Elementary slopes of plane-wide breakout

TL;DR

This paper models a breakout game using a mathematical framework, identifying conditions for periodic orbits based on the initial slope of the ball, and introduces elementary blocks to describe these orbits.

Contribution

It provides a novel mathematical description of periodic orbits in a breakout game using elementary blocks and slope conditions.

Findings

Characterization of slopes producing periodic orbits

Introduction of elementary blocks for orbit description

Necessary and sufficient conditions for orbit periodicity

Abstract

We consider a mathematical model of the breakout game where $\mathbb{R}^2$ is covered by unit square bricks everywhere except in one place. We introduce elementary blocks (particular sets of bricks) in order to describe a family of periodic (in a breakout-appropriate sense) orbits. Necessary and sufficient conditions for slopes (the initial direction of the ball) to produce these orbits are given.