Original description by Steven H. Cullinane:

Square model of Fano 3-space

This square model of PG(3,2), the projective 3-space over the 2-element Galois field GF(2), has its origin in the Cullinane diamond theorem.

The black squares in each picture depict a two-dimensional vector subspace (plane) within the vector 4-space over GF(2).

A vector 4-space over GF(2) becomes a projective 3-space when the origin of the space is deleted. In the resulting projective 3-space, vector-space planes through the origin become projective lines, with 3 rather than 4 points because the origin has been deleted. In the above image, the origin (top left subsquare) is retained, for the sake of visual symmetry. For details of the underlying geometry, see (for instance) Geometry and Symmetry by Paul B. Yale.

Original caption:
The square model of PG(3,2), the Fano 3-space.
Sets of black squares represent the 35 lines.