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We use th product form for many purposes . It is a design form

So i+ i is a combinatorial form but i*i is a composition or a product form . What is composition?
It is a larger conception of arrangment, usually more complex and more artistically designed .
Combining objects or elements has a simple notion, that of gathering or collecting together . However if a patterned arrangement is introduced into this simple gathering then we may begin to apprehend composition.

The first notion of composition is in fact factorisation. This is dividing a form into a pattern of blocks or objects that can refit back together to recreate the original form. Thus the original form is represent able as as a multiple form . These sub foms are the factors of the original.
However we cn go evn further in our design, so thst the factors carry a pattern in 2 or more independent " directions" . It is these directions in the patterning that enable us to compose or recompose th form.

Hermann Grassmnn identified this notion of direction as being key to his system designs for products . This notion of direction is clearly spaciometric . Most of us will be stuck if asked to think of morevthan3 directions in which to pattern a structure, but fortunately Justus Grassmann found a mineralogy study at the same time as he was piecing out his combinatorial ideas. The patterning in a mineral , like a diamond actually shows how nature can pattern in more than 3 directions , and how faceted crystals are !
These multiple directions reveal a deeper decomposition of direction, into orientation and translation along a line segment or arc segment in that orientation.
So i gives us an orientation and a translation path. The orientation however is not that of a line segment but that off plane, and the translation is not solely along an arc but also around the centre of that arcs enveloping circle as a translation or a pure rotation.

The basic combination of arc segments will be a bigger arcs of a Circle or even whole perimeters and more. However if I want to rotate a pure arc around its centre that is a change in orientation, and in Grassmann product design any change in orientation requires a sign clearly indicating that.
So i is a quarter arc i*i is a change in orientation of that uarter arc by a quarter arc in the plane that i identifies.

We cn view this orientation changed in many ways: a plane rotates carrying the quarter arc, the quarter arc rotates in the plane  around its centre , the observer rotates relative to th plane by a quarter arc.
So now 5*i is. Form with several possible definitionS: a big arc of more than one perimeter, a proportional arc segment on a circle with 5 times the radius

5*i is often defined as equal to i*5, that is that commutativity is defined as part of this composition. However Grassmann does not in general see commutativity as a fundamental characteristic of any product design. In his methodical analysis, commutativity has to be demonstrated. Associativity is the only general compositional form that appears to be fundamental .

Thus when we blithely write 5i*5i = 25*i*i we are often writing something that is physically unfounded!, or spaciometrically inane. This often comes about through a confusion about Number.

The concept of Number has changed over the centuries from Arithmos a mosaic of regular sub forms and thus a multiple form, to a pattern of Arabic numerals. Dedekind tried to ground number y a concept call the Number line which was a revision of Wallis's measuring or measurement line, which in turn was the pythagorean segmnted "good" line, often translated as straight, but surprisingly that is not necessarily its only meaning.

Thus Dedekind defined number as an abstract notion often called length or the magnitude of extension in one orientation.
Everyone of a philosophical mind knows length or extension does not exist in space on its own! In fact it only exists in our proprioceptive decomposition of our experiential continuum.

Arithmoi as mosaics do exist in space and have all manner of oriented extension. From these we may select one as primary or principal. When this is done then the spatial object imposes conditions on how all other orientations and their extensive quality may be combined! Justus in his careful combinatorial study of Minerals was forced into these realisations, and published little read pamphlets on the topic