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{SECT 0 {PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 261 
36 "This Maple file shows symmetries of " }}{PARA 265 "" 0 "" {TEXT 
260 25 "the prism and the pyramid" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT 256 45 "First load the plots and  plotto
ols packages:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(plots)
:with(plottools):" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 58 "`tetra` i
s a tetrahedron and `slice1` is a symmetry plane:" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 270 "a:=sqrt(3):\ntetra:=polygonplot3d([\n[[1/2,0
,-a/2],[-1/4,a/4,-a/2],[-1/4,-a/4,-a/2]],\n[[1/2,0,-a/2],[-1/4,a/4,-a/
2],[0,0,0]],\n[[1/2,0,-a/2],[-1/4,-a/4,-a/2],[0,0,0]],\n[[-1/4,a/4,-a/
2],[-1/4,-a/4,-a/2],[0,0,0]]],\nscaling=constrained,thickness=3,\nstyl
e=wireframe,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 
"slice1:=polygonplot3d(\n[[0,0,0],[1/2,0,-a/2],[-1/4,0,-a/2]],\nscalin
g=constrained,thickness=2,style=patch,\ncolor=aquamarine):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "display(\{tetra,slice1\},\norienta
tion=[-161,76],\ntitlefont=[COURIER,BOLD,18],\ntitle=`Symmetry plane o
f a tetrahedron`);" }}}}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "
" 0 "" {TEXT -1 119 "Conclusion: Since each permutation is the product
 of transpositions, the full symmerty group ot the tetrahedron is S_4.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 
208 "We now define more slices. (Because of polygonal intersection pro
blem, we split the new slice into two halves.) Composition of reflecti
ons to two symmetry planes gives symmetry rotatations of the tetrahedr
on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "rslice2:=polygonplot
3d([\n[-1/4,0,-a/2],[-1/4,-a/4,-a/2],[1/4,0,-a/4]],\nscaling=constrain
ed,thickness=2,\nstyle=patch,color=cyan):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 127 "lslice2:=polygonplot3d([\n[-1/4,0,-a/2],[-1/4,a/4,
-a/2],[1/4,0,-a/4]],\nscaling=constrained,thickness=2,\nstyle=patch,co
lor=cyan):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "display3d(\{
tetra3,slice1,lslice2,rslice2\},\norientation=[-129,60],\ntitlefont=[C
OURIER,BOLD,18],\ntitle=`Composition of Reflections I`);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "lslice3:=polygonplot3d([\n[-1/6,0,
-1/a],[-1/8,-a/8,-a/4],[1/2,0,-a/2]],\nscaling=constrained,thickness=2
,\nstyle=patch,color=cyan):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
127 "rslice3:=polygonplot3d([\n[-1/6,0,-1/a],[-1/4,a/4,-a/2],[1/2,0,-a
/2]],\nscaling=constrained,thickness=2,\nstyle=patch,color=cyan):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "display3d(\{tetra,slice1,ls
lice3,rslice3\},\norientation=[145,60],\ntitlefont=[COURIER,BOLD,18],
\ntitle=`Composition of Reflections II`);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 105 "display3d(\{tetra,slice1,lslice3,rslice3\},\norien
tation=[-13,120],\ntitle=`Composition of Reflections III`);" }}}{SECT 
0 {PARA 4 "" 0 "" {TEXT 259 32 "We now insert the symmetry axes:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "symmax1:=polygonplot3d([\n[-
1/4,0,-a/2],[1/4,0,-a/4]],\nscaling=constrained,thickness=5,color=blue
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "display3d(\{tetra,sy
mmax1\},\norientation=[-162,74],\ntitlefont=[COURIER,BOLD,18],\ntitle=
`Half turn around the axis`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 100 "symmax2:=polygonplot3d([\n[1/2,0,-a/2],[-1/6,0,-1/a]],\nscaling
=constrained,thickness=5,color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 123 "display3d(\{tetra,symmax2\},\norientation=[133,69],
\ntitlefont=[COURIER,BOLD,18],\ntitle=`120 degree rotation around the \+
axis`);" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 103 "We now make an ani
mation  showing a half-turn around a horizontal symmetry axis  in a pe
ntagonal prism." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p:=(i,j)-
>array([cos(2*Pi*i/5),sin(2*Pi*i/5),j]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 221 "pentaprism:=display(\{\npolygonplot3d([seq(p(i,1),i=
1..5)],\nstyle=wireframe),\npolygonplot3d([seq(p(i,-1),i=1..5)],\nstyl
e=wireframe),\nseq(polygonplot3d([p(i,1),p(i,-1)]),i=1..5)\n\},\nscali
ng=constrained,color=red,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 126 "display([\nseq(rotate(pentaprism,k*Pi/10,0,0),k=0..1
0)],\ninsequence=true,\norientation=[-42,74], axes=normal,tickmarks=[0
,0,0]);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 124 "The following anima
tion shows that half turn around a horizontal symmetry axis of the bas
e is NOT a symmetry of the pyramid." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 171 "pentapyr:=display(\{\npolygonplot3d([seq(p(i,0),i=1.
.5)],\nstyle=wireframe),\nseq(polygonplot3d([p(i,0),array([0,0,1])]),
\ni=1..5)\},\nscaling=constrained,color=red,thickness=3):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "display([\nseq(rotate(pentapyr,k*P
i/10,0,0),k=0..10)],\ninsequence=true,\norientation=[-42,74], axes=nor
mal,tickmarks=[0,0,0]);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 146 "The
 following animation shows that the symmery rotations of the base (wit
h vertical axis) extend to symmetries of both the pyramid and the pris
m. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "display([\nseq(rotat
e(pentaprism,0,0,-k*Pi/25),k=0..10)],\ninsequence=true,\norientation=[
-42,74],\naxes=normal,tickmarks=[0,0,0]);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 125 "display([\nseq(rotate(pentapyr,0,0,-k*Pi/25),k=0..
10)],\ninsequence=true,\norientation=[-30,74],\naxes=normal,tickmarks=
[0,0,0]);" }}}{PARA 268 "" 0 "" {TEXT -1 124 "Conclusions: 1. The posi
tive symmery group of the pyramid is cyclic. 2.The positive symmery gr
oup of the prism is dihedral. " }}}}{MARK "12" 0 }{VIEWOPTS 1 1 0 3 2 
1804 }