> a quaternion is a 4D representation of a rotation in 3D space

A unit quaternion represents a rotation in R³.

A quaternion represents the quotient of two vectors in R³. That's what Hamilton had in mind.

  /* Produce a quaternion from an axis-angle representation.*/
  function quat_from_axis(theta: number, x: number, y: number, z: number) {    
    const halfcos = Math.cos(theta * 0.5);    
    const halfsin = Math.sin(theta * 0.5);    

    return new Quaternion(
        halfcos,
        axis[0] * halfsin,
        axis[1] * halfsin,
        axis[2] * halfsin,
    );    
  }