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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 38 "\n      Generating a Sie
pinski gasket.\n" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 241 "First, we have a procedure that takes one triangle as in
put, and returns three triangles with a side length of half the origin
al.\nThis is the basic generation step for the gasket.\n\nNote that it
 returns 3 triangles, NOT a list of 3 triangles." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 384 "OneToThree:=proc(tri)\n  local mid1, mid2, mid3;\n\n
  mid1:=[(tri[1][1]+tri[2][1])/2,(tri[1][2]+tri[2][2])/2];\n  mid2:=[(
tri[2][1]+tri[3][1])/2,(tri[2][2]+tri[3][2])/2];\n  mid3:=[(tri[3][1]+
tri[1][1])/2,(tri[3][2]+tri[1][2])/2];\n\n  RETURN( [tri[1], mid1,   m
id3,  tri[1]  ],\n                   [mid1,   tri[2], mid2,  mid1   ],
\n                   [mid2,   tri[3], mid3,  mid2   ]);\nend:\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "gasket takes a list of triangles \+
as input, and applies OneToThree to each triangle.  It then repeats th
e process on THAT list, numsteps times. \n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 214 "gasket:=proc(start,numsteps)\n   local triangles, i,
 k;\n\n   triangles:=start;\n   for i from 1 to numsteps do\n     tria
ngles:=[seq(OneToThree(triangles[k]), k=1..nops(triangles))];\n   od;
\n\n   RETURN(triangles);\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
135 "Finally, we have a handy utility routine that allows us to plot s
everal disjoint curves (our triangles) on the same plot, with no axes.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "PlotCurves:= proc(curvelist)\n
   local i;\n   plots[display](\n       ( seq(plot(curvelist[i]),i=1..
nops(curvelist))),\n       axes=none,scaling=constrained);\nend:\n" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "\nNow we are ready to make a Sie
rpinski gasket.  \n\nFirstTri is an equlateral triangle of side length
 2, which we use for our \"level 0\" gasket." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "FirstTri := evalf([ [0,0], [1,sqrt(3)], [2,0], [0,0] \+
]):\nPlotCurves([FirstTri]);\n" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURV
ESG6$7&7$\"\"!F(7$$\"\"\"F($\"1+++330K<!#:7$$\"\"#F(F(F'-%'COLOURG6&%$
RGBG$\"#5!\"\"F(F(-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG" 
2 415 415 415 2 0 1 0 2 9 0 1 1 1.000000 45.000000 45.000000 0 0 0 0 
0 0 0 0 0 0 0 255 0 0 255 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 59 "We can now refine this a few times to get
 a level 3 gasket:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "l3:= gasket([
FirstTri], 3):\nPlotCurves(l3);\n" }}{PARA 13 "" 1 "" {INLPLOT "6?-%'C
URVESG6$7&7$\"\"!F(7$$\"1++++++]7!#;$\"1+++5N1l@F,7$$\"1+++++++DF,F(F'
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F_t7$F`rF=FhsF_tF2-F$6$7&F\\t7$$\"1++++++v=F_pF-7$FfsF-F\\tF2-F$6$7&Fg
t7$$\"\"#F(F(7$F]tF(FgtF2-F$6$7&Fau7$F\\sF(FjtFauF2-F$6$7&Feu7$F`rF(7$
FerF-FeuF2-F$6$7&FiuFbo7$FjqF-FiuF2-F$6$7&F^vFctFjuF^vF2-%*AXESSTYLEG6
#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG" 2 415 415 415 2 0 1 0 2 9 0 1 1 
1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 255 0 0 255 0 0 1 
0 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "\nIf w
e want, we can use the previous figure (l3) as input, and refine it fu
rther.  Thus, the following should give us a level 6 gasket:\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "l6:= gasket(l3 , 3):\nPlotCurves(l6
);" }}{PARA 13 "" 1 "" {INLPLOT "6ghm-%'CURVESG6#7&7$\"\"!F(7$$\"+++]i
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TYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG" 2 415 415 415 2 0 1 0 2 9 
0 1 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
1 0 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "\n\n
Had we been so inclined, we could have written the gasket procedure  r
ecursively instead of iteratively.  Here is one way to do that:\n" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 193 "cheese:=proc(trilist,n)\n  local i;\n\n \+
 if (n<=0) then \n   RETURN(op(trilist));\n  else\n   RETURN( seq(chee
se([OneToThree(trilist[i])], n-1),\n                    i=1..nops(tril
ist)));\n  fi;\n\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 645 "This
 works by using OneToThree to triple each triangle in the list, as gas
ket does.  It then calls itself on the resulting list of 3 triangles, \+
asking for one level less of decoration.  When the number of  levels t
o go gets to zero, the input is returned (stripped of one level of nes
ting).\n\nAs a consequence, notice that the output is a bunch of trian
gles,  NOT a list of triangles as in gasket.  This means that we have \+
to  wrap the result in brackets before we can feed it into PlotCurves.
\n\n\"cheese\" is not a better way than \"gasket\" to make the Sierpin
ski gasket; they are just two different ways of doing the same thing.
\nHere is the result." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "PlotCurves
([cheese([FirstTri],6)]);\n" }}{PARA 13 "" 1 "" {INLPLOT "6ghm-%'CURVE
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