#Andrew Sommese 2005 restart;with(geom3d):with(plots):
Warning, the assigned name polar now has a global binding Warning, the name changecoords has been redefined
point(A,cos(0),sin(0),0),point(B,cos(2*Pi/3),sin(2*Pi/3),0),point(C,cos(4*Pi/3),sin(4*Pi/3),0): midpoint(mAB,A,B);midpoint(mBC,B,C);midpoint(mCA,C,A); NiNJJG1BQkc9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJLWdlb20zZDotZm9ybUc2JEYnL0krbW9kdWxlbmFtZUdGJUknZ2VvbTNkRzYkRidJKF9zeXNsaWJHRiVJKHBvaW50M2RHRiVJJ2Nvb3Jkc0dGJTclIyIiIiIiJSwkKiQiIiQjRjQiIiNGMyIiIQ== NiNJJG1CQ0c9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJLWdlb20zZDotZm9ybUc2JEYnL0krbW9kdWxlbmFtZUdGJUknZ2VvbTNkRzYkRidJKF9zeXNsaWJHRiVJKHBvaW50M2RHRiVJJ2Nvb3Jkc0dGJTclIyEiIiIiIyIiIUY2 NiNJJG1DQUc9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJLWdlb20zZDotZm9ybUc2JEYnL0krbW9kdWxlbmFtZUdGJUknZ2VvbTNkRzYkRidJKF9zeXNsaWJHRiVJKHBvaW50M2RHRiVJJ2Nvb3Jkc0dGJTclIyIiIiIiJSwkKiQiIiQjRjQiIiMjISIiRjUiIiE= triangle(T1,[A,B,C]); NiNJI1QxRz02IkkmZmFsc2VHSSpwcm90ZWN0ZWRHRidFXFtsJUkmZ2l2ZW5HRiU3JUkiQUc9RiVGJkVcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSIiIiIiIUY5SSJCRz1GJUYmRVxbbCNGLkY1RjY3JSMhIiIiIiMsJCokIiIkI0Y4RkBGREY5SSJDRz1GJUYmRVxbbCNGLkY1RjY3JUY+LCRGQkY+RjlGLkkrdHJpYW5nbGUzZEdGJUkxZ2VvbTNkOi12ZXJ0aWNlc0dGLzclRjdGPUZISSdtZXRob2RHRiVJJ3BvaW50c0dGJQ== form(T1); NiNJK3RyaWFuZ2xlM2RHNiI= map(coordinates,DefinedAs(T1)); NiM3JTclIiIiIiIhRiY3JSMhIiIiIiMsJCokIiIkI0YlRipGLkYmNyVGKCwkRixGKEYm point(c,cos(Pi/3),sin(Pi/3),sqrt(2));point(a,cos(Pi),sin(Pi),sqrt(2));point(b,cos(5*Pi/3),sin(5*Pi/3),sqrt(2)); NiNJImNHPTYiSSZmYWxzZUdJKnByb3RlY3RlZEdGJ0VcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSMiIiIiIiMsJCokIiIkRjNGMyokRjVGMw== NiNJImFHPTYiSSZmYWxzZUdJKnByb3RlY3RlZEdGJ0VcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSEiIiIiISokIiIjIyIiIkY2 NiNJImJHPTYiSSZmYWxzZUdJKnByb3RlY3RlZEdGJ0VcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSMiIiIiIiMsJCokIiIkRjMjISIiRjUqJEY1RjM= midpoint(mab,a,b);midpoint(mbc,b,c);midpoint(mca,c,a); NiNJJG1hYkc9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJLWdlb20zZDotZm9ybUc2JEYnL0krbW9kdWxlbmFtZUdGJUknZ2VvbTNkRzYkRidJKF9zeXNsaWJHRiVJKHBvaW50M2RHRiVJJ2Nvb3Jkc0dGJTclIyEiIiIiJSwkKiQiIiQjIiIiIiIjRjMqJEY7Rjk= NiNJJG1iY0c9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJLWdlb20zZDotZm9ybUc2JEYnL0krbW9kdWxlbmFtZUdGJUknZ2VvbTNkRzYkRidJKF9zeXNsaWJHRiVJKHBvaW50M2RHRiVJJ2Nvb3Jkc0dGJTclIyIiIiIiIyIiISokRjVGMw== NiNJJG1jYUc9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJLWdlb20zZDotZm9ybUc2JEYnL0krbW9kdWxlbmFtZUdGJUknZ2VvbTNkRzYkRidJKF9zeXNsaWJHRiVJKHBvaW50M2RHRiVJJ2Nvb3Jkc0dGJTclIyEiIiIiJSwkKiQiIiQjIiIiIiIjI0Y6RjUqJEY7Rjk= triangle(T2,[a,b,c]); NiNJI1QyRz02IkkmZmFsc2VHSSpwcm90ZWN0ZWRHRidFXFtsJUkmZ2l2ZW5HRiU3JUkiYUc9RiVGJkVcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSEiIiIiISokIiIjIyIiIkY7SSJiRz1GJUYmRVxbbCNGLkY1RjY3JUY8LCQqJCIiJEY8I0Y4RjtGOkkiY0c9RiVGJkVcW2wjRi5GNUY2NyVGPCwkRkNGPEY6Ri5JK3RyaWFuZ2xlM2RHRiVJMWdlb20zZDotdmVydGljZXNHRi83JUY3RkFGSUknbWV0aG9kR0YlSSdwb2ludHNHRiU= form(T2); NiNJK3RyaWFuZ2xlM2RHNiI= map(coordinates,DefinedAs(T2)); NiM3JTclISIiIiIhKiQiIiMjIiIiRig3JUYpLCQqJCIiJEYpI0YlRihGJzclRiksJEYtRilGJw== segment(aBC,mBC,a);segment(bCA,mCA,b);segment(cAB,mAB,c);segment(Abc,mbc,A);segment(Bca,mca,B);segment(Cab,mab,C); NiNJJGFCQ0c9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJJmdpdmVuR0YlNyRJJG1CQ0c9RiVGJkVcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSMhIiIiIiMiIiFGO0kiYUc9RiVGJkVcW2wjRi5GNUY2NyVGOUY7KiRGOiMiIiJGOkYuSSpzZWdtZW50M2RHRiU= NiNJJGJDQUc9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJJmdpdmVuR0YlNyRJJG1DQUc9RiVGJkVcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSMiIiIiIiUsJCokIiIkI0Y5IiIjIyEiIkY6IiIhSSJiRz1GJUYmRVxbbCNGLkY1RjY3JUY+LCRGPCNGQUY/KiRGP0Y+Ri5JKnNlZ21lbnQzZEdGJQ== NiNJJGNBQkc9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJJmdpdmVuR0YlNyRJJG1BQkc9RiVGJkVcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSMiIiIiIiUsJCokIiIkI0Y5IiIjRjgiIiFJImNHPUYlRiZFXFtsI0YuRjVGNjclRj4sJEY8Rj4qJEY/Rj5GLkkqc2VnbWVudDNkR0Yl NiNJJEFiY0c9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJJmdpdmVuR0YlNyRJJG1iY0c9RiVGJkVcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSMiIiIiIiMiIiEqJEY6RjhJIkFHPUYlRiZFXFtsI0YuRjVGNjclRjlGO0Y7Ri5JKnNlZ21lbnQzZEdGJQ== NiNJJEJjYUc9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJJmdpdmVuR0YlNyRJJG1jYUc9RiVGJkVcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSMhIiIiIiUsJCokIiIkIyIiIiIiIyNGP0Y6KiRGQEY+SSJCRz1GJUYmRVxbbCNGLkY1RjY3JSNGOUZALCRGPEY+IiIhRi5JKnNlZ21lbnQzZEdGJQ== NiNJJENhYkc9NiJJJmZhbHNlR0kqcHJvdGVjdGVkR0YnRVxbbCNJJmdpdmVuR0YlNyRJJG1hYkc9RiVGJkVcW2wjSS1nZW9tM2Q6LWZvcm1HNiRGJy9JK21vZHVsZW5hbWVHRiVJJ2dlb20zZEc2JEYnSShfc3lzbGliR0YlSShwb2ludDNkR0YlSSdjb29yZHNHRiU3JSMhIiIiIiUsJCokIiIkIyIiIiIiI0Y4KiRGQEY+SSJDRz1GJUYmRVxbbCNGLkY1RjY3JSNGOUZALCRGPEZGIiIhRi5JKnNlZ21lbnQzZEdGJQ== draw([T1(color=red),T2(color=green),aBC(color=blue),bCA(color=blue),cAB(color=blue),Abc(color=blue),Bca(color=blue),Cab(color=blue)]); 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 sign(cos(Pi/2)); NiMiIiI= h := t -> sqrt( 2-cos(t)^2+cos(t)-sin(t)^2); NiM+SSJoRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3FydEc2JEkqcHJvdGVjdGVkR0YvSShfc3lzbGliR0YlNiMsKiIiIyIiIiokLUkkY29zR0YuNiM5JEYzISIiRjZGNCokLUkkc2luR0YuRjhGM0Y6RiVGJUYl expand(9/4- ((cos(Pi+t)+1/2 )^2 + sin(Pi+t)^2));lprint(%); NiMsKiIiIyIiIiokLUkkY29zRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHNiI2I0kidEdGLEYkISIiRidGJSokLUkkc2luR0YpRi1GJEYv 2-cos(t)^2+cos(t)-sin(t)^2
