Hamiltonian surfaces in the 4-cube, 4-bit gray codes and Venn diagrams

We study Hamiltonian surfaces in the d-dimensional cube Id as intermediate objects useful for comparative analysis of Venn diagrams and Gray cycles. In particular we emphasize the importance of 0-Hamiltonian spheres and the "sphericity" of Gray odes in the context of reducible Venn diagrams. For illustration we show that precisely two, out of the nine known types of 4-bit Gray cycles, are not spherical. The unique, balanced Gray cycle is spherical, which in turn leads to a new construction of a reducible Venn diagram with 5 ellipses (originally constructed by P. Hamburger and R.E. Pippert).