On intersection of two embedded spheres in 3-space

TL;DR

This paper characterizes the conditions under which two or three embedded polyhedral spheres in 3D space intersect, focusing on the connectivity and neighbor relationships of their intersection components.

Contribution

It provides necessary and sufficient conditions for the existence of such intersecting polyhedral spheres based on specified neighbor sequences.

Findings

Derived conditions for two sphere intersections

Extended analysis to three sphere intersections

Applicable to high-school mathematical education

Abstract

We study intersection of two polyhedral spheres without self-intersections in 3-space. We find necessary and sufficient conditions on sequences x = x_1,x_2,...,x_n, y = y_1,y_2,...,y_n of positive integers, for existence of 2-dimensional polyhedra f,g in R^3 homeomorphic to the sphere and such that * f-g has n connected components, of which the i-th one has x_i neighbors in f and * g-f has n connected components, of which the i-th one has y_i neighbors in g. Analogously we study intersection of three polyhedral spheres without self-intersections in 3-space. Russian version is accessible to high-school teachers and students interested in mathematics.