Article overview
This mathematical exploration starts with paths on a square and builds connections to rational slopes and visible points on an integer lattice. Coprime pairs provide a bridge between elementary number theory and questions about how paths can be represented and counted.
As the counting problem grows, asymptotic reasoning offers a way to investigate large collections of paths. The article places these accessible mathematical ideas alongside the broader mathematical legacy of Maryam Mirzakhani, showing how simple geometric questions can lead toward deeper structures.