On pinned billiard balls and foldings

TL;DR

This paper analyzes systems of fixed-position balls with pseudo-velocities, providing explicit bounds on the maximum number of collisions, and explores foldings as a mathematical analogy for paper folding along creases.

Contribution

It introduces bounds on the number of pseudo-collisions in pinned ball systems and connects these to the mathematical concept of foldings.

Findings

Derived explicit upper bounds for pseudo-collisions in n pinned balls in d dimensions.

Established a link between pseudo-collision systems and foldings.

Provided a mathematical framework for analyzing collision sequences in fixed-position systems.

Abstract

We consider systems of "pinned balls," i.e., balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times of collisions for different pairs of pinned balls are chosen in an exogenous way. We give an explicit upper bound for the maximum number of pseudo-collisions for a system of $n$ pinned balls in a $d$-dimensional space, in terms of $n$, $d$ and the locations of ball centers. As a first step, we study foldings, i.e., mappings that formalize the idea of folding a piece of paper along a crease.