Mass in the Hyperbolic Plane∗

Archimedes computed the center of mass of several regions and bodies [Dijksterhuis], and this fundamental physical notion may very well be due to him. He based his investigations of this concept on the notion of moment as it is used in his Law of the Lever. A hyperbolic version of this law was formulated in the nineteenth century leading to the notion of a hyperbolic center of mass of two point-masses [Andrade, Bonola]. In 1987 Galperin proposed an axiomatic definition of the center of mass of finite systems of point-masses in Euclidean, hyperbolic and elliptic n-dimensional spaces and proved its uniqueness. His proof is based on Minkowskian, or relativistic, models and evades the issue of moment. A surprising aspect of this work is that hyperbolic mass is not additive. Ungar [2004] used the theory of gyrogroups to show that in hyperbolic geometry the center of mass of three point-masses of equal mass coincides with the point of intersection of the medians. Some information regarding the centroids of finite point sets in spherical spaces can be found in [Fog, Fabricius-Bjerre]. In this article we offer a physical motivation for the hyperbolic Law of the Lever and go on to provide a model-free definition and development of the notions of center of mass, moment, balance and mass of finite point-mass systems in hyperbolic geometry. All these notions are then extended to linear sets and laminae. Not surprisingly, the center of mass of the uniformly dense hyperbolic triangle coincides with the intersection of the triangle’s medians. However, it is pleasing that a hyperbolic analog of Archimedes’s mechanical method can be brought to bear on this problem. The masses of uniform disks and regular polygons are computed in the Gauss model and these formulas are very surprising. Other configurations are examined as well.