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{SECT 0 {PARA 215 "" 0 "" {TEXT 216 21 "MAPLE WORKSHEET #4: \n" }
{TEXT 216 48 "Pyramids, Prisms, Antiprisms and the Icosahedron" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT 220 37 "Load the plots and plottools pac
kages" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(p
lottools):" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 220 33 "Rotation matrix \+
around the z-axis" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Rz := t
heta->Matrix([ \n" }{MPLTEXT 1 0 28 "[cos(theta),-sin(theta),0],\n" }
{MPLTEXT 1 0 27 "[sin(theta),cos(theta),0],\n" }{MPLTEXT 1 0 9 "[0,0,1
] \n" }{MPLTEXT 1 0 3 "] \n" }{MPLTEXT 1 0 2 "):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "Rz(theta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "Rz(theta).Vector([a,b,c]):" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 220 
43 "Triangle in space with given vertex-vectors" }}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 34 "tri := (x1,x2,x3)->polygonplot3d(\n" }{MPLTEXT 
1 0 41 "map( (x->convert(x,list)), [x1,x2,x3] ),\n" }{MPLTEXT 1 0 14 "
thickness=2 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "display3d
(\n" }{MPLTEXT 1 0 30 "tri([1,0,0],[0,1,0],[0,0,1]),\n" }{MPLTEXT 1 0 
52 "orientation=[-34,63],axes=normal,tickmarks=[0,0,0],\n" }{MPLTEXT 
1 0 20 "lightmodel=light1);\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 220 
93 "We now build a pentagonal pyramid with equilateral triangular face
s (side length=2 sin(Pi/5))" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 221 95 "h
 is the height of the pyramid; each side is obtained by rotating an in
itial side by Rz(theta):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "
h:=sqrt((2*sin(Pi/5))^2-1):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 19 "sides := display3d(" }{MPLTEXT 1 0 3 "\{ \n" }
{MPLTEXT 1 0 10 "seq( tri(\n" }{MPLTEXT 1 0 33 "Rz(2*Pi/5)^(k).Vector(
[1,0,0]), \n" }{MPLTEXT 1 0 42 "Rz(2*Pi/5)^(k+1).Vector([1,0,0]), \nVe
ctor(" }{MPLTEXT 1 0 9 "[0,0,h])\n" }{MPLTEXT 1 0 12 "), k=1..5) \n" }
{MPLTEXT 1 0 4 "\}): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "di
splay3d(sides,\n" }{MPLTEXT 1 0 34 "style=patch, orientation=[61,43],
\n" }{MPLTEXT 1 0 21 "scaling=constrained);" }}}}{SECT 1 {PARA 4 "" 0 
"" {TEXT 221 69 "Another way to do this is to use `rotate' from the pl
ottools package:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rotsides
 := display3d(\{ \n" }{MPLTEXT 1 0 6 "seq( \n" }{MPLTEXT 1 0 9 "rotate
( \n" }{MPLTEXT 1 0 50 "tri([1,0,0],[cos(2*Pi/5),sin(2*Pi/5),0],[0,0,h
]),\n" }{MPLTEXT 1 0 16 "0,0,2*k*Pi/5), \n" }{MPLTEXT 1 0 9 "k=1..5) \+
\n" }{MPLTEXT 1 0 3 "\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "display3d(rotsides,\n" }{MPLTEXT 1 0 34 "style=patch, orientation=
[61,43],\n" }{MPLTEXT 1 0 21 "scaling=constrained);" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT 221 28 "We add the base and combine:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "base:=polygonplot3d(\n[" }{MPLTEXT 
1 0 63 "seq(\nconvert(Rz(2*Pi/5)^(k).Vector([1,0,0]),list), \nk=1..5)]
 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display3d(\{sides,ba
