{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 128 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 14 128 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 128 128 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 1 14 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 14 128 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 14 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 128 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 1 14 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 1 14 128 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 278 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 14 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 14 128 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 1 14 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 14 128 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 290 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 296 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 301 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 50 "High School Modul es > Geometry by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 256 40 "How Many Ways Are There to Paint a Cube?" }}{PARA 0 " " 0 "" {TEXT -1 57 "\nAn exploration of colorings and symmetries of th e cube.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execute the \+ Code Resource section first. Although there will be no output immediat ely, these definitions are used later in this worksheet.]" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1059 "#-------------------------------- --------------\n# 1 is pink, 2 is green, 3 is blue, 4 is yellow, 5 is red, 6 is white\n\nc1 := COLOR(RGB, .97,.58,.67):\nc2 := COLOR(RGB,.2 ,.7,.2):\nc3 := COLOR(RGB,.3,.3,.7):\nc4 := COLOR(RGB,.99,.7, 0):\nc5 \+ := COLOR(RGB,.9, .1, .1):\nc6 := COLOR(RGB,.85,.8, .7):\nC1 := 'style= patchnogrid, color = c1':\nC2 := 'style=patchnogrid, color = c2':\nC3 \+ := 'style=patchnogrid, color = c3':\nC4 := 'style=patchnogrid, color = c4':\nC5 := 'style=patchnogrid, color = c5':\nC6 := 'style=patchnogri d, color = c6':\n\nd1 := COLOR(RGB, .8,.5,.6):\nd2 := COLOR(RGB, .1,.4 ,.1):\nd3 := COLOR(RGB, .2,.2,.4):\nd4 := COLOR(RGB, .95,.6, 0):\nd5 : = COLOR(RGB, .72, 0, 0):\nd6 := COLOR(RGB, .72,.64, .5):\nD1 := 'style =patchnogrid, color = d1':\nD2 := 'style=patchnogrid, color = d2':\nD3 := 'style=patchnogrid, color = d3':\nD4 := 'style=patchnogrid, color \+ = d4':\nD5 := 'style=patchnogrid, color = d5':\nD6 := 'style=patchnogr id, color = d6':\n\n\ncb := 'COLOR(RGB,.1,.1,.1)':\nls := 'STYLE(LINE) ,THICKNESS(2),LINESTYLE(3)':\n\n#------------------------------------- ---------" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "#============ =================================================\nColoredSheets := pr oc( fr, bk, le, ri, tp, bt )\n # colors for : front, back, left, righ t, top, bottom ....\n local k,Z,PP,y;\n \n for k from 1 to 6 do\n y := args[k];\n Z := C||y; \n PP||k := polygonplot( [[k-1,0],[k ,0],[k,k],[k-1,k],[k-1,0]], Z) :\n QQ||k := polygonplot( [[k-1 ,0],[k,0],[k,k],[k-1,k],[k-1,0]], color = black) :\n od;\n\n display ( [seq(PP||k, k = 1..6),seq(QQ||k, k = 1..6)],\n scaling=co nstrained,axes=none );\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1083 "#==== =========================================================\nColorCubeOp en := proc( fr, bk, le, ri, tp, bt )\n # colors for : front, back, le ft, right, top, bottom ....