Appears in collection : The Pentagram Map and Ramifications

An n4 configuration is a collection of points and lines in the plane such that exactly 4 points lie on each line and exactly 4 lines pass through each point. The first such example, a 214 configuration, was found by F. Klein in 1878, but it was in the complex plane, and no projective transformation would make it real. The first real example of 214 configuration was discovered by B. Grünbaum and J. Rigby only in 1990. Their construction was based on the symmetries of a regular heptagon, and it was unknown whether it was rigid (modulo projective transformations). I shall present constructions of movable n4 configurations based on Poncelet polygons and relate this subject with commuting deeper diagonal pentagrams maps, the Poncelet grid, and integrable billiards. This is joint work with L. Berman, G. Gévay, and J. Richer-Gebert (recently published in Focus of Math., Sigma).