File:Fano 3-space rectilinear.png

English: A purely rectilinear, tetrahedral model of the Fano 3-space, PG(3,2), which has 15 points, 35 lines, and 15 planes. Points A, B, C are at infinity (and so are points K, L, M, and O). Line AB may be thought of as being a horizon. Points D, E, H, F, G, I, N, J are the vertices of a parallelepiped seen in perspective. Triangles HFD and NJG are subtended by a pencil of lines issuing from C, therefore by the Desargues Theorem the intersection of DH and GN (K) is collinear with the intersections of HF and NJ (A), and GJ and DF (B). (This is just one example of the application of the Desargues Theorem in this diagram.) The colored line segments join the edge-points of Fano subplanes; they would be expected to be circular or elliptical or curved in a non-purely-rectilinear model. Points labeled with primed labels are meant to be identified with their corresponding non-prime-labeled points.
Fano subplanes: (1) AKBMCLO, (2) AEDGCLI, (3) AEDFBKH, (4) BFDGCMJ, (5) BHEICMN, (6) AHFJCLN, (7) AIGJBKN, (8) KHDGCON, (9) MJDEAON, (10) LIDFBON, (11) DHKLMJI, (12) AIGFMOH, (13) BHEGLOJ, (14) CJFEKOI, (15) MLKEFGN.
There are two possible 3D interpretations. One is that EDFHIGJN is a parallelepiped and points A, B, C are vanishing points, so the whole thing is seen in perspective. The other interpretation is that it is a projection (possibly orthographic) of a tetrahedron ABCD whose dimensions are entirely bounded.