In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Interested about Quaternions & rotations go here to earn more..

1. hamilton walk

On October 16th in the year 1843, Sir William Rowan Hamilton departed from his residence at Dunsink Observatory on a walk into Dublin. On this walk, Hamilton had a sudden moment of insight at the site of Broom Bridge which led to his invention of a new breed of numbers called quaternions. This walk has become significant to mathematicians, many of whom treat it as a mathematical pilgrimage. This video is my homage to the Hamilton Walk.

2. william rowan

hamilton

Nice! Youtubers! Lin-Manuel Miranda would be proud.

3. quaternions:

If you are confused by the equation i^2 = j^2 = k^2 = ijk = -1 , go to the next video. The reason it looks so weird is because you lose the commutative property when you go from 2D rotation to 3D rotation, the property stating that ab = ba .

This means that the order of multiplication matters, and that if you reorder them, you get a different result. If you'll imagine for a minute, when you rotate an object in 2D space, you can do more than one rotation, and the order of those rotations wouldn't matter; it'll end up in the same ending position. But if you're rotating an object in 3D space, then the order of the rotations absolutely matters! Turning an object 90 deg counterclockwise then 90 deg away from you (if that makes any sense...) is not the same as turning it 90 deg away from you then 90 deg counterclockwise.

4. Stereographic

projection

One thing that makes quaternions so challenging is that they live and act in four dimensions, which is extremely hard (impossible?) to visualize. Luckily, we can build an intuition for quaternion multiplication and how it computes rotation in 3d just by focusing on unit quaternions, the ones which sit a distance 1 from the origin.