\n local Fr,Bk,Le,Ri,Tp,Bt,x,y,R,theta,a a,Z, Bf,Rf, clb;\n clb := 'color = COLOR(RGB,.1,.1,.1)':\n R := .6; \+ theta := Pi/4;\n x := evalf(R*cos(theta)); y := evalf(R*sin(theta) );\n aa:= .25;\n Z := C||fr;\n Fr := polygonplot( [[0,0],[1,0],[1,1 ],[0,1],[0,0]], Z) :\n Z := D||bk;\n Bk := polygonplot( [[-x, y],[1-x,y],[1-x,y +1],[-x,y+1],[-x,y]], Z):\n Z := C||le;\n Le : = polygonplot( [[0,0],[-x,y],[-x,y +1],[0,1],[0,0]], Z):\n Z := D|| ri;\n Ri := polygonplot( [[1,0],[1-x,y],[1-x,y +1],[1,1],[1,0]], Z): \n Z := C||tp;\n Tp := polygonplot( [[-x,y+1],[1-x,y+1],[1,y+y+1],[0 ,y+y+1],[-x,y+1]], Z):\n Z := D||bt;\n Bt := polygonplot( [[-aa,y-1+ aa],[1-aa,y-1+aa],[1-x,y ],[-x,y],\n [-aa,y-1+aa ]], Z ):\n\n Bf := polygonplot( [[1-x,1],[1-x,y +1]],clb):\n Rf \+ := polygonplot( [[0,0],[0,1]],clb):\n\n display( [ Fr,Le,Tp,Bt,Ri,Bk, Bf,Rf\n], scaling = constrained,axes=none );\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 921 "#================================= ============================\nColorCube := proc( fr, bk, le, ri, tp, b t )\n local XY0,XY1,XZ0,XZ1,YZ0,YZ1,Z;\n Z := d||bk;\n XY0 := PLOT3D( POLYGONS([[0,0,0],[1,0,0],[1,1,0],[0,1,0]]),\n \+ STYLE(PATCH), Z):\n Z := c||fr;\n XY1 := PLOT3D( POLYGONS ([[0,0,1],[1,0,1],[1,1,1],[0,1,1]]),\n STYLE(PATCH), Z):\n\n Z := d||le;\n XZ0 := PLOT3D( POLYGONS([[0,0,0],[1,0,0], [1,0,1],[0,0,1]]),\n STYLE(PATCH), Z):\n Z := c|| ri;\n XZ1 := PLOT3D( POLYGONS([[0,1,0],[1,1,0],[1,1,1],[0,1,1]]),\n STYLE(PATCH), Z):\n\n Z := d||bt;\n YZ0 := PL OT3D( POLYGONS([[0,0,0],[0,1,0],[0,1,1],[0,0,1]]),\n \+ STYLE(PATCH), Z):\n Z := c||tp;\n YZ1 := PLOT3D( POLYGONS([[1,0 ,0],[1,1,0],[1,1,1],[1,0,1]]),\n STYLE(PATCH), Z):\n display(XY0,XY1,XZ0,XZ1,YZ0,YZ1, scaling = constrained);\nend proc :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3182 "ColorCubeRotation := proc( case )\n local XY0,XY1,XZ0,XZ1,YZ0,YZ1,b,d,A1,A2,O; \n XY0 := PLOT3D( POLYGONS([[0,0,0],[1,0,0],[1,1,0],[0,1,0]]),\n \+ STYLE(PATCH), c1):\n XY1 := PLOT3D( POLYGONS([[0,0,1] ,[1,0,1],[1,1,1],[0,1,1]]),\n STYLE(PATCH), c1):\n \+ XZ0 := PLOT3D( POLYGONS([[0,0,0],[1,0,0],[1,0,1],[0,0,1]]),\n \+ STYLE(PATCH), c5):\n XZ1 := PLOT3D( POLYGONS([[0,1,0],[ 1,1,0],[1,1,1],[0,1,1]]),\n STYLE(PATCH), c5):\n \+ YZ0 := PLOT3D( POLYGONS([[0,0,0],[0,1,0],[0,1,1],[0,0,1]]),\n \+ STYLE(PATCH), c6):\n YZ1 := PLOT3D( POLYGONS([[1,0,0],[1, 1,0],[1,1,1],[1,0,1]]),\n STYLE(PATCH), c6):\n b \+ := 1.65; d := 1-b;\n #------------ face axes ----------------- ------------\n if (case = 1) then\n A1 := PLOT3D( CURVES( [[.5 , .5, 1],[.5,.5, b]]), cb,ls):\n A2 := PLOT3D( CURVES( [[.5, .5, 0],[.5,.5, d]]), cb,ls): \n O:=[30,72]; fi;\n if (case = 2) \+ then\n A1 := PLOT3D( CURVES( [[.5, 1, .5],[.5, b,.5]]), cb,ls): \n A2 := PLOT3D( CURVES( [[.5, 0, .5],[.5, d,.5]]), cb,ls): \n \+ O:=[38,72]; fi;\n if (case = 3) then\n A1 := PLOT3D( CURVE S( [[ 1, .5, .5],[ b, .5,.5 ]]), cb,ls):\n A2 := PLOT3D( CURVES( \+ [[ 0, .5, .5],[ d, .5,.5 ]]), cb,ls): \n O:=[50,72]; fi;\n #-- ---------- EDGES -----------------------------\n if (case = 4) then \n A1 := PLOT3D( CURVES( [[ 1, .5, 1], [ b, .5, b ]]), cb,ls):\n \+ A2 := PLOT3D( CURVES( [[ 0, .5, 0], [ d, .5, d ]]), cb,ls): \n \+ O:=[44,72]; fi;\n if (case = 5) then\n A1 := PLOT3D( CURVES ( [[ 0, .5, 1], [ d, .5, b ]]), cb,ls):\n A2 := PLOT3D( CURVES( [ [ 1, .5, 0], [ b, .5, d ]]), cb,ls): \n O:=[44,72]; fi;\n if ( case = 6) then\n A1 := PLOT3D( CURVES( [[.5, 1, 1], [.5, b, b ] ]), cb,ls):\n A2 := PLOT3D( CURVES( [[.5, 0, 0], [.5, d, d ]]), cb,ls): \n O:=[44,72]; fi;\n if (case = 7) then\n A1 := \+ PLOT3D( CURVES( [[ .5, 0, 1], [ .5, d, b ]]), cb,ls):\n A2 := PLO T3D( CURVES( [[ .5, 1, 0], [ .5, b, d ]]), cb,ls): \n O:=[44,72]; fi;\n if (case = 8) then\n A1 := PLOT3D( CURVES( [[ 1, 1, .5] , [ b, b, .5 ]]), cb,ls):\n A2 := PLOT3D( CURVES( [[ 0, 0, .5], [ d, d, .5 ]]), cb,ls): \n O:=[16,82]; fi;\n if (case = 9) then \n A1 := PLOT3D( CURVES( [[ 0, 1, .5], [ d, b, .5 ]]), cb,ls):\n \+ A2 := PLOT3D( CURVES( [[ 1, 0, .5], [ b, d, .5 ]]), cb,ls): \n \+ O:=[30,65]; fi;\n #------------ CORNERS ----------------------- ------\n if (case = 10) then\n A1 := PLOT3D( CURVES( [[ 1, 1, \+ 1], [ b, b, b ]]), cb,ls):\n A2 := PLOT3D( CURVES( [[ 0, 0, 0], [ d, d, d ]]), cb,ls): \n O:=[20,80]; fi;\n if (case = 11) then \n A1 := PLOT3D( CURVES( [[ 1, 0, 1], [ b, d, b ]]), cb,ls):\n \+ A2 := PLOT3D( CURVES( [[ 0, 