Just as the unit couples or complex numbers form a circle around the origin so the unit quaternions form a sphere. The unit quaternions have to be understood as unit vectors around the origin, and this is somewhat confusing when someone keeps telling you quaternions are 4 dimensional! However this is due to a confusion about 4 dimensions that Hamilton never had, nor did any serious scientist or mathematician of the time. The question was what use could the extra facility used for evaluating the unit quaternions be usefully put to. Hamilton naturally favoured that it be used to record time, others wanted to use it to record relative position of one space to another.
The unit quaternions occupy a dual role as do the unit couples. Conjuncting them always produces a unit quaternion. This property is used to define a rotation operator acting on a general quaternion vector Q :uQu, just as for couples it is uCu. Because couples are also commutative uCu is the same as u2C which is always evaluated as -C if u =i. The confusion arises because Hamilton leaves the I,j,k as fundamental axes vectors while also using the procedural call I which is a call to evaluate i by rotating it to a defined axis through pi/2 radians. It in fact evaluates to another unit couple or quaternion. Now these are also valid axes but what is not clear is how they relate to the original axes in terms of rotation.
There are 2 scenarios: the original axes have rotated in space to new axial positions in the unit sphere. Or the unit sphere has oriented itself along different axial conventions relative to the original axial position. One is the axes rotating in space, the other is space rotating relative to the axes.
Hamilton actually chose to rotate a quaternion vector thus identifying the unit quaternions as bilateral operators on vectors. He did establish appropriate notation for the operators, but we all think of I 2 as = -1. In the quaternion group this really must be rewritten as IQi =-1 when Q is the unit quaternion (1,0,0,0) etc.
Now the absolutely crucial associativity rule for unit quaternions ijk=-1, without which I was lost in my exploration of polynomial rotations, and Hamilton was unhappy with his triples!
Which states ij can pre operate on k or k can post operate on ij, but this must give the same resultant . In addition I can pre operate on jk or jk can post operate on I and these must conform to the previous 2 and to each other. But how do we evaluate this associative rule?
Fundamentally we have to accept the formalism, because that encodes a restriction on the unit quaternions in space. The I unit quaternion can only act in the ij plane on the i axis and the j axis. You will need to concentrate here, because the unit quaternion in the i vector direction acts as an operator on the i vector and the j vector rotating the i vector to the j vector and the j vector to the -I vector, so 2 actions of the I unit quaternion is like acting on any I or j vector with -1.
Now the j unit quaternion acts only in the jk plane, that is the plane defined by the j vector and the k vector. So again concentrate carefully: the j unit quaternion acts on the j vector rotating it to the k vector, and acts on the k vector rotating it to the -j vector. So 2 applications of the j unit quaternion is like acting on j with -1.
So finally the k unit quaternion acts on the plane described by the k vector and the I vector. Thus it rotates the k vector to the I vector and the I ve tor to thr -k vector, and is like acting on the k vector with -1 when 2 actions are applied.
You of course realise that these unit quaternions act on the whole of their planes only , this they are cyclic in their planes and help to generate the general rotation of the sphere in combination .
Thus the unit quaternions denoted by ijk have a particular action on the vectors in space, and have to follow he formal sequence. This means that using quaternions involves quantized motion in a sequence, and it actually matters whether the sequenc starts pre or post the vector. Thus ijk can pre act on I or k vectors but not on j vectors. However it can post act on I and j vectors but not on k vectors, in each case the action of the associative quotients is equivalent to acting on the vector with -1
There is a subtle error here, and it was one Maxwell made and it is the reason why he gave quaternions the heave no. In this highlighted foregoing explanation i have not included the 4th calculation axis! I will proceed to highlight the correction using a notation change to make it clearer.
The 4 axes vectors are e,I,j,k and their four unit quaternions are 1,q1,q2,q3.
Q1 acts on e commutativity to take it to q1i, q1 then acts on i to take it to -1e. Finally q1acts on -1e to take it to – q1I . Thus we see that q1 acts in the ei plane, and it acts on the unit vector through its unit quaternion. Q1q1=-1 this is defined as an anticlockwise rotation. However q1actspre to q2 or q2 acts post q1. These operators only commute with the vector notation, not the quaternion.
