Mathematical and physical billiard in pyramids

TL;DR

This paper investigates billiard trajectories within triangular pyramids, establishing conditions for the existence of 4-cycles, formulating conjectures, and analyzing the impact of physical factors like gravity on these cycles.

Contribution

It introduces new conditions for 4-cycle existence in pyramids, proves specific cases, and explores physical billiards with gravity, advancing understanding of billiard dynamics in pyramidal shapes.

Findings

Pyramids with two orthogonal faces have at most two 4-cycles

Conditions for the existence of 4-cycles are established

Gravity influences billiard trajectories in pyramids

Abstract

In this experimental work we study billiard trajectories in triangular pyramids and try to establish conditions that guarantee the existence (or absence) of 4-cycles (there can be not more, than three of them). We formulate conjectures and prove some statements. For example, if a pyramid has two orthogonal faces, then it has not more than two 4-cycles. Also we study 4-cycles of the "physical" billiard in pyramids, i.e. in the presence of gravity. Here we present our observations for a generic case.