{VERSION 1 0 "X11/Motif" "1.0"}{GLOBALS 3 1}{FONT 0 "-adobe-helve tica-bold-r-normal--*-140-*" "helvetica" "Helvetica-Bold" 8 14 0 "Helvetica-Bold" 12}{FONT 1 "-adobe-times-medium-r-normal--*-140- *" "times" "Times-Roman" 4 14 64 "Times-Roman" 12}{FONT 2 "-adobe -courier-medium-r-normal--*-140-*" "courier" "Courier" 4 14 192 " Courier" 12}{FONT 3 "-adobe-helvetica-bold-r-normal--*-180-*-*-*- *-*-*" "helvetica" "Helvetica-Bold" 8 18 0 "Helvetica-Bold" 18} {SCP_R 1 0 28{COM_R 2 0{TEXT 3 66 "\015\015 Mak ing a fractal in R3, a Sierpinski pyramid\015"}}{COM_R 3 0{TEXT 1 0 ""}}{COM_R 4 0{TEXT 1 142 "First, we define some useful routin es. Tetra takes a list of the verticies of a tetrahedron, and \+ returns a list of its faces (4 triangles)."}}{INP_R 5 0 "> "{TEXT 0 198 "Tetra := proc(verts) \012 local a,b,c,d; \012 \+ a := verts[1]; \012 b := verts[2]; \012 c := verts[3 ];\012 d := verts[4];\012 RETURN( [[a,b,c], [a,b,d], [a,c,d], [b,c,d]]);\012 end:\012"}}{COM_R 6 0{TEXT 1 64 "Midpo int returns the midpoint of the segment joining two points."}} {INP_R 7 0 "> "{TEXT 0 90 "Midpoint := proc(a,b) \012 RETURN( \+ [(a[1]+b[1])/2, (a[2]+b[2])/2, (a[3]+b[3])/2]); \012 end:"}} {COM_R 8 0{TEXT 1 19 "\015Now for the guts.\015"}}{SEP_R 9 0} {COM_R 10 0{TEXT 1 141 "\015Refine takes a tetrahedron (a list of 4 points) and returns the 4 smaller tetrahedra defined by those \+ points and the midpoints of each edge."}}{INP_R 11 0 "> "{TEXT 0 403 "Refine := proc (tetra) \012 local a,b,c,d; \012 a: =tetra[1]; \012 b:=tetra[2]; \012 c:=tetra[3]; \012 \+ d:=tetra[4]; \012 RETURN( [a, Midpoint(a,b), Midpoint(a,c ), Midpoint(a,d)], \012 [b, Midpoint(a,b), Midpoin t(b,c), Midpoint(b,d)], \012 [c, Midpoint(c,b), Mid point(a,c), Midpoint(c,d)], \012 [d, Midpoint(d,b), Midpoint(d,c), Midpoint(a,d)]); \012 end:\012"}}{COM_R 12 0 {TEXT 1 170 "RefineAll applies Refine to each element of a list o f tetrahedra. Note that it uses map. we could have written it wi th seq, but this is shorter and has the same effect. "}}{INP_R 13 0 "> "{TEXT 0 67 "RefineAll := proc(tlist) \012 RETURN(map(Re fine,tlist)); \012 end: \012"}}{COM_R 14 0{TEXT 1 147 "Recurse \+ is a general purpose routine that computes the n-fold composition of f applied to the second argument. that is, f(f(f(f(...f(inpu t)...))))"}}{INP_R 15 0 "> "{TEXT 0 144 "Recurse := proc(f, input , n) \012 if ( n <= 0) then \012 RETURN(input); \012 \+ else \012 RETURN(Recurse(f, f(input), n-1)); \012 fi; \0 12 end:"}}{COM_R 16 0{TEXT 1 0 ""}}{SEP_R 17 0}{COM_R 18 0{TEXT 1 152 "Now we can put it all together. \012FirstT is our initial tetrahedron, equilateral of side length 2. I know the height lo oks funny, but thats it. Really."}}{INP_R 19 0 "> "{TEXT 0 76 "Fi rstT := [[-1,-1,0],[0,sqrt(3)-1,0],[1,-1,0],[0,sqrt(3)/3-1,2*sqrt (15)/5]]:"}}{COM_R 20 0{TEXT 1 203 "\015Here we go. Here's the 3 rd stage. (Notice that we have to map Tetra onto\012each element \+ of the list that comes out of the recursion, so that we will\012h ave triangles instead of the verticies of tetrahedra."}}{INP_R 21 0 "> "{TEXT 0 12 "with(plots):"}}{INP_R 22 0 "> "{TEXT 0 88 "set options3d(axes=none, scaling=constrained,shading=XY,style=patch,o rientation=[60,48]);"}}{INP_R 23 0 "> "{TEXT 0 80 "polygonplot3d( map(Tetra,[Refine(FirstT)]),title=`Sierpinski Pyramid, level 1`); "}}{GRH_R 24 0{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00000000000000000000000000000000000000000000}}{COM_R 28 0{TEXT 1 288 "\015What would you suspect the dimension of the limiting obj ect is? (remember we started with solid pyramids.) If you do th e calculation, you find it is log 4/log2, that is, its 2! so we have a fractal with an integer dimension (its just the \"wrong\" one, from a topological perspective)."}}{COM_R 29 0{TEXT 1 0 ""} }}{END}