On Wed, 29 Oct 2003 reid@math.arizona.edu quoted and wrote:
Are all such patterns known? Say, given a mapping of a hypergraph to the plane, where vertices are mapped to points, and the images of the vertices of each hyperedge are cocircular, can we tell what other subsets of vertices are forced to map to cocircular points?
interesting question! i might have speculated that any such theorem (that a particular pattern holds) could be deduced by looking at the angles. but i don't see that conway's configuration (see below) can be. maybe i just didn't see it.
I have long been interested in theorems of this type. Some particularly interesting ones are the well-known Clifford chain (which is really just about circles), and the following (which aren't): 1. If two each vertex of the Petersen graph one assigns a line in 3-space, in such a way that all but one of the edges correspond to pairs of lines that intersect at right angles, then so does the last edge. 2. If to each vertex of a cube one assigns a conic in the plane in such a way that every edge but one corresponds to a pair of conics having double contact, then so does the last edge. [I know a few more.] JHC