se\},\n" }{MPLTEXT 1 0 34 "style=patch, orientation=[61,43],\n" }
{MPLTEXT 1 0 22 "scaling=constrained, \n" }{MPLTEXT 1 0 56 "titlefont=
[TIMES,ROMAN,22], title=`Pentagonal Pyramid`);" }}}}}{SECT 1 {PARA 3 "
" 0 "" {TEXT 220 35 "Create a  double pentagonal pyramid" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sides1 := display3d(" }{MPLTEXT 1 
0 3 "\{ \n" }{MPLTEXT 1 0 11 "seq( tri( \n" }{MPLTEXT 1 0 33 "Rz(2*Pi/
5)^(k).Vector([1,0,0]), \n" }{MPLTEXT 1 0 42 "Rz(2*Pi/5)^(k+1).Vector(
[1,0,0]), \nVector(" }{MPLTEXT 1 0 10 "[0,0,-h])\n" }{MPLTEXT 1 0 12 "
), k=1..5) \n" }{MPLTEXT 1 0 3 "\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "display3d(\{sides, sides1\},\n" }{MPLTEXT 1 0 55 "sty
le=patch, orientation=[57,75], scaling=constrained,\n" }{MPLTEXT 1 0 
63 "titlefont=[TIMES,ROMAN,22], title=`Double Pentagonal Pyramid`);" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 220 29 "Create a pentagonal antiprism
" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 221 36 "The top and bottom pentagona
l faces:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "height:=simplify
(3*(sin(Pi/5))^2-(1-cos(Pi/5))^2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "bottom := polygonplot3d(\n" }{MPLTEXT 1 0 69 "[seq( \+
\nconvert(Rz(2*Pi/5)^(k).Vector([1, 0, -1/2]),list), \nk=1..5)]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "top := polygonplot3d(\n" }
{MPLTEXT 1 0 77 "[seq( \nconvert(Rz(Pi/5).Rz(2*Pi/5)^(k).Vector([1, 0,
 1/2]),list), \nk=1..5)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "display3d(\{bottom, top\}, scaling=constrained,\n" }{MPLTEXT 1 0 
33 "style=patch, orientation=[90,0]);" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT 221 54 "Using translate and rotate from the plottools package:" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "newtop := \n" }{MPLTEXT 1 
0 11 "translate(\n" }{MPLTEXT 1 0 33 "rotate(bottom,0,0,Pi/5), 0,0,1):
\n" }{MPLTEXT 1 0 28 "display3d(\{bottom, newtop\},\n" }{MPLTEXT 1 0 
53 "style=patch, orientation=[90,0],scaling=constrained);" }}}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT 221 28 "Insert the upward triangles:" }}{SECT 
1 {PARA 5 "" 0 "" {TEXT 222 26 "First the initial triange:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "faceup := tri(\n" }{MPLTEXT 1 0 44 
"[1, 0, -1/2], Rz(Pi/5).Vector([1, 0, 1/2]),\n" }{MPLTEXT 1 0 34 "Rz(2
*Pi/5).Vector([1, 0, -1/2]) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 54 "display3d(\{bottom, top, faceup\}, scaling=constrained,\n" }
{MPLTEXT 1 0 34 "style=patch, orientation=[37,75]);" }}}}{SECT 1 
{PARA 5 "" 0 "" {TEXT 222 26 "Then the rest by rotation:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "faceups := k->tri( \n" }{MPLTEXT 1 
0 32 "op(map((x->Rz(2*Pi/5).x)@@ k,  \n" }{MPLTEXT 1 0 24 "[Vector([1,
 0, -1/2]), \n" }{MPLTEXT 1 0 30 "Rz(Pi/5).Vector([1, 0,1/2]), \n" }
{MPLTEXT 1 0 35 "Rz(2*Pi/5).Vector([1, 0, -1/2])]  \n" }{MPLTEXT 1 0 
6 ")) \n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display3d(\{b
ottom, top, seq(faceups(k), k=1..5)\},\n" }{MPLTEXT 1 0 54 "style=patc
h,scaling=constrained, orientation=[37,75]);" }}}}{SECT 1 {PARA 5 "" 0
 "" {TEXT 222 34 "Using the plottools package again:" }}{EXCHG {PARA 0
