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MATLAB December 24th

 rho = pi*(1+sqrt(5))/2;
 n = 70000;
 x = 0:n;
 z = cumsum(exp(i*(rho*x.*x+.4)));
 [m,k] = max(imag(z));
 plot(z,'g')
 axis image
 hold on
 plot(z(1:n/100:n),'ro','MarkerSize',4,'LineWidth',2)
 plot(z(k),'y*','MarkerSize',10,'LineWidth',2)
 axis off