The latest version of the paper elaborates on the concept of the popular mathematical puzzle “What is the shortest path between the ant and the drop of honey?”. In this paper, the author fully solves this problem for regular prisms and cylinders, providing a detailed mathematical insight into geometry and the analysis of shortest paths.
ABSTRACT
An ant sits on one base of a regular n-prism and a drop of honey on the opposite base, both on the bisector of their base and at equal distance from the rim. The “obvious” path – across one base, up a lateral face, across the other base – has length D = h+b, independent of where along the bisector the two points sit. We ask when a shorter path exists, how many there are, how long they are, and at what angle the ant must set off.
The problem is solved completely for every n. A single pair of formulas gives the length and the direction of every candidate path on every regular prism; the cylinder follows as the limit n → ∞. Along the way three phenomena appear that are easy to miss when prisms are treated one at a time: some paths exist only when the ant is far enough from the rim, some prisms admit no shorter path at all unless they are tall enough, and the number of shorter
paths behaves very differently depending on whether helical paths are allowed.
Three conjectures are stated in Section 10 and settled in the appendices. The first is proved; the second turns out to be false except at one critical value; the third is confirmed, together with an asymptotic growth law.