1, 0], [ d, b, d ]]), cb,ls): \n O :=[18,72]; fi;\n if (case = 12) then\n A1 := PLOT3D( CURVES( [ [ 0, 1, 1], [ d, b, b ]]), cb,ls):\n A2 := PLOT3D( CURVES( [[ 1, \+ 0, 0], [ b, d, d ]]), cb,ls): \n O:=[20,80]; fi;\n \n #------- ------------------------------------------\n\n display(XY0,XY1,XZ0, XZ1,YZ0,YZ1,A1,A2, \n scaling = constrained, orientation = \+ O);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 263 36 "1. Painting Anot her Six Sided Object" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 273 "It will turn out, after a fascinating exploration, tha t coloring a cube involves subtles of the symmetries of a cube. Before we get into all of that, lets look at a simpler figure which also has six faces, but does not \"suffer\" from symmetries which complicate t he matter.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColoredSheet s( 1,2,4,1,3,2);" }}}{PARA 0 "" 0 "" {TEXT -1 232 "\nEach of the six f aces in this case is of a different size, and rotating or flipping the shape will result in a shape which looks the same. This makes for a f airly easy computation of how many ways this shape can be colored.\n\n \n " }{TEXT 264 33 "1. Number of Ways of Painting : " }{TEXT 265 13 "Using 1 Color" }{TEXT -1 67 "\n\nUsing only one color, there is on ly one way to paint the figure.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "1^6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Col oredSheets( 3,3,3,3,3,3);" }}}{PARA 0 "" 0 "" {TEXT -1 5 "\n\n " } {TEXT 266 33 "2. Number of Ways of Painting : " }{TEXT 267 14 "Using \+ 2 Colors" }{TEXT -1 128 "\n\nEach face has two choices for a color. Th ere are six faces, so 2 x 2 x 2 x 2 x 2 x 2 = 2^6. Lets look at severa l of these 64.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2^6;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColoredSheets( 3,3,3,4,4,4 ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColoredSheets( 3,4,3,4 ,3,4 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColoredSheets( 3 ,4,3,4,4,4 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColoredShe ets( 3,4,4,3,4,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Color edSheets( 4,4,3,4,3,4 );" }}}{PARA 0 "" 0 "" {TEXT -1 4 "\n " } {TEXT 268 33 "3. Number of Ways of Painting : " }{TEXT 269 14 "Using \+ 3 Colors" }{TEXT -1 44 "\n\nEach face has three choices for a color. \+ \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "3^6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColoredSheets( 1,1,2,2,3,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColoredSheets( 3,3,1,1,2,2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColoredSheets( 1,2,3,1,2,3 ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColoredSheets( 1,2,1,3 ,1,1);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "\n " }{TEXT 270 33 "4. Number of Ways of Painting : " }{TEXT 271 14 "Using 4 Colors" }{TEXT -1 43 "\n\nEach face has four choices for a color. \n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "4^6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ColoredSheets( 3, 4, 5, 6, 4, 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColoredSheets( 5,5,5,4,6,3);" }}}{PARA 0 "" 0 "" {TEXT -1 4 "\n " }{TEXT 272 33 "5. Number of Ways of Painting : " } {TEXT 273 14 "Using 5 Colors" }{TEXT -1 43 "\n\nEach face has give cho ices for a color. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "5^6; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColoredSheets( 1,2,3,4, 5,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColoredSheets( 4,3 ,1,2,1,5);" }}}{PARA 0 "" 0 "" {TEXT -1 5 "\n\n " }{TEXT 274 33 "6. \+ Number of Ways of Painting : " }{TEXT 275 14 "Using 6 Colors" }{TEXT -1 67 "\n\nUsing only one color, there is only one way to paint the fi gure.