Well, the way quaternions model rotation is by unit quaternions (the unit vectors in 4-space) acting on pure imaginary quaternions ai + bj + ck. The unit quater

Well, the way quaternions model rotation is by unit quaternions (the unit vectors in 4-space) acting on pure imaginary quaternions ai + bj + ck. The unit quaternions form a 3 dimensional space called the 3-sphere or SU(2). But 3 variables isn't enough parameters to describe the 3-sphere completely (it will always miss at least 1 point). So there's half a fix to your objection!