{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 69 137 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 69 137 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning " 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 45 "Movie of tiling an Icosahedral \+ \"Soccer Balls\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 49 "Load packages and define functions and procedures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart;\nwith(linalg): \nwith(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "geoeps := 10^(-6) ;" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'geoepsG#\"\"\"\"(+++\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Function \+ definitions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "spherical distance " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "spheredist := proc(v1,v2);\n \+ RETURN(evalf(arccos(dotprod(v1,v2)/sqrt(dotprod(v1,v1)*dotprod(v2,v2)) ))):\nend:" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 47 "select the nearest point from an antipodal pair" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 146 "nearestfromApair := proc(v,ap1)\n if sphered ist(v,ap1) <= evalf(Pi/2) \n then RETURN(copy(ap1));\n else RE TURN(evalm(-ap1));\n fi;\n end:" }}{PARA 0 "" 0 "" {TEXT -1 66 "T he great circle defined by the plane perpendicular to the vector " } {TEXT 256 2 "v." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 560 "greatcircle := \+ proc(v,t)\nlocal e1,e2,e3,u1,u2,m,vlist,i;\n e1 :=vector([1,0,0]);\n e2 :=vector([0,1,0]);\n e3 :=vector([0,0,1]);\n m :=max(seq(abs (v[i]),i=1..3));\n if abs(v[1]) = m then u1 := e2; u2 :=e3; fi; \n \+ if abs(v[2]) = m then u1 := e1; u2 :=e3; fi;\n if abs(v[3]) = m \+ then u1 := e1; u2 :=e2; fi;\n u1 := evalf(u1); u2 :=evalf(u2);\n v list := GramSchmidt([v,u1,u2]);\n u1 := evalm((1/sqrt(dotprod(vlist[ 2],vlist[2])))*vlist[2]);\n u2 := evalm((1/sqrt(dotprod(vlist[3],vli st[3])))*vlist[3]);\n RETURN(evalm(cos(t)*u1+sin(t)*u2));\n end:\n " }}{PARA 0 "" 0 "" {TEXT -1 57 "The great circle which is the fixed p oint of a reflection" }{TEXT 262 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "greatcircleofR := proc(R,t)\nlocal e,v;\n e := vector([1,0,0] );\n v := evalm(R&*e-e);\n if not isnegligible(v) then RETURN(grea tcircle(v,t)); fi;\n e := vector([0,1,0]);\n v := evalm(R&*e-e);\n if not isnegligible(v) then RETURN(greatcircle(v,t)); fi;\n e := \+ vector([0,0,1]);\n v := evalm(R&*e-e);\n if not isnegligible(v) th en RETURN(greatcircle(v,t)); fi;\n end:\n" }}{PARA 0 "" 0 "" {TEXT -1 26 "The arc of a circle from " }{TEXT 257 2 "v1" }{TEXT -1 4 " to " }{TEXT 258 2 "v2" }{TEXT -1 28 ", vectors of the same length " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "arc := proc(v1,v2,t)\n local v3,a,b,eq1,eq2,ab,th;\n if epsclose(v1,v2) then RETURN([copy(v1),t= 0..0.1]); fi;\n v3 := evalm(a*v1+b*v2" }{TEXT -1 0 "" }{MPLTEXT 1 0 356 ");\n eq1 := dotprod(v1,v3,'orthogonal') = 0;\n eq2 := dotprod (v3,v3,'orthogonal') = dotprod(v1,v1);\n ab := [fsolve(\{eq1,eq2\}, \{a,b\})];\n v3 := map2(subs,ab[1],v3);\n if dotprod(v2,v3) < 0 \+ then v3 := evalm(-v3); fi;\n th := evalf(arccos(dotprod(v1,v2)/sqrt (dotprod(v1,v1)*dotprod(v2,v2))));\n RETURN([evalm(cos(t)*v1+sin(t)* v3),t=0..th]);\n end:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "endpo intsofarc := proc(arc)\n local arcform,t,a,b,e1,e2;\n arcform := op( 1,arc); \n t := op([2,1],arc);\n a := op([2,2,1],arc);\n b := op([2 ,2,2],arc);\n e1 := map(evalf,map2(subs,t=a,arcform));\n e2 := map(e valf,map2(subs,t=a,arcform));\n RETURN(copy(e1),copy(e2));\n end: \+ \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plotarc := proc(arc)\n RETUR N(spacecurve(arc[1],arc[2],colour = black,thickness=2));\n end: " }} {PARA 0 "" 0 "" {TEXT -1 34 "The rotation matrix through angle " } {TEXT 260 4 "ang " }{TEXT -1 22 "and fixing the vector " }{TEXT 259 4 "axis" }{TEXT -1 1 "." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 702 "rotationmatrix := proc(axis,ang)\n local e1,e2,e 3,u1,u2,m,vlist,i,Q,rot;\n rot :=matrix([[1,0,0],[0,cos(ang),-sin( ang)],[0,sin(ang),cos(ang)]]);\n e1 :=vector([1,0,0]);\n e2 := vector([0,1,0]);\n e3 :=vector([0,0,1]);\n m :=max(seq(abs(axi s[i]),i=1..3));\n if abs(axis[1]) = m then u1 := e2; u2 :=e3; fi; \n if abs(axis[2]) = m then u1 := e1; u2 :=e3; fi;\n if abs( axis[3]) = m then u1 := e1; u2 :=e2; fi;\n vlist:= GramSchmidt([a xis,u1,u2]);\n u1 := evalm((1/sqrt(dotprod(vlist[2],vlist[2])))*vl ist[2]);\n u2 := evalm((1/sqrt(dotprod(vlist[3],vlist[3])))*vlist[ 3]);\n Q := matrix([axis,u1,u2]);\n Q := evalm(transpose(Q)&*r ot&*Q);\n RETURN(Q);\n end: " }}{PARA 0 "" 0 "" {TEXT -1 43 "T he matrix for reflection along the vector " }{TEXT 261 2 "v." }{TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 327 "refl ectionmatrix := proc(v)\nlocal e1,e2,e3,f1,f2,f3,r;\n e1 :=vector([1 .,0,0]);\n e2 :=vector([0,1.,0]);\n e3 :=vector([0,0,1.]);\n r : = dotprod(v,v);\n f1 := evalm(e1-(2/r)*dotprod(e1,v)*v); \n f2 := \+ evalm(e2-(2/r)*dotprod(e2,v)*v);\n f3 := evalm(e3-(2/r)*dotprod(e3,v )*v); \n RETURN(matrix(3,3,[f1,f2,f3]));\n end: \n" }}{PARA 0 "" 0 "" {TEXT -1 22 "vector of a reflection" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "vectorofreflection := proc(R)\nlocal e,v;\n e := v ector([1,0,0]);\n v := evalm(R&*e-e);\n if not isnegligible(v) the n RETURN(evalm(v/sqrt(dotprod(v,v)))); fi;\n e := vector([0,1,0]);\n v := evalm(R&*e-e);\n if not isnegligible(v) then RETURN(evalm(v/ sqrt(dotprod(v,v)))); fi;\n e := vector([0,0,1]);\n v := evalm(R&* e-e);\n if not isnegligible(v) then RETURN(evalm(v/sqrt(dotprod(v,v) ))); fi;\n end:\n" }}{PARA 0 "" 0 "" {TEXT -1 30 "intersection of \+ fixed ciricles" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "intersectionoffi xedlines := proc(p,q)\nlocal v1,v2,A,v,r,c;\n v1 := vectorofreflectio n(p);\n v2 := vectorofreflection(q);\n A := transpose(concat(v1,v2)) ;\n v := linsolve(A,vector([0,0]),r,'c');\n v := map2(subs,c[1]=1,v) ;\n RETURN(evalm(v/sqrt(dotprod(v,v))));\nend:" }}{PARA 0 "" 0 "" {TEXT -1 16 "compare matrices" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "i snegligible := proc(A)\nlocal e;\nglobal geoeps; \n if add(abs(e[1]), e = [entries(A)]) < geoeps \n then RETURN(true);\n else RETUR N(false);\n fi;\nend: \n\nepsclose := proc(A,B)\n RETURN(isnegligi ble(evalm(copy(A)-copy(B))));\n end:\n " }}{PARA 0 "" 0 "" {TEXT -1 12 "compare arcs" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "arcsareequal : = proc(arc1,arc2)\nlocal