27-line Comments - a 3-Dimensional Configuration:135 pt 27 line configuration is the 27 lines on a cubic surface.The 6 red and 6 green lines are the Schlafli double-six.The 12 interior blue lines, along with 3 blue lines at infinity,are the 15 lines corresponding to pairs of a set of 6 lines,corresponding to the 15 dimensions of Spin(6).There are 72 properly related sets of 6 lines among the 27 lines.Each of the 27 lines is incident on 10 points,and each of the 135 points is incident on 2 lines.Each red or green line intersects: 5 red or green lines, 4 near the outside and one in the middle; 4 interior blue lines; and 1 blue line at infinity.Each interior blue line intersects: 4 red or green lines; 5 interior blue lines, 4 near the outside and one parallel with intersection point at infinity; and 1 blue line at infinity.Each blue line at infinity intersects: 4 red or green lines; 4 interior blue lines, parallel to the planes of the red or green lines that intersect; and 2 blue lines at infinity, the other 2 of a triangle at infinity.45 pt 27 line configuration is the dual of the 135 pt 27 line.The 27 lines are incident on 5 points, and the 45 points are incident on 3 lines. The double six itself is self-dual.