xyz Projective Plane

This is an xyz drawing of a graph representing the Real Projective Plane.  Notice that there are 12+4=16 balls representing vertices and 12+12=24 struts representing edges.  12 of the struts lie in the facial planes of a cube, coming in 6 pairs, each forming a false intersection at the center of a corresponding face of the cube.  It is called an "xyz drawing" because every edge is parallel to one of the three coordinate axes.

One should also imagine that there are 3+6 faces.  To be more specific, there are 3 square faces and 6 self-intersecting hexagons.  The squares are easiest to identify; each is comprised of 4 balls and 4 struts and the three of them are mutually perpendicular, lying in three planes of symmetry of the cube.  Each of the six hexagons, comprised of 6 balls and 6 struts, lies in a face of the cube and is self-intersecting.

Collecting these data, notice that the Euler characteristic of this is v-e+f=16-24+9=1.  The only compact 2-dimensional manifold with Euler characteristic 1 is the Real Projective Plane.  (One may also imagine this model as a discrete version of the Roman Surface....)