e1,e2,e3,e4;\n e1,e2 := endpointofarc(arc1) ;\n e3,e4 := endpointofarc(arc2);\n if (isneglible(evalm(e1-e3)) a nd isneglible(evalm(e2-e4)))\n or\n (isneglible(evalm(e1-e4) ) and isneglible(evalm(e2-e3)))\n then RETURN(true);\n else \+ RETRUN(false);\n fi;\nend: \n" }}{PARA 0 "" 0 "" {TEXT -1 24 " construct tiling circles" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1727 "tilin gcircles := proc(p,q,r,t)\nlocal boundarytilelist,surroundedtilelist,i ,j,k,tile,tile2,reflectionlist,\n rlist,A,Av,loc,found,R,S,foundR ,circlelist;\n rlist := [p,q,r];\n reflectionlist := [copy(p),copy(q ),copy(r)];\n boundarytilelist :=[[diag(1,1,1),\{\}]];\n surroundedt ilelist :=[];\n while nops(boundarytilelist) > 0 do\n tile := boun darytilelist[1]; \n boundarytilelist := subsop(1=NULL,boundarytilel ist);\n surroundedtilelist :=[op(surroundedtilelist), tile[1]];\n \+ for j in \{1,2,3\} do\n if not member(j,tile[2],'loc')\n \+ then A := evalm(tile[1]&*rlist[j]);\n R := evalm(A&*rl ist[j]&*transpose(A));\n foundR := false;\n \+ for S in reflectionlist do\n if epsclose(S,R) then foun dR := true; break; fi;\n od;\n if not foun dR then reflectionlist := [op(reflectionlist),copy(R)]; fi; \n \+ found := false;\n for k from 1 to nops(boundarytil elist) do\n tile2 := boundarytilelist[k]; \n \+ if epsclose(A,tile2[1]) \n then found := true ;\n tile2 := [tile2[1],\{op(tile2[2]),j\}];\n \+ if nops(tile2[2]) >2 \n \+ then boundarytilelist := subsop(k=NULL,boundarytilelist);\n \+ else boundarytilelist := subsop(k=tile2,boundarytile list);\n fi;\n break; \n fi;\n od; \n if not \+ found \n then boundarytilelist := [op(boundarytilelist ),[copy(A),\{j\}]];\n fi;\n fi; \n \+ od;\n od;\n RETURN([seq(greatcircleofR(R,t),R=reflectionlis t)]);\nend: \n" }}{PARA 0 "" 0 "" {TEXT -1 19 "arcs for soccerb all" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1501 "soccerballarcs := proc(p,q ,r,v,t)\nlocal boundarynodelist,surroundednodelist,i,j,k,\n node, node2,arclist,rlist,A,Av,loc,found;\n rlist := [p,q,r];\n arclist := [];\n boundarynodelist :=[[diag(1,1,1),copy(v),\{\}]];\n surrounded nodelist :=[];\n while nops(boundarynodelist) > 0 do\n node := bou ndarynodelist[1]; \n boundarynodelist := subsop(1=NULL,boundarynode list);\n surroundednodelist :=[op(surroundednodelist), [node[1], no de[2]]];\n for j in \{1,2,3\} do\n if not member(j,node[3],'l oc')\n then A := evalm(node[1]&*rlist[j]);\n Av : = evalm(A&*v);\n arclist := [op(arclist),arc(node[2],Av,t )];\n found := false;\n for k from 1 to nops (boundarynodelist) do\n node2 := boundarynodelist[k]; \+ \n if epsclose(A,node2[1]) \n then f ound := true;\n node2 := [node2[1],node2[2], \{ op(node2[3]),j\}];\n if nops(node2[3]) >2 \n \+ then boundarynodelist := subsop(k=NULL,boundar ynodelist);\n else boundarynodelist := subso p(k=node2,boundarynodelist);\n fi;\n \+ break;\n fi;\n od; \+ \n if not found \n then boundarynodelis t := [op(boundarynodelist),[copy(A),copy(Av),\{j\}]];\n \+ fi;\n fi; \n od;\n od;\n RETURN( surroundednodelist,arclist);\nend: \n" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Construction \+ of the icosahedral soccer balls" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "sphere and icosahedral reflection matrices" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "rad :=0.995; \nsphereparam := [rad*sin(u)*cos(v ),rad*sin(u)*sin(v),rad*cos(u)];\nsphere:= plot3d(sphereparam,u=0..Pi, v=0..2*Pi,color=yellow,orientation=[90,0], \n \+ style=patchnogrid,scaling=constrained,grid=[30,30]):\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$radG$\"$&**!