N:=60;k:='k';z:=array([],0..N);
NiM+SSJORzYiIiNn NiM+SSJrRzYiRiQ= NiM+SSJ6RzYiLUkmYXJyYXlHSSpwcm90ZWN0ZWRHRig2JDsiIiEiI2c3Ig== for k from 0 to N/4-1 do point(cc,cos(Pi/3+k*4*Pi/N),sin(Pi/3+k*4*Pi/N),h(k*4*Pi/N));
point(aa,cos(Pi+k*4*Pi/N),sin(Pi+k*4*Pi/N),h(k*4*Pi/N));
point(bb,cos(5*Pi/3+k*4*Pi/N),sin(5*Pi/3+k*4*Pi/N),h(k*4*Pi/N));
midpoint(mab,aa,bb);midpoint(mbc,bb,cc);midpoint(mca,cc,aa);
triangle(U,[aa,bb,cc]);form(U);
segment(aBC,mBC,aa);segment(bCA,mCA,bb);segment(cAB,mAB,cc);segment(Abc,mbc,A);segment(Bca,mca,B);segment(Cab,mab,C);
z[k]:=draw([T1(color=red),U(color=green),aBC(color=blue),bCA(color=blue),cAB(color=blue),Abc(color=blue),Bca(color=blue),Cab(color=blue)]);
od: for k from N/4 to N/2-1 do point(cc,cos(Pi/3+k*4*Pi/N),sin(Pi/3+k*4*Pi/N),-h(k*4*Pi/N));
point(aa,cos(Pi+k*4*Pi/N),sin(Pi+k*4*Pi/N),-h(k*4*Pi/N));
point(bb,cos(5*Pi/3+k*4*Pi/N),sin(5*Pi/3+k*4*Pi/N),-h(k*4*Pi/N));
midpoint(mab,aa,bb);midpoint(mbc,bb,cc);midpoint(mca,cc,aa);
triangle(U,[aa,bb,cc]);form(U);
segment(aBC,mBC,aa);segment(bCA,mCA,bb);segment(cAB,mAB,cc);segment(Abc,mbc,A);segment(Bca,mca,B);segment(Cab,mab,C);
z[k]:=draw([T1(color=red),U(color=green),aBC(color=blue),bCA(color=blue),cAB(color=blue),Abc(color=blue),Bca(color=blue),Cab(color=blue)]);
od: for k from N/2 to 3*N/4-1 do point(cc,cos(Pi/3+k*4*Pi/N),sin(Pi/3+k*4*Pi/N),-h(k*4*Pi/N));
point(aa,cos(Pi+k*4*Pi/N),sin(Pi+k*4*Pi/N),-h(k*4*Pi/N));
point(bb,cos(5*Pi/3+k*4*Pi/N),sin(5*Pi/3+k*4*Pi/N),-h(k*4*Pi/N));
midpoint(mab,aa,bb);midpoint(mbc,bb,cc);midpoint(mca,cc,aa);
triangle(U,[aa,bb,cc]);form(U);
segment(aBC,mBC,aa);segment(bCA,mCA,bb);segment(cAB,mAB,cc);segment(Abc,mbc,A);segment(Bca,mca,B);segment(Cab,mab,C);
z[k]:=draw([T1(color=red),U(color=green),aBC(color=blue),bCA(color=blue),cAB(color=blue),Abc(color=blue),Bca(color=blue),Cab(color=blue)]);
od: for k from 3*N/4 to N do point(cc,cos(Pi/3+k*4*Pi/N),sin(Pi/3+k*4*Pi/N),h(k*4*Pi/N));
point(aa,cos(Pi+k*4*Pi/N),sin(Pi+k*4*Pi/N),h(k*4*Pi/N));
point(bb,cos(5*Pi/3+k*4*Pi/N),sin(5*Pi/3+k*4*Pi/N),h(k*4*Pi/N));
midpoint(mab,aa,bb);midpoint(mbc,bb,cc);midpoint(mca,cc,aa);
triangle(U,[aa,bb,cc]);form(U);
segment(aBC,mBC,aa);segment(bCA,mCA,bb);segment(cAB,mAB,cc);segment(Abc,mbc,A);segment(Bca,mca,B);segment(Cab,mab,C);
z[k]:=draw([T1(color=red),U(color=green),aBC(color=blue),bCA(color=blue),cAB(color=blue),Abc(color=blue),Bca(color=blue),Cab(color=blue)]);
od: display(seq(z[j], j=0..N),view=[-1..1, -1..1, -1.6..1.6], insequence=true); 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