How do we represent a rotation in 3 dimensions? This is a surprisingly deep question, and one we can learn a lot from. Our goal will be to “rediscover” quaternions from first principles, building them up by exploring different representations and fixing the failures along the way.Finding safe representations for 3D rotations has applications in a wide array of areas: 3D modeling software, robot path planning, space navigation, and medicine (e.g., representing the rotation of a scalpel in a robot...