On intersection of two embedded spheres in 3-space

TL;DR

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

Contribution

It establishes necessary and sufficient conditions for the existence of such intersecting polyhedral spheres with specified neighbor counts, extending to three spheres.

Findings

Conditions for two sphere intersections are characterized.

Conditions for three sphere intersections are analyzed.

Results are accessible to high-school teachers and students.

Abstract

This article is covered by the article arxiv.1012.0925 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.