{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 0 0 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 } {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 "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 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 19 "2-4-4 \"Soccer Tori\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Load pa ckages 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 0 {PARA 4 "" 0 "" {TEXT -1 20 "Function definitions" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 21 "geometry and graphing" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 22 "The line segment from " }{TEXT 256 2 "p1" }{TEXT -1 4 " to " }{TEXT 257 3 "p2." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plane arc := proc(p1,p2,t)\n RETURN([evalm((1-t)*p1+t*p2),t=0..1]);\nend:\n " }}{PARA 0 "" 0 "" {TEXT -1 46 "an line segment which must satisfy co nstraints" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 350 "boundarc := proc(pt,b ounds)\n local eqset,solt,a,b;\n eqset := \{\};\n for b in bounds do\n eqset := \{op(eqset), subs(b[1]=pt[1],b[2]=pt[2],b[3])\};\n od; \n solt := solve(eqset,t);\n if lhs(solt[1]) = t then b := rhs(solt[1] ) else a := lhs(solt[1]); fi;\n if lhs(solt[2]) = t then b := rhs(solt [2]) else a := lhs(solt[2]); fi;\n RETURN([pt,t=a..b]);\nend:\n\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "endpointsofarc := proc(arc)\n loc al 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(eval f,map2(subs,t=a,arcform));\n e2 := map(evalf,map2(subs,t=a,arcform)); \n RETURN(copy(e1),copy(e2));\n end: \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plotplanearc := proc(arc)\n RETURN(plot([arc[1][1], arc[1][2],arc[2]],colour = black,thickness=2));\n end: " }}{PARA 0 " " 0 "" {TEXT -1 13 "\ncompare 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 (isnegli ble(evalm(e1-e3)) and isneglible(evalm(e2-e4)))\n or\n (isne glible(evalm(e1-e4)) and isneglible(evalm(e2-e3)))\n then RETURN( true);\n else RETRUN(false);\n fi;\nend: \n" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "transformations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The transf ormation matrix for a reflection along the vector " }{TEXT 258 1 "v" } {TEXT -1 15 " fixing point " }{TEXT 259 1 "p" }{TEXT -1 1 " " } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "reflectiontran s := proc(v,p)\n local e1,e2,f1,f2,r,A,b;\n e1 :=vector([1.,0] );\n e2 :=vector([0,1.]);\n r := dotprod(v,v);\n f1 := eva lm(e1-(2/r)*dotprod(e1,v)*v); \n f2 := evalm(e2-(2/r)*dotprod(e2,v )*v);\n A := matrix(2,2,[f1,f2]);\n b := evalm(p-A&*p);\n \+ RETURN(matrix(3,3,[A[1,1],A[1,2],b[1],A[2,1],A[2,2],b[2],0,0,1]));\n \+ end: \n" }}{PARA 0 "" 0 "" {TEXT -1 25 "Parts of a transformation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ApartofR := proc(R);\n RETURN(su bmatrix(R,1..2,1..2));\nend:" }{TEXT -1 0 "" }{MPLTEXT 1 0 64 "\n\nbpa rtofR := proc(R);\n RETURN(vector([R[1,3],R[2,3]]));\nend: \n" }} {PARA 0 "" 0 "" {TEXT -1 30 "The action of a transformation" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "movepoint := p roc(R,p)\nlocal u,v;\n u := vector(3,[p[1],p[2],1]);\n v := evalm(R& *u);\n RETURN(vector(2,[v[1],v[2]])):\nend:\n" }}{PARA 0 "" 0 "" {TEXT -1 16 "compare matrices" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "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" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "eps close := proc(A,B)\n RETURN(isnegligible(evalm(copy(A)-copy(B))));\n e nd:\n " }}{PARA 0 "" 0 "" {TEXT -1 108 "The line which is the fixed \+ point of a reflection, given in parametric form with unit speed parame trization." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 323 "reflectionlineofR := proc(R,t)\nlocal A,b,p,u,e;\n A := ApartofR(R);\n b := bpartofR(R );\n p := evalm(0.5*b);\n e := vector(2,[1,0]);\n u := evalm(A&* e+e);\n if isnegligible(u)\n then e := vector(2,[0,1]);\n \+ u := evalm(A&*e+e);\n fi;\n u := evalm(u/sqrt(dotprod(u,u) ));\n RETURN(evalm(p+t*u));\nend:\n" }}{PARA 0 "" 0 "" {TEXT -1 34 " test when a point satisfies bounds" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "testbounds := proc(v,bounds)\nlocal b;\n for b in bounds do\n \+ if not evalb(subs(b[1]=v[1],b[2]=v[2],b[3])) then RETURN(false) fi; \n od;\n RETURN(true);\nend:\n" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 20 "tiling constructions" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "intersection \+ of reflection lines " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "intersectionoffixedlines := proc(p,q)\nlocal s,t,L,M,st;\n \+ L := reflectionlineofR(p,s);\n M := reflectionlineofR(q,t);\n st : = solve(\{L[1]=M[1],L[2]-M[2]\},\{s,t\});\n RETURN(map2(subs,st,L)); \nend:\n" }}{PARA 0 "" 0 "" {TEXT -1 24 "centre of mass of a tile" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "centreoftile := proc(p,q,r)\nlocal v1,v2,v3;\n v1 := intersectionoffixedlines(p,q);\n v2 := intersecti onoffixedlines(q,r);\n v3 := intersectionoffixedlines(r,p);\n RETURN (evalm((v1+v2+v3)/3)); \nend:\n \n" }}{PARA 0 "" 0 "" {TEXT -1 22 "con struct tiling lines" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1871 "tilingline s := proc(p,q,r,t,sn,bounds)\nlocal boundarytilelist,surroundedtilelis t,i,j,k,tile,tile2,reflectionlist,\n rlist,A,Av,loc,found,R,S,fou ndR,arclist;\n rlist := [p,q,r];\n reflectionlist := [copy(p),copy(q ),copy(r)];\n boundarytilelist :=[[diag(1,1,1),copy(sn),\{\}]];\n su rroundedtilelist :=[];\n while nops(boundarytilelist) > 0 do\n til e := boundarytilelist[1]; \n boundarytilelist := subsop(1=NULL,boun darytilelist);\n surroundedtilelist :=[op(surroundedtilelist), tile [1]];\n for j in [1,2,3] do\n if not member(j,tile[3],'loc ')\n then A := evalm(tile[1]&*rlist[j]);\n R := e valm(A&*rlist[j]&*inverse(A));\n Av := movepoint(A,v); \+ \n foundR := false;\n for S in reflectionlis t do\n if epsclose(S,R) then foundR := true; break; fi; \n od;\n if not foundR then reflectionlist := [op(reflectionlist),copy(R)]; fi; \n found := false; \n for k from 1 to nops(boundarytilelist) do\n \+ tile2 := boundarytilelist[k]; \n if epsclose(A,t ile2[1]) \n then found := true;\n \+ tile2 := [tile2[1],tile2[2],\{op(tile2[3]),j\}];\n \+ if nops(tile2[3]) > 2 \n then bound arytilelist := subsop(k=NULL,boundarytilelist);\n \+ else boundarytilelist := subsop(k=tile2,boundarytilelist);\n \+ fi;\n break;\n \+ fi;\n od; \n if (not found) and \+ testbounds(Av,bounds)\n then boundarytilelist := [op(b oundarytilelist),[copy(A),copy(Av),\{j\}]];\n fi;\n \+ fi; \n od;\n od;\n arclist := [seq(boun darc(reflectionlineofR(R,t),bounds),R=reflectionlist)];\n RETURN(arcl ist);\nend: \n" }}{PARA 0 "" 0 "" {TEXT -1 19 "arcs for soccerb all" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1669 "soccerballarcs := proc(p,q ,r,v,t,bounds)\nlocal boundarynodelist,surroundednodelist,i,j,k,\n \+ node,node2,arclist,rlist,A,Av,loc,found,v0,Av0;\n v0 := centreoftil e(p,q,r);\n rlist := [p,q,r];\n arclist := [];\n boundarynodelist : =[[diag(1,1,1),copy(v0),copy(v),\{\}]];\n surroundednodelist :=[];\n \+ while nops(boundarynodelist) > 0 do\n node := boundarynodelist[1]; \n boundarynodelist := subsop(1=NULL,boundarynodelist);\n surro undednodelist :=[op(surroundednodelist), [node[1], node[2]]];\n for j in [1,2,3] do\n if not member(j,node[4],'loc')\n then A := evalm(node[1]&*rlist[j]);\n Av := movepoint(A,v); \+ \n Av0 := movepoint(A,v0); \n arclist := [o p(arclist),planearc(node[3],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 found := true;\n \+ node2 := [node2[1],node2[2], node2[3],\{op(node2[4]),j\}];\n \+ if nops(node2[4]) > 2 \n then boundarynodelist := subsop(k=NULL,boundarynodelist);\n \+ else boundarynodelist := subsop(k=node2,boundarynodelist); \n fi;\n break;\n \+ fi;\n od; \n if not found and testbounds(Av0,bounds)\n then boundarynodelist : = \n [op(boundarynodelist),[copy(A),copy(Av0),co py(Av),\{j\}]];\n fi;\n fi; \n \+ od;\n od;\n RETURN(surroundednodelist,arclist);\nend: \+ \n" }}}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Construction of \+ the 2-4-4 soccer tori" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "torus an d 2-4-4 reflection matrices" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 323 "bigR :=5; smallR := 3;\nTorusMap := proc(R,r,u,v)\n RETURN( [(R+r*cos(2*Pi*v))*cos(2*Pi*u),(R+r*cos(2*Pi*v))*sin(2*Pi*u), r*sin(2* Pi*v)]);\n end:\ntorus:= plot3d(TorusMap(bigR,smallR,u,v),u=0..1, v=0..1,color=yellow,orientation=[0,45], \n \+ style=patchnogrid,scaling=constrained, grid=[30,30]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "N := 12.;\nbounds := [[x,y,0<=x],[x,y,x<=1],[x, y,0<=y],[x,y,y<=1]]; \nv1 := vector([1,-1.]): p1 := vector([0,0]): \n v2 := vector([0,1.]): p2 := vector([0,1/N]):\nv3 := evalm(-v1-v2): p3 := vector([0,0]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "p := reflecti ontrans(v1,p1):\nq := reflectiontrans(v2,p2):\nr := reflectiontrans(v3 ,p3):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%bigRG\"\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'smallRG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG$\"#7\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'boundsG7&7%% \"xG%\"yG1\"\"!F'7%F'F(1F'\"\"\"7%F'F(1F*F(7%F'F(1F(F-" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "tiling and walls for the plane and surfa ce" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "v := centreoftile(p,q ,r);\nR := intersectionoffixedlines(p,q):\nP := intersectionoffixedlin es(q,r):\nQ := intersectionoffixedlines(r,p):\nwalls := [planearc(P,Q, t),planearc(Q,R,t),planearc(R,P,t)]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "planewalls := seq(plotplanearc(L),L=walls):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "Rlines := tilinglines(p,q,r,t,v,bounds):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "planetiling := seq(plot([L[1][1],L[1][2], L[2]], color=red,linestyle=2),L=Rlines):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "display(\{planetiling,planewalls\},scaling = constrai ned,axes=none);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'MATRIXG6#7 $7#$\"+zxxxF!#67#$\"+dbbbbF," }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6Y-%'CURVESG6%7S7$\"\"!$\"1+++MLLLL!#;7$$\"1nmm;arz@! #Y2aF+F 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