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,sphereparamG7%,$*&-%$sinG6#%\"uG\"\"\"-%$cosG6#%\"vG F,$\"$&**!\"$,$*&F(\"\"\"-F)F/F,F1,$-F.F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "v1 := vector([1,0,0]);\nv2 := vector([0,1,0]);\nv 3 := vector([x,y,z]);\neq1 := dotprod(v1,v3,'orthogonal') = -cos(Pi/3) ;\neq2 := dotprod(v2,v3,'orthogonal') = -cos(Pi/5);\neq3 := x^2+y^2+z^ 2 = 1;\nxyz := fsolve(\{eq1,eq2,eq3\},\{x,y,z\});\nv3 := map2(subs,xyz ,v3);\n\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G-%'MATRIXG6#7%7#\" \"\"7#\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G-%'MATRIXG6#7%7 #\"\"!7#\"\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v3G-%'MATRIXG6# 7%7#%\"xG7#%\"yG7#%\"zG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/%\" xG#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/%\"yG,&*$-%%s qrtG6#\"\"&\"\"\"#!\"\"\"\"%F.\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$eq3G/,(*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF*F+F,*$)%\"zGF*F+F,F," } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$xyzG<%/%\"xG$!+++++]!#5/%\"yG$!+X* p,4)F*/%\"zG$!+S*p,4$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v3G-%'MA TRIXG6#7%7#$!+++++]!#57#$!+X*p,4)F,7#$!+S*p,4$F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "p := reflectionmatrix(v1);\nq := reflectionma trix(v2);\nr := reflectionmatrix(v3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG-%'MATRIXG6#7%7%$!\"\"\"\"!F,F,7%F,$\"\"\"F,F,7%F,F,F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG-%'MATRIXG6#7%7%$\"\"\"\"\"!F,F, 7%F,$!\"\"F,F,7%F,F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'MAT RIXG6#7%7%$\"+++++]!#5$!+X*p,4)F,$!+S*p,4$F,7%F-$!*&*p,4$!\"*$!+&***** **\\F,7%F/F5$\"+[*p,4)F," }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 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"6#>%\"PG-%'MATRIXG6#7%7#$!+=6Jd_!#57#\"\"!7#$\"+)33l])F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'MATRIXG6#7%7#\"\"!7#$!+$* 3AoN!#57#$\"+#fB%\"vG-%'MATR IXG6#7%7#!\"\"7#$F*\"\"!7#$\"%su!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"vG-%'MATRIXG6#7%7#$!+uR)\\J\"!#5F)7#$\"+a-cD)*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%testG7%$!+uR)\\J\"!#5F&$!+D(G\\J\"F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "walls := plotarc(arc(P,Q,t) ,color=black,linestyle=1,thickness = 2),\n plotarc(arc(Q,R,t), color=black,linestyle=1,thickness = 2),\n plotarc(arc(R,P,t),c olor=black,linestyle=1,thickness = 2):\nnode := pointplot3d(\{evalm(1. 0*v)\},style=POINT,symbol=CROSS,colour=black):\ndisplay3d(\{tiling,wal ls,node\},scaling = constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 686 581 581 {PLOTDATA 3 "68-%'CURVESG6%7T7%\"\"!F($\"\"\"F(7%$!+c&>(H6!#6F ($\"+Z=O****!#57%$!+E\\HfAF.F($\"+nuW(***F17%$!+S>e)Q$F.F($\"+2rD%***F 17%$!+mkVMF6F1F($\"+<=DO**F17%$!+o<_R7F1F($\"+I 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