  On any compact space one can consider the map which sends a point to the set
of points farthest from this point. In nice cases, there is just a single point
farthest from a given point and so by restricting the domain slightly one can
form a dynamical system on the space based on this map. We give a complete
characterization of this map, including explicit formulas, when the metric
space is the regular octahedron equipped with its intrinsic flat cone metric.
