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.