Prince Rupert's Cube and A Local Theory of Rupert Polyhedra
Abstract: Given two cubes of equal size, it is possible -- against all odds -- to bore a hole through one which is large enough to pass the other straight through. This preposterous property of the cube was first noted by Prince Rupert of the Rhine in the 17th century. Surprisingly, the cube is not alone -- many other polyhedra have this property, which we call being Rupert.
There is an open conjecture of Jerrard, Wetzel, and Yuan, which suggests that all convex polyhedra are Rupert. Aiming towards this conjecture, my recent paper develops a theory of so-called "local" passages, passages where the "passed through" and "passing through" polyhedra have arbitrarily similar orientations. By restricting to the local theory, we can vastly simplify the analysis of polyhedra which have two particular types of simple substructure. This simplification allows us to develop two "by-eye" tests to show that a polyhedron is Rupert, and builds structure to support future analysis.
This talk will aim to offer an overview of the background work, explain the key aspects of the theory in the paper, and suggest some directions for future research.
As usual, refreshments will be served at 3:30pm in BH 300. Hope to see many of you there.