Thanks to Kujonai and Tim i started out on the road to complex algebras, and indeed the process algebra by investigating sign and polysigns. My take on them was they were unary operators that rotated axes, and that took me to De Moivre and the roots of unity, the children of Shunya, yoked yet dynamic and free. I little suspected that this would lead to Grassmann and to Euclid, and finally to Dirac, who let the children of Shunya be named positrons and electrons. This yoked pair is the nearest we have got to the fundaments of Shuny,. the intestines and guts of our very visceral universe.(or if you prefer, the beating heart of it).
And of course quaternions ae their extended family of relations.
I identified the problem with sign, as being a procedural call itself a member of the roots of unity. but the symbol obscured this, and the conventions hid this and encouraged one big difficulty, which i why the triples could not be found satisfactorily, and why √-1 ws such a problem after the introduction of the negative or misfortunate numbers.
None of this is Brahmagupta's or Bombelli"s fault. the problem lies in gluing together relations that ought not to be glued.
We all intuitively know that contra and opposite require a rotation through pi. This is why a mirror image is so profoundly odd. it represents stack upon stack of these rotations through a common centre, but we are used to thinking of a synchronous rotation about one common centre. Because objects are more or less in fixed relationships within themselves, these internal constraints mean the body rotates as a whole, but a mirroe image is built up of billions of photons that are reflected at a surface, or absorbed and retransmitted, which subjectively appear to be coming from behind the mirror. The reflection constitutes a rotation at a point, and these point rotations are not bound by the internal constraints within the light source, which is the object emitting light to the mirror. These photons do not have to rotate as if they are constrained, they only have to rotate by the reflection or refraction laws. Thus these reflections through a "point" or rotations at a "point" are subjectively processed as coming from behind the mirror, and this pofoundly alters the whole set of relationships in the reflected image. A mirror image is a totally constructed image by the image processing centre in the subjective processing centre.
Given this level of complexity , why would we call it a negative?
As Hamilton shows in his couples the real plane and the real space involves a complex set of relationships, not an imaginary one and certainly not a negative or contra one. The use of the term contra is so specific as to be useless in general and at worse misleading. | for example, after all my life thinking that i was restricted to operating in the plane of the complex numbers, i finally got the associative rule in quaternions. i,j k are not restricted. it is just that the planes are special cases, and as such helpful to orient oneself, but useless to describe the general behaviour of quatenions. The general behaviour of any group has to be defined by the associative rule for at least 3 elements! this was what Hamilton suddenly realised, and this is why he called the 4th axis the calculation axis. It is probably more important to call it the associative rule axis ,
So i have this toy, and i suddenly realised it was demonstrating the dynamic quaernion relationships! it is in fact a nice model of a Quaternion reference frame. having made that observation i notice that the Hamilton triples were clearly within the model. Could i solve Hamilton's triple problem from this insight?
The answer was yes!
And then i hit on the problem: seeming contradictions especially when pre and post multiples were tried. The clear associative rule should be ij =-i. but this maked -j = to 1
so if i tried -1ij=-1/+1 ij has to be ±1 respectively which leads to a similar problem.
I have to go back to Euclidean duality notions to find the problem. We equate or dual two forms by picking them up(translation) and rotating them(rotation) to see if they fit(duality). thus duality involves translation and rotation. in this particular instance i am dealing with rotations. Thus the identity rotation 1 is in fact differnt to the rotation called -1. it is not the one its the minus. The minus is in fact a rotation in quaternions. To distinguish this in polynomial rotations i called it sign and uses an exponential to differnetiate the 2 things called sign sign 0 (+) and sign1(-). sign 0 is the quatenion identity and does absolutely nothing! sign 1 however is a 2nd root of unity quaternion and it rotates through π. Thus under the associativity rules it cannot be allowed to commute like 1!
This means that -i is not the same as i-! we fooled ourselves by allowing the minus to commute with the 1 when we said that -1 *a = a*-1 = -a
In Euclid we would try to fit one object on top of the other only to find that as we rotated one figure the other was mysteriously linked to it and it rotated too! We could never fit it on the other figure to say it was dual, so we just defined it away. However, if we left the minus where it was and took the one through we could rotate it independently and then we would see that a has to be defined as a = a- to get duality, but they are clearly Not dual.
Hamilton discusses this issue at the beginning of his work on the couples. His solution is not to equate them but for us to reorient our reference frames for each situation, that is we do a rotation. Thus we formally identify a- as rotated a or r(a) we then equate a- = r(a) and -1*a is then = r(a) and -1* r(a) = a. r(a) is clearly the same terminology as -a
Now what about a*-1? This is a*r(1). This means we rotate our orientation relative to the 1 this time. In real space we have an infinite choice of reorientations, but the sign1 quaternion always has to have the effect of rotating through π relative to the original effective step, but if i use Euclid's shorthand notation i have to either separate the line segments and deal with them individually or deal with them as a joined line. Applying sign to a joined segmented line is applying it globally to both segments. When the rectangle is formed from this line it looks no different to the original, thus -(a*1)=-1*-a
However if we separate the segments and apply sign1 to one or the other , then can we join them together again to represent a rectangle? We blithely assume we can, but we can not. Now suppose we denote the line segments by directed lines radiating from an origin, what is the difference? The diference is the direction of the lines. We now cannot just describe the rectangle as being contained between the 2 segments, we now have 4 possible rectangles. The process of duality is now radically different. We can pick up and rotate, but the only match is when the rotation is of a fixed relation between the directed lines. The match is diagonally across the origin. We have identified one the other is -a*1 =-1*a.
We define this by Bombelli's rules(Brahmagupta's originals), but we have 4 distinct cases. In the second two cases the 1 does change its relative position but the minus does not. This is because the minus is a fixed direction in this space.
Now these minus's are direction indicators, not sign1. They are the vectors to the quaternion sign1, thus we make the mistake of taking the direction vector through with the 1 when doing the duality process.
We can now look at how complex sign 1 really is.