Werner Boy's Surface

In 1901, David Hilbert assigned Werner Boy the task of proving once and for all that it was impossible to embed the Projective Plane into 3-space without singularities. Instead Boy came back with this surface. It has a triple point, but no singularity.

n = 128;
u = linspace(0,pi,n);
v = linspace(0,pi,n);

u = repmat(u,n,1);
v = repmat(v',1,n);

x = cos(v).*sin(u);
y = sin(v).*sin(u);
z = cos(u);
f = 1/2*((2*x.^2-y.^2-z.^2) + 2*y.*z.*(y.^2-z.^2) +  ...
    z.*x.*(x.^2-z.^2) + x.*y.*(y.^2-x.^2));
g = sqrt(3)/2 * ((y.^2-z.^2) + z.*x.*(z.^2-x.^2) + x.*y.*(y.^2-x.^2));
h = (x+y+z).*((x+y+z).^3 + 4*(y-x).*(z-y).*(x-z));

clf
s = surf(f,g,h/10,u, ...
    'LineStyle','none', ...
    'FaceLighting','gouraud', ...
    'FaceColor','interp');
colormap jet;

axis off;
daspect([1 1 1]);
l1 = light;
l2 = light;
lightangle(l1,70,-40);
lightangle(l2,-30,80);
view(-40,32);
camzoom(1.5);