Spheresallowing

Spheresallowing is a term used in speculative geometry to describe a property of a collection of spheres in Euclidean space. Informally, a configuration of spheres is called spheresallowing if the spheres can be deformed, via a continuous family of ambient transformations, into a standard reference arrangement without changing the pattern of how the spheres touch or intersect.

Formally, given a finite set of spheres in R^n, the configuration is spheresallowing if there exists an isotopy of the ambient space that carries each sphere to a corresponding reference sphere and preserves the tangency graph, i.e., the incidence relations among spheres, throughout the deformation.

Relation to existing concepts: The idea mirrors notions in circle packing and tangency graphs. Spheresallowing generalizes these ideas to higher dimensions but remains informal in most sources, lacking a universally accepted formal definition. Some discussions restrict the property to configurations where spheres touch but do not overlap, while others allow controlled overlaps.

History and usage: The term appears in speculative discussions and pseudo-curriculum literature exploring spatial constraint satisfaction problems. It is not widely adopted in formal mathematics and should be treated as a heuristic concept rather than a proven theorem.

See also: related topics include circle packing, sphere packing, tangency graphs, and isotopy.