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import graph3;
size(400);
currentlight.background=palegreen+opacity(0.2);
linegranularity=1;
real c=(1+sqrt(5))/2;
triple[] z={(c,1,0),(-c,1,0),(-c,-1,0),(c,-1,0)};
triple[] x={(0,c,1),(0,-c,1),(0,-c,-1),(0,c,-1)};
triple[] y={(1,0,c),(1,0,-c),(-1,0,-c),(-1,0,c)};
triple[][] Q={
{z[0],y[1],x[3],x[0],y[0],z[3]},
{z[1],x[0],x[3],y[2],z[2],y[3]},
{z[2],z[1],y[2],x[2],x[1],y[3]},
{z[3],z[0],y[0],x[1],x[2],y[1]},
{x[0],x[3],z[1],y[3],y[0],z[0]},
{x[1],x[2],z[2],y[3],y[0],z[3]},
{x[2],x[1],z[3],y[1],y[2],z[2]},
{x[3],x[0],z[0],y[1],y[2],z[1]},
{y[0],y[3],x[1],z[3],z[0],x[0]},
{y[1],y[2],x[2],z[3],z[0],x[3]},
{y[2],y[1],x[3],z[1],z[2],x[2]},
{y[3],y[0],x[0],z[1],z[2],x[1]}
};
path3 p=arc(O,Q[0][0],Q[0][1]);
real R=abs(point(p,reltime(p,1/3)));
triple[][] P;
for(int i=0; i < Q.length; ++i){
P[i]=new triple[] ;
for(int j=0; j < Q[i].length; ++j){
P[i][j]=Q[i][j]/R;
}
}
surface sphericaltriangle(triple center, triple A, triple B, triple C,
int nu=3, int nv=nu) {
path3 tri1=arc(center,A,B);
path3 tri2=arc(center,A,C);
path3 tri3=arc(center,B,C);
triple tri(pair p) {
path3 cr=arc(O,relpoint(tri2,p.x),relpoint(tri3,p.x));
return relpoint(cr,p.y);
};
return surface(tri,(0,0),(1-sqrtEpsilon,1),nu,nv,Spline);
}
for(int i=0; i < P.length; ++i){
triple[] pentagon=sequence(new triple(int k) {
path3 p=arc(O,P[i][0],P[i][k+1]);
return point(p,reltime(p,1/3));
},5);
pentagon.cyclic(true);
draw(sequence(new path3(int k) {
return arc(O,pentagon[k],pentagon[k+1]);},5),linewidth(2pt));
triple M=unit(sum(pentagon)/5);
for(int i=0; i < 5; ++i){
surface sf=sphericaltriangle(O,pentagon[i],M,pentagon[i+1]);
draw(sf,black);
}
}
for(int i=0; i < P.length; ++i){
for(int j=1; j <= 5; ++j){
triple K=P[i][0];
triple A=P[i][j];
triple B=P[i][(j % 5)+1];
path3[] p={arc(O,K,A),arc(O,A,B),arc(O,B,K)};
draw(subpath(p[0],reltime(p[0],1/3),reltime(p[0],2/3)),linewidth(4pt));
triple[] hexagon={point(p[0],reltime(p[0],1/3)),
point(p[0],reltime(p[0],2/3)),
point(p[1],reltime(p[1],1/3)),
point(p[1],reltime(p[1],2/3)),
point(p[2],reltime(p[2],1/3)),
point(p[2],reltime(p[2],2/3))};
hexagon.cyclic(true);
triple M=unit(sum(hexagon)/6);
for(int i=0; i < 6; ++i){
surface sf=sphericaltriangle(O,hexagon[i],M,hexagon[i+1]);
draw(sf,white);
}
}
}
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