What is the shape of a uniformly massive object that generates a gravitational
potential equivalent to that of two equal point-masses? If the weight of each point-mass is
sufficiently small compared to the distance between the points then the answer is a pair of balls
of equal radius, one centered at each of the two points, but otherwise it is a certain domain of
revolution about the axis passing through the two points. The existence and uniqueness of such
a domain is known, but an explicit parameterization is known only in the plane where the region
is referred to as a Neumann oval. We construct a four-dimensional “Neumann ovaloid”, solving
explicitly this inverse potential problem.