This picture is showing (the real points of a part of) the
Barth-sextic, an algebraic
surface in complex three-dimensional projective space with
65 double points.
4(t^2 x^2 - y^2)(t^2 y^2 - z^2)(t^2 z^2 - x^2) -
(1+2t)(x^2 + y^2 + z^2 - 1)^2 = 0
t = 0.5*(1 + sqrt(5))
Here is the PoV3-source I used