Pentagram map

In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paper though it seems that the special case, in which the map is defined for pentagons only, goes back to an 1871 paper of Alfred Clebsch and a 1945 paper of Theodore Motzkin. The pentagram map is similar in spirit to the constructions underlying Desargues' Theorem and Poncelet's porism. It echoes the rationale and construction underlying a conjecture of Branko Grünbaum concerning the diagonals of a polygon.

Suppose that the vertices of the polygon P are given by ${\displaystyle P_{1},P_{3},P_{5},\ldots }$ The image of P under the pentagram map is the polygon Q with vertices ${\displaystyle Q_{2},Q_{4},Q_{6},\ldots }$ as shown in the figure. Here ${\displaystyle Q_{4}}$ is the intersection of the diagonals ${\displaystyle (P_{1}P_{5})}$ and ${\displaystyle (P_{3}P_{7})}$, and so on.

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