Projective Plane of Order 3

This represents the projective plane of order 3.  The idea is to map the 13 points and 13 lines onto Steiner's "Roman Surface" in an arrangement that is as symmetrical as possible, mimicking a similar, highly satisfactory realization of the Fano plane.  In this case, the symmetry of the shape coincides with the tetrahedral group.  The points are represented by balls and the lines are represented by 4 circles, 6 ellipses, and 3 struts.  Although all 13 lines are distinctly represented, three of the points coincide at the origin.