File:Fano Space From Cube.png

English: A Fano space (finite projective 3-space over the finite field of order 2) with the affine points arranged in a cube. The four-bit strings are shorthand for homogeneous coordinates. Example: 0101 stands for [0:1:0:1]. Points whose coordinates end at zero are at infinity, forming a Fano plane. The space has fifteen points: (2^4 - 1)/(2 - 1) = 2^3 + 2^2 + 2 + 1 = 15. Eight (2^3) of those points are affine, seven (2^2 + 2 + 1) are at infinity, three (2+1) are at infinity's infinity (with coordinates ending with 00) forming a projective line at infinity within the plane at infinity, and one is at infinity's infinity's infinity (with coordinates ending with 000). Each line has three points and there is a unique line through every two points, so there are 15*14/3! = 35 lines. There are two Fano planes for each point at infinity plus the Fano plane at infinity itself, making 15 Fano planes. The third point in a line can be obtained from the coordinates of the other two points through XORing.