgelF1-
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$6$7$7$F[qFdglFc]pF1-F$6$7$7$FfpFdglFb^pF1-F$6$7$FienFixF1-F$6$7$7$FAF
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goF1-F$6$7$7$F__nFdu7$F__nFhoF1-F$6$7$7$Fi_nFduF]`pF1-F$6$7$7$F^uFfclF
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Fc`nF^wF`apF1-F$6$7$Fc_oFdapF1-F$6$7$7$Fi_nF\\blFc_oF1-F$6$7$Fg^oF[bpF
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F1-F$6$7$FhhlFifpF1-F$6$7$F`gmF`gpF1-F$6$7$F]do7$F_oFLF1-F$6$7$7$Fi`oF
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"" 0 "" {TEXT 260 1 "4" }{TEXT -1 203 ".  (optional for extra credit).
  Create one or more other fractals.  Discuss, guess at, or compute th
eir dimensions.  The amount of extra credit will be based on originali
ty, creativity, and correctness." }}}}{MARK "2 2 0" 0 }{VIEWOPTS 0 0 
0 1 1 1803 }