Now q2 also acts on 1e but in a different plane, and not necessarily orthogonal until the third constraint is in place.
The first question is how to define anticlockwise. This can only be done by defining the face of the "clock". We usually use a hand rule. The first thing is to define the ei clock face with the left hand rule: thumb points down at face fingers curl anti clockwise. In thumb down position From that position I rotate the thumb to point at 2 pi/3 to the ei plane in the thumbs to right and positioned at an angle. The fingers curl away from me anticlockwise , the. Thumb points to the face of the ej plane. Extend the index finger to point at the direction of the -1e, then rotate the thumb to past the thumbs up position to point to the face of the ek plane anticlockwise curling away from you
So q2 rotates 1e to q2j, and the j to -1e and q2q2=-1,
Next q3 acts on 1e taking it to q3k and k to -1e thus q3 q3 =-1.
The different geometry is clear and explains the confusion due to being wedded to the Cartesian frame.
Now since q1q2q3=-1 we have to see that as a cross plane action. The planes intersect only in 1e and-1e, so q1q2 can can get from i to -j. How do we get to q3k? My notion that q1q2 can only act in the plane they define is clearly unfounded and limiting. This has been a long held assumption which is here shown to be wrong, and obscuring of the actual behaviours. The actual behaviours are not forces rotating the axes in a spherical surface but combinatorial Forms that interact with any other such form in a relative and comparative way that generates a resultant quantity which is a vector as well as a quaternion unit. Hamilton described this action as rotating a cone around its apex axis, which means that pre and post multiplying the quaternion vector is fundamentally different to acting on the calculation vector e. this vector is a special unit identity vector in the quaternion group and it is commutative. However, to help distinguish the geometrical action, I change it's name under each action . Thus the e,I ,j,k vectors are this e vector attached to each quaternion in its position. So instead of writing q2e I write q2j where j is a special quaternion identity vector. There is only one special quaternion identiyt vector, so giving it this infinite variety of names would be confusing, except for relativity. If I let it have multiple names then I can think of the myself as moving in space to these axis vectors, ie my axes moves relative to the fixed sphericall axes. However if I insist on it having only one name, then I am demanding that all axes, or quaternions move to it, that is spherical space moves relative to me.
So finally I can make the intuitive suggestion that the 3 quaternions q1q2q3 act as if on each other to rotate the ijk vectors around the e vector as an axis. By choosing the e vector carefully we can encode asymmetrical rotation around this axis. The e axis in fact is not suitable for a time dimension, as Hamilton wanted, it actually represents the axis of rotation.
One other aspect of the e axis is its scaling effect. All none unit quaternions scale, but the e axis scales every other quaternion in one go, thus it can be a powerful door into the microscopic or macroscopic worlds encoded by quaternions.
Their are many other unit quaternion associative actions that can now be calculated, but the point is a set of rotator actions have been identified, and linear combinations of these can be formed to describe any general, none destructive rotation of the axes vectors I,j k. These I have called polynomial rotations, and they have to be applied to the basis vectors of a 4d space a vector partitioning of a spherical space. How they could be applied is through a dot product, a cross product or a wedge product, or products involving determinants, or any other product rule we care to create.
Having understood the principles now I wish to change the notation to minimise this long lasting confusion.
There is one more important convention: when evaluating any association tha action is always away from the resultant. Thus ij has no meaning, and had no meaning until the associative rule gave it a context. However il acting on a vector a either in pre or post position has a meaning and a direction of action: away from the resultant. Ij acting on a in post position first gets a result from ja , and then proceeds from there . aij is not a acting on i but i acting on a in the post position and that is commutative for the element a a vector, but not if a is a quaternion. If all the items are quaternions as in ijk the. associative rule is understood to always precede from left to right Thus k acting on I is the same whether written ki or ik. The latter is confuse able with I acting on k, hence the convention