On discrete differential geometry in twistor space

In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in S4S4 to complex values of a generalized cross-ratio by considering S4S4 as a real section of the complex Plücker quadric, realized as the space of two-spheres in S4S4. We develop the geometry of the Plücker quadric by examining the novel contact properties of two-spheres in S4S4, generalizing classical Lie geometry in S3S3. Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. We define discrete principal contact element nets for the Plücker quadric and prove several elementary results. Employing a second real structure, we show that these results generalize previous results by Bobenko and Suris (2007) [18] on discrete differential geometry in the Lie quadric.