This article has some of the story behind the paper https://www.theguardian.com/science/2023/apr/03/new-einstein-shape-aperiodic-monotile https://www.theguardia

This article has some of the story behind the paper https://www.theguardian.com/science/2023/apr/03/new-einstein-shape-aperiodic-monotile https://www.theguardian.com/science/2023/apr/03/new-einstein... Short answer to your question: David Smith discovered the hat shape by experimentation.