Displayed below are some matrices that I have been studying.  These are the incidence matrices for skew lines in PG(3,q).  More precisely, the rows and columns of these matrices correspond to the lines in projective 3-dimensional space over a field of q elements.  Place a one (black pixel) in the (i,j)-position of the matrix if line i and line j do not intersect.  Place a zero (white pixel) in all other positions.  By studying the incidence matrix (using linear algebra, for example) you can learn about the geometry.

The geometry depends on the number q, which is always a prime power.  The number of lines (and the size of the matrix) is (q^2+1)(q^2+q+1).  In particular, the number of lines in each of these geometries is finite.  The pictures correspond to the following values of q:  2, 3, 4, 5, 7, 8.   Larger values than this quickly become very difficult for my computer to handle.

What structure or patterns do you see in these matrices?





q = 2,    35 lines









q = 3,    130 lines









q = 4,    357 lines  (This is my favorite picture.  Not too big, not too small...)









q = 5,    806 lines









q = 7,    2850 lines









q = 8,    4745 lines






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