On the mathematics table problem

In this paper we study the mathematical table problem from a geometric-topological point of view.

We prove a zero-existence theorem on a cylinder, which gives a new proof of Fenn's square-table theorem under Fenn's boundary conditions, and establish a variant under different boundary conditions.

We also prove that every square table admits a horizontal placement on saddle surfaces.

Finally, we show that almost every level set of a smooth Fenn graph contains a rectangle similar to any prescribed rectangle and an orientation-preserving similar copy of every prescribed cyclic quadrilateral.