 "> " 0 "" {MPLTEXT 1 0 46 "newfaceups := k-> rotate(faceup,0,0,2*k*Pi
/5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display3d(\{bottom,
 top, seq(newfaceups(k), k=1..5)\},\n" }{MPLTEXT 1 0 54 "style=patch,s
caling=constrained, orientation=[37,75]);" }}}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT 221 30 "Insert the downward triangles:" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "facedown := tri(\n" }{MPLTEXT 1 0 31 "Rz(Pi/5).V
ector([1, 0, 1/2]), \n" }{MPLTEXT 1 0 33 "Rz(2*Pi/5).Vector([1, 0, -1/
2]),\n" }{MPLTEXT 1 0 33 "Rz(3*Pi/5).Vector([1, 0, 1/2]) ):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display3d(\{bottom, top, fac
eup, facedown\},\n" }{MPLTEXT 1 0 55 "style=patch, scaling=constrained
, orientation=[37,75]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
facedowns := k->tri(\n" }{MPLTEXT 1 0 29 "op(map((x->Rz(2*Pi/5).x)@@ k
," }{MPLTEXT 1 0 32 "\n[Rz(Pi/5).Vector([1, 0, 1/2]),\n" }{MPLTEXT 1 
0 33 "Rz(2*Pi/5).Vector([1, 0, -1/2]),\n" }{MPLTEXT 1 0 33 "Rz(3*Pi/5)
.Vector([1, 0, 1/2])] \n" }{MPLTEXT 1 0 5 "))\n):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 50 "newfacedowns := k-> rotate(facedown,0,0,2*k*
Pi/5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "display3d(\{botto
m, top, seq(newfacedowns(k), k=1..5), seq(faceups(k), k=1..5)\},\n" }
{MPLTEXT 1 0 55 "style=patch, scaling=constrained, orientation=[37,75]
,\n" }{MPLTEXT 1 0 58 "titlefont=[TIMES,ROMAN,22], title=`Pentagonal A
ntiprism`);" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 220 64 "Put pentagonal
 prisms on the top and bottom faces to obtain the\n" }{TEXT 220 11 "ic
osahedron" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "h:=sqrt((2*sin(
Pi/5))^2-1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "cap1:=displ
ay3d(\n" }{MPLTEXT 1 0 10 "\{seq(tri(\n" }{MPLTEXT 1 0 45 "Rz(Pi/5).Rz
(2*Pi/5)^(k).Vector([1, 0, 1/2]),\n" }{MPLTEXT 1 0 55 "Rz(Pi/5).Rz(2*P
i/5)^(k+1).Vector([1, 0, 1/2]), \nVector(" }{MPLTEXT 1 0 17 "[0, 0, h+
1/2])),\n" }{MPLTEXT 1 0 12 "k=1..5) \} ):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "cap2:=display3d(\n" }{MPLTEXT 1 0 10 "\{seq(tri(\n"
 }{MPLTEXT 1 0 37 "Rz(2*Pi/5)^(k).Vector([1, 0, -1/2]),\n" }{MPLTEXT 
1 0 47 "Rz(2*Pi/5)^(k+1).Vector([1, 0, -1/2]), \nVector(" }{MPLTEXT 1 
0 18 "[0, 0, -h-1/2])),\n" }{MPLTEXT 1 0 12 "k=1..5) \} ):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display3d(\{cap1, cap2\},\n" }
{MPLTEXT 1 0 55 "style=patch, scaling=constrained, orientation=[37,68]
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "display3d(\{ seq(face
downs(k), k=1..5), seq(faceups(k), k=1..5), cap1, cap2\},\n" }{MPLTEXT
 1 0 55 "style=patch, scaling=constrained, orientation=[37,68],\n" }
{MPLTEXT 1 0 49 "titlefont=[TIMES,ROMAN,22], title=`Icosahedron`,\n" }
{MPLTEXT 1 0 17 "transparency=.5);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
220 42 "A tetrahedron in space with given vertices" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 25 "tet := (x1,x2,x3,x4)->\{ \n" }{MPLTEXT 1 0 
6 "seq( \n" }{MPLTEXT 1 0 6 "tri( \n" }{MPLTEXT 1 0 35 "op(subsop(k=NU
LL, [x1,x2,x3,x4])) \n" }{MPLTEXT 1 0 4 "), \n" }{MPLTEXT 1 0 9 "k=1..
