size(7.5cm);

pair        pO=(0,0);
int         n=100;
pen         p1=.5bp+green, p2=palegreen, p3=blue, p4=paleyellow;
transform   t=scale(.4);
pair[]      z;

z[1]=(1,0);

for(int i=2; i<=n; ++i) {
    z[i]=rotate(90,z[i-1])*(z[i-1]+dir(z[i-1]));
}
for(int i=n; i>=2; --i) {
    real taux=1-i/(2*n);
    path ch=pO--z[i]--z[i-1];
    fill(ch--cycle,taux*p2);
    draw(ch,p1);
    draw(t*format("$\sqrt{%i}$",i), ellipse,z[i],p3,Fill(p4));
}
draw(pO--z[1],p1);
draw(t*"$\sqrt1$", ellipse,z[1],p3,Fill(p4));