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "6^6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ColoredSheets(1, 2, 3, 4, 5, 6);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ColoredSheets(1, 6, 3, 5, \+ 2, 4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 18 "2. Painting a Cube" }}{PARA 0 "" 0 "" {TEXT -1 318 "\nLets now look at a cube. Cubes also have six sides, bu t they are obviously three dimensional shapes rather than planar figur es. To make it easy to see all of the sides in a single glance we can \+ use this \"exploded\" view of cube where the top and bottom faces are \+ pryed open a bit, so we can see all six sides at once.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 6,6,6,6,6,6);" }}} {PARA 0 "" 0 "" {TEXT -1 202 "\nThe faces of a cube can be painted in \+ various ways too. In fact any paint combination used above can also be used on a cube. (You may wish to grab the image below and turn it to \+ look at all six faces.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "C olorCube( 1,2,3,4,5,6 );\n`Grab this cube with the mouse, and turn it \+ to see the other sides`;" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 139 "Here are a few paintings of a cube. Please ins pect them from all angles and compare the exploded view with the live \+ three dimensional viw.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 " ColorCube( 1,2,3, 1,2, 3 );\n`Grab this cube with the mouse, and turn \+ it to see the other sides`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,2,3,1,2,3);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "ColorCube( 3,6,5, 2,3,5 ); \n`Grab this cube with the mouse, and turn it to see the other sides`; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ColorCubeOpen( 3,6,5, 2 ,3,5 );" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 31 "3. \+ Problems with Cube Paintings" }}{PARA 0 "" 0 "" {TEXT -1 205 "\nLets l ook at these three colorings of the shape we were dealing with above. \+ Notice that each of the three colorings has two yellow sides and 4 blu e sides. But the three results are completely different. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColoredSheets( 6,6,2,2,3,3 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColoredSheets( 2,2,3,3,6,6 ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColoredSheets( 3,3,6,6 ,2,2 );" }}}{PARA 0 "" 0 "" {TEXT -1 133 "\n\nNow look at these three \+ similar colorings of a cube. There is something similar about these th ree. Can you explain the commonality?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColorCubeOpen( 6,6,2,2,3,3 );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "ColorCubeOpen( 2,2,3,3,6,6 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColorCubeOpen( 3,3,6,6,2,2 );" }}} {PARA 0 "" 0 "" {TEXT -1 513 "\n The three different colorings of the \+ plane figure were three distinct colorings. However, the three colorin gs of the cube we just perfromed are actually the same coloring - sinc e you can rotate a cube painted the first of the three ways, to get an y of the other two. Another way of looking at it, is if you had three \+ cubes and painted them the three ways we just painted them, and these \+ cubes were thrown into a bottle and then shaken up, then thrown onto a table top, could you discern which cube is which? No." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 297 24 "4. Rotations of the Cube " }}{PARA 0 "" 0 "" {TEXT -1 381 "\nIts helpful if we examine some of \+ the ways that one can rotate a cube. This is important in our current \+ endeavor because two different paintings of the cube may turn out to b e equivalent, after rotations.\n\nEach rotation has an axis of rotatio n. Each of the three dimensional diagrams below can be turned using th e mouse. Try to rotate the cube, keeping the indicated axis fixed.