A Star-Shaped Spherical Heptagon Whose Development Self-Intersects

Problem 13 in G. Rote's list of open problems from the 2009 Oberwolfach workshop Discrete Differential Geometry, posed by J. O'Rourke, asks when a simple piecewise-geodesic curve on the sphere, developed onto the plane with the same edge lengths and turning angles, does not intersect itself. The source records that closed convex polygons develop simply and asks: what about star-shaped polygons? A closed polygon has to be cut open at a point to be developed. The theorems of O'Rourke and Schevon (1989) and O'Rourke (2003) on closed curves name no cut point, and O'Rourke and Vîlcu (2011) single out the closed curves whose development is simple for every cut point; we read the question in this way, as the source states no convention. We give a simple spherical heptagon with rational vertices, contained in the open hemisphere about the north pole \(N\) and star-shaped with respect to \(N\), whose development cut at one of its vertices is not simple: its first and last edges cross transversally. So star-shaped polygons need not develop without self-intersection, even if only cuts at vertices are considered, whereas on a cone closed curves that are star-shaped from the apex do (O'Rourke and Vîlcu). The proof is a certificate in exact rational and interval arithmetic, checked by a short program. The developments cut at the other six vertices are simple, and a second certified heptagon shows that cutting at the vertex farthest from \(N\) can fail as well. Whether every star-shaped spherical polygon has some cut point with a simple development remains open; numerical searches found no polygon all of whose cut points are bad. General conditions for a simple development are not treated. This is an unrefereed note.