Quaternion

Quaternions are used frequently in Houdini point wrangles as a convient way or orienting geometry (the orient attribute is a quarternion). Unit quarternions (called versors) provide a convenient way to represent orientation or rotation of geometry in 3D.
Sir William Rowan Hamilton defined quaternions as an extension of complex numbers. A rotation can be defined as a vector4 (w + xi + yj + zk) instead of a 3x3 matrix (Euler).

For example:

http://deborahrfowler.com/HoudiniResources/WrangleNodeExampleVexFunctions.html
http://deborahrfowler.com/HoudiniResources/WrangleNodeExampleRandomRotate.html

Quarternions, in a simplified explanation, are a way of rotating an object and avoiding gimbal lock which occurs in Euler rotations.

Here is an excellent illustration of gimbal lock: https://www.youtube.com/watch?v=zc8b2Jo7mno&t=413s

Here are three videos explaining quaternions:

• https://www.youtube.com/watch?v=d4EgbgTm0Bg&vl=en
• https://www.youtube.com/watch?v=3BR8tK-LuB0
• https://www.youtube.com/watch?v=SCbpxiCN0U0

The top of the three videos listed is from 3Blue1Brown which has some really excellent videos on mathematical principles you may find very useful.

For example, this one on vectors.