\n\n " }{TEXT 298 38 " I. Axes Through Opposite Faces" }{TEXT -1 1 " \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(3);" }} }{PARA 0 "" 0 "" {TEXT -1 490 "\nWhat kinds of colorings would be fixe d by one or more of these kinds of rotations? \n - If the axis of rota tion passes through two opposite faces, and the four remaining faces a re the same color, then 1, 2, or 3 rotations yield the same coloring. \+ \n- If one of the remaining faces is one color, and the other three an other, then a double rotation of this sort will yield the same result \n- If opposite pairs of the remaining four are the same color, double rotations also fix this coloring.\n\n" }{TEXT 301 41 "\n II. A xes Through Opposite Edges\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ColorCubeRotation(9);" }}}{PARA 0 "" 0 "" {TEXT 300 1 "\n" }{TEXT -1 314 "What kinds of colorings would be fixed by one or more of these kinds of rotations?\n\nOne type of coloring fixed by this sort of rot ation is when the two faces which touch the edge through which the axi s passes are the same color, and the opposite two faces are the same, \+ and then two remaining sides are the same.\n\n" }{TEXT 299 45 "\n \+ III. Axes Through Opposite Vertices\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ColorCubeRotation(10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ColorCubeRotation(11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ColorCubeRotation(12);" }}}{PARA 0 "" 0 "" {TEXT -1 188 "\n \nWhat kinds of colorings would be fixed by one or more of the se kinds of rotations? \n\nIf the three faces which touch each vertex \+ of the axis are the same, then the coloring is the same.\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 278 38 "5. Painting a Cube wit h 1 or 2 Colors" }}{PARA 0 "" 0 "" {TEXT -1 8 "\n \n " }{TEXT 281 33 "1. Number of Ways of Painting : " }{TEXT 282 13 "Using 1 Colo r" }{TEXT -1 67 "\n\nUsing only one color, there is only one way to pa int the figure.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "1^6;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 4,4,4,4,4,4) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,1,1,1 ,1,1);" }}}{PARA 0 "" 0 "" {TEXT -1 5 "\n\n " }{TEXT 279 33 "2. Numb er of Ways of Painting : " }{TEXT 280 14 "Using 2 Colors" }{TEXT -1 255 "\n\nEach face has two choices for a color. At first glance, it mi ght appear that there are 2 x 2 x 2 x 2 x 2 x 2 = 2^6 different colori ngs as we saw before. However, this is not true because various colori ngs are no different from others - after rotations.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2^6;" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n " }{TEXT 294 8 " 1-5 " }{TEXT -1 126 "\n\nFor example, there are si x ways you can paint one side yellow, and five sides blue. All six of \+ the colorings are equivalent." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ColorCubeOpen(4,3,3,3,3,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ColorCubeOpen(3,3,3,4,3,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ColorCubeOpen(3,3,3,3,4,3);" }}}{PARA 0 "" 0 "" {TEXT -1 3 "\n\n\n" }{TEXT 295 8 " 2-4 " }{TEXT -1 271 "\n\nWhen tw o sides are painted one color, and the remaining four sides the other \+ color, a number of colorings which appear to be different are actually the same. For example, if the two green sides are always on opposite \+ faces... How many of this first type are equivalent?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColorCubeOpen( 2,2,1,1,1,1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,1,2,2,1,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColorCubeOpen( 1,1,1,1,2,2 ) ;" }}}{PARA 0 "" 0 "" {TEXT -1 203 "\nThere is another type of colorin g which is not equivalent to those above but which also has two green \+ and and four pink sides - where the two green sides share an edge. How many of these are equivalent?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 2,1,1,2,1,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,1,1,2,1,2);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 296 8 " 3-3 " }{TEXT -1 115 "\n\nWhat abou t three white sides and 3 red sides? How many different colorings are \+ possible which are not equivalent?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ColorCubeOpen(6,5,6,5,6,5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "ColorCubeOpen(5,5,5,6,6,6);" }}}{PARA 0 "" 0 "" {TEXT -1 109 "\nPainting one color on four sides and another color on \+ two was handled above, as was painting five and one.\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 283 33 "6. Painting a Cube with \+ 3 Colors" }}{PARA 0 "" 0 "" {TEXT -1 118 "\nEach face has three choic es for a color but again there are symmetries. How many non-equivalent colors can we find?\n\n" }{TEXT 287 9 " 1-1-4" }{TEXT -1 1 "\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,3,2,3,3,3); " }}}{PARA 0 "" 0 "" {TEXT 288 11 " \n 1-2-3" }{TEXT -1 1 "\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,3,2,2,3,3); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,2,2,3, 3,3);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 289 9 " 2-2-2\n" }} {PARA 0 "" 0 "" {TEXT -1 53 "Each pair of colors can be painted on opp osite faces." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColorCubeOpe n( 1,1,2,2,3,3 );" }}}{PARA 0 "" 0 "" {TEXT -1 59 "\nOr each pair of s ides of the same color can share an edge." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColorCubeOpen( 1,2,1,3,3,2 );" }}}{PARA 0 "" 0 "" {TEXT -1 150 " \nIs there any other way of coloring 2 sides which each of three colors? Is there a way of having only one pair of opposite s ides with the same color?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ColorCubeOpen( 1,2,1,2,3,3 );" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }} }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 284 33 "7. Painting a C ube with 4 Colors" }}{PARA 0 "" 0 "" {TEXT -1 46 "\nHow many non-equi valent colors can we find?\n\n" }{TEXT 290 11 " 1-1-1-3" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 6,4, 5,3,3,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 6,3,5,3,3,4);" }}}{PARA 0 "" 0 "" {TEXT -1 3 "\n\n\n" }{TEXT 291 12 " 1-1-2-2\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOp en( 6,3,4,4,5,5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorC ubeOpen( 6,5,3,4,5,4);" }}}{PARA 0 "" 0 "" {TEXT -1 118 "\n\nEach face has three choices for a color but again there are symmetries. How man y non-equivalent colors can we find?\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 285 33 "8. Painting a Cube with 5 Colors" }} {PARA 0 "" 0 "" {TEXT -1 46 "\nHow many non-equivalent colors can we f ind?\n\n" }{TEXT 292 14 " 1-1-1-1-2\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen(1,2,3,4, 5,5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen(1,5,3,5, 4,2);" }}}{PARA 0 "" 0 " " {TEXT -1 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 277 33 "9. Painting a Cube with \+ 6 Colors" }}{PARA 0 "" 0 "" {TEXT -1 119 "\n\nEach face has three cho ices for a color but again there are symmetries. How many non-equivale nt colors can we find?\n\n" }{TEXT 293 15 " 1-1-1-1-1-1" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,2, 3,4,5,6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ColorCubeOpen( 1,2,4,3,5,6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 286 39 "10. Total Number of Colorings of a Cube" }}{PARA 0 "" 0 "" {TEXT -1 317 "\nIts not clear from the discussion above, but it turns out there are 57 ways of painting a cube with three colors. \+ This is usually shown with a result from more advanced math (abstract \+ algebra) called Burnsides' theorem, which allows you to count the pain tings based on the transformations that fix certain paintings." }} {PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{PARA 0 "" 0 "" {TEXT 259 36 "\n \+ \251 2002 Waterloo Maple Inc " }}}{MARK "0 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }