Interpenetrating nets

One of the simplest constructions in projective geometery is to start with a line (the "horizon line"), and mark three arbitrary points on it. Draw a line through each of the points, such that the three lines now mark a triangle (i.e. for this exercise they can't be parallel). All further lines are then drawn from any of the three points through a point intersected by the other two lines. By continuing to draw lines, a network of hexagons emerges on the plane.

The three initial points constrain, or determine, all of the nodes and connections for the network. "Move any or all of the three original points into any of the infinite number of positions on the horizon line," Olive Whicher writes in her introductory text Projective Geometry: Creative Polarities in Space and Time, "and the network will always arise, each time with a different form and measure." One can see other forms in the resulting pattern. As Whicher describes, "The network is like a matrix in which other interpenetrating nets are to be seen."

In this deceptively simple model, nodes (intersection points) and connections (lines) are shared by different networks. Each network grows out of the same initial determinants (the points on the horizon line). Each network though has its own set of rules for adding lines -- its own law system as it were. The steps above create hexagons, but within the pattern quadrangles can be seen; the quadrangles are completed by adding the missing diagonal. Curiously, the missing diagonal for all of the quadrangles drawn from the same three starting points will pass through the horizon line at the same fourth point.

There is some insight in there -- that phenomena as processes of interconnected, interacting nodes ordered by some lawfulness, have multiple dimensions, and these dimensions interpenetrate. Something like that?

jd