4) \n" }{MPLTEXT 1 0 2 "\}:" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 220 39 
"Identity and half-turns around the axes" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "H(0) := Matrix(3,shape=identity):\n" }{MPLTEXT 1 0 
47 "H(1) := Matrix([[1,0,0], [0,-1,0], [0,0,-1]]):\n" }{MPLTEXT 1 0 
47 "H(2) := Matrix([[-1,0,0], [0,1,0], [0,0,-1]]):\n" }{MPLTEXT 1 0 
46 "H(3) := Matrix([[-1,0,0], [0,-1,0], [0,0,1]]):" }}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT 220 31 "Vertices of regular tetrahedron" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "a:=k->H(k).Vector([1,1,1]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display3d(tet( seq(a(k), k=0
..3)),\n" }{MPLTEXT 1 0 44 "style=patch, axes=boxed, tickmarks=[0,0,0]
);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 220 69 "Cutting the tetrahedron \+
into non-mirror symmetric congruent polyhedra" }}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 43 "transtet:=display3d(tet(seq(a(k),k=0..3)),\n" }
{MPLTEXT 1 0 26 "style=patch,transparency=1" }{MPLTEXT 1 0 2 "):" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "mirror:
=display3d(\{\n" }{MPLTEXT 1 0 51 "polygonplot3d([[1,0,0],[0,1,0],[-1,
0,0],[0,-1,0]],\n" }{MPLTEXT 1 0 28 "transparency=0.5,color=red)\n" }
{MPLTEXT 1 0 3 "\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "dis
play(\{transtet,mirror\},orientation=[1,79],\n" }{MPLTEXT 1 0 29 "axes
=boxed,tickmarks=[0,0,0]," }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 36 "view
=[-1.5..1.5,-1.5..1.5,-1.5..1.5]" }{MPLTEXT 1 0 2 ");" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "trtet:=display3d(\{\n" }{MPLTEXT 1 
0 42 "polygonplot3d([[1,0,0],[0,1,0],[1,1,1]]),\n" }{MPLTEXT 1 0 46 "p
olygonplot3d([[-1,0,0],[0,-1,0],[-1,-1,1]]),\n" }{MPLTEXT 1 0 53 "poly
gonplot3d([[1,1,1],[1,0,0],[0,-1,0],[-1,-1,1]]),\n" }{MPLTEXT 1 0 16 "
polygonplot3d([[" }{MPLTEXT 1 0 35 "1,1,1],[0,1,0],[-1,0,0],[-1,-1,1]]
)" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 20 "\},transparency=0.5):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(\{transtet,mirror,tr
tet\},orientation=[85,77],\n" }{MPLTEXT 1 0 29 "axes=boxed,tickmarks=[
0,0,0]," }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 36 "view=[-1.5..1.5,-1.5..
1.5,-1.5..1.5]" }{MPLTEXT 1 0 2 ");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "display3d(trtet,scaling=constrained,orientation=[-73,
79],\n" }{MPLTEXT 1 0 29 "axes=boxed,tickmarks=[0,0,0]," }{MPLTEXT 1 
0 1 "\n" }{MPLTEXT 1 0 36 "view=[-1.5..1.5,-1.5..1.5,-1.5..1.5]" }
{MPLTEXT 1 0 2 ");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "displ
ay3d(\{trtet,\n" }{MPLTEXT 1 0 24 "rotate(trtet,Pi,0,Pi/2)\n" }
{MPLTEXT 1 0 3 "\},\n" }{MPLTEXT 1 0 42 "scaling=constrained,orientati
on=[-73,79],\n" }{MPLTEXT 1 0 29 "axes=boxed,tickmarks=[0,0,0]," }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 36 "view=[-1.5..1.5,-1.5..1.5,-1.5..
1.5]" }{MPLTEXT 1 0 2 ");" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
 216 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 220 10 "Wireframes" }}
{SECT 0 {PARA 4 "" 0 "" {TEXT 221 35 "Line in space with given endpoin
ts:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "lin:=(x1,x2)->polygon
plot3d(\n" }{MPLTEXT 1 0 40 "[convert(x1, list), convert(x2, list)],\n
" }{MPLTEXT 1 0 25 "style=line, thickness=3):" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT 221 51 "The plottools package contains a routine for lines:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "with(plottools):\n" }
{MPLTEXT 1 0 32 "display(line([0,0,0], [3,4,5], \n" }{MPLTEXT 1 0 39 "
thickness=3, color=red, linestyle=3), \n" }{MPLTEXT 1 0 31 "axes=norma
l,tickmarks=[3,3,3],\n" }{MPLTEXT 1 0 21 "orientation=[20,30]);" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT 221 22 "Wireframe tetrahedron:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "wiretet:=(x1,x2,x3,x4)->\n" 
}{MPLTEXT 1 0 81 "\{ lin(x1, x2), lin(x1, x3), lin(x1, x4), lin(x2, x3
), lin(x2, x4), lin(x3, x4) \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 40 "display3d(wiretet( seq(a(k), k=0..3) ),\n" }{MPLTEXT 1 0 31 "a
xes=boxed, tickmarks=[0,0,0]);" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
220 16 "Pentagonal prism" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 221 46 "A pa
ralleogram in space with 3 given vertices:" }}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 34 "par := (x0,x1,x2)->polygonplot3d(\n" }{MPLTEXT 1 0 
27 "map((x->convert(x,list)), \n" }{MPLTEXT 1 0 30 "[x1,x0,x2,evalm(x1
+x2-x0)] ),\n" }{MPLTEXT 1 0 13 "thickness=2):" }}}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 46 "display3d(par([0,0,0], [1,1/2,0], [0,1/2,1]),\n
" }{MPLTEXT 1 0 26 "style=patch, axes=normal,\n" }{MPLTEXT 1 0 40 "tic
kmarks=[0,0,0], orientation=[18,66]);" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT 221 41 "Creating the faces of a pentagonal prism:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ptop := polygonplot3d(\n" }{MPLTEXT
 1 0 27 "[ seq( Rz(2*Pi/5)^k.Vector(" }{MPLTEXT 1 0 22 "[1,0,1]), k=1.
.5) ] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pbottom := poly
gonplot3d(\n" }{MPLTEXT 1 0 52 "[ seq( Rz(2*Pi/5)^(k).Vector([1,0,-1])
, k=1..5) ] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "pside := \+
par([1,0,-1], evalm(Rz(2*Pi/5)&*[1,0,-1]), [1,0,1]):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "display3d(\{pbottom, ptop, pside\},\n" }
{MPLTEXT 1 0 13 "style=patch);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "pfaces := k->par(\n" }{MPLTEXT 1 0 42 "op( map( (x->evalm(Rz(2
*Pi/5)&*x)) @@ k, \n" }{MPLTEXT 1 0 49 "[ [1,0,-1], Rz(2*Pi/5)&*[1,0,-
1], [1,0,1] ] )) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "disp
lay3d(\{pbottom, ptop, seq(pfaces(k), k=1..5)\},\n" }{MPLTEXT 1 0 34 "
style=patch, scaling=constrained,\n" }{MPLTEXT 1 0 54 "titlefont=[TIME
S,ROMAN,22], title=`Pentagonal Prism`);" }}}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT 220 47 "A parallelepiped in space with 4 given vertices" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "parpip := (x0,x1,x2,x3)->\n"
 }{MPLTEXT 1 0 54 "\{ par(x0, x1, x2), par(x0, x2, x3), par(x0, x1, x3
), \n" }{MPLTEXT 1 0 43 "par(x1, evalm(x1+x2-x0), evalm(x1+x3-x0)),\n"
 }{MPLTEXT 1 0 43 "par(x2, evalm(x1+x2-x0), evalm(x2+x3-x0)),\n" }
{MPLTEXT 1 0 44 "par(x3, evalm(x1+x3-x0), evalm(x2+x3-x0)) \}:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display3d(parpip([0,0,0],[1,
0,0],[1/2,1/2,1],[0,1,0]),\n" }{MPLTEXT 1 0 14 "style=patch, \n" }
{MPLTEXT 1 0 52 "axes=boxed, tickmarks=[0,0,0], orientation=[70,69]);"
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "colpar := (x0,x1,x2,c1,c
2,c3)->polygonplot3d(\n" }{MPLTEXT 1 0 28 "map( (x->convert(x,list)), \+
\n" }{MPLTEXT 1 0 30 "[x1,x0,x2,evalm(x1+x2-x0)] ),\n" }{MPLTEXT 1 0 
40 "thickness=2, color=COLOR(RGB,c1,c2,c3)):" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 38 "colparpip := (x0,x1,x2,x3,c1,c2,c3)->\n" }
{MPLTEXT 1 0 32 "\{ colpar(x0, x1, x2,c1,c2,c3), \n" }{MPLTEXT 1 0 30 
"colpar(x0, x2, x3,c1,c2,c3), \n" }{MPLTEXT 1 0 30 "colpar(x0, x1, x3,
c1,c2,c3), \n" }{MPLTEXT 1 0 55 "colpar(x1, evalm(x1+x2-x0), evalm(x1+
x3-x0),c1,c2,c3),\n" }{MPLTEXT 1 0 55 "colpar(x2, evalm(x1+x2-x0), eva
lm(x2+x3-x0),c1,c2,c3),\n" }{MPLTEXT 1 0 56 "colpar(x3, evalm(x1+x3-x0
), evalm(x2+x3-x0),c1,c2,c3) \}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 63 "display3d(colparpip([1,1,1],[-1,1,1],[1,-1,1],[1,1,-1],1,1,0)
,\n" }{MPLTEXT 1 0 14 "style=patch, \n" }{MPLTEXT 1 0 52 "axes=boxed, \+
tickmarks=[0,0,0], orientation=[70,69]);" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT 220 22 "Steelframe tetrahedron" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "with(plots):steeltri:=" }{MPLTEXT 1 0 23 "(x1,x2,x3,s
)->display(\{" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 13 "polygonplot3d" }
{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 27 "map((x->convert(x,list)), \n" }
{MPLTEXT 1 0 1 "[" }{MPLTEXT 1 0 6 "x1,x2," }{MPLTEXT 1 0 26 "(1-s).x2
+s.(x1+x2+x3)/3, \n" }{MPLTEXT 1 0 7 "       " }{MPLTEXT 1 0 23 "(1-s)
.x1+s.(x1+x2+x3)/3" }{MPLTEXT 1 0 2 "]\n" }{MPLTEXT 1 0 4 ")),\n" }
{MPLTEXT 1 0 41 "polygonplot3d(map((x->convert(x,list)), \n" }{MPLTEXT
 1 0 7 "[x2,x3," }{MPLTEXT 1 0 26 "(1-s).x3+s.(x1+x2+x3)/3, \n" }
{MPLTEXT 1 0 7 "       " }{MPLTEXT 1 0 25 "(1-s).x2+s.(x1+x2+x3)/3]\n"
 }{MPLTEXT 1 0 4 ")),\n" }{MPLTEXT 1 0 41 "polygonplot3d(map((x->conve
rt(x,list)), \n" }{MPLTEXT 1 0 7 "[x3,x1," }{MPLTEXT 1 0 26 "(1-s).x1+
s.(x1+x2+x3)/3, \n" }{MPLTEXT 1 0 7 "       " }{MPLTEXT 1 0 25 "(1-s).
x3+s.(x1+x2+x3)/3]\n" }{MPLTEXT 1 0 2 "))" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 2 "\})" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "steeltri([1,0,0],[0,1,0],[0,0,1],.2
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "steeltet := (x1,x2,x3
,x4,s)->\{ \n" }{MPLTEXT 1 0 6 "seq( \n" }{MPLTEXT 1 0 5 "steel" }
{MPLTEXT 1 0 6 "tri( \n" }{MPLTEXT 1 0 37 "op(subsop(k=NULL, [x1,x2,x3
,x4])),s \n" }{MPLTEXT 1 0 4 "), \n" }{MPLTEXT 1 0 9 "k=1..4) \n" }
{MPLTEXT 1 0 2 "\}:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "display(steeltet(" }{MPLTEXT 1 0 22 "seq(a(k),k=0..3)
,.4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "movesteeltet:=t->
\n" }{MPLTEXT 1 0 38 "display(steeltet(seq(a(k),k=0..3),t)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "movesteeltet(.5);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display([seq(movesteeltet(k/
20),k=0..20)],\n" }{MPLTEXT 1 0 17 "insequence=true);" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1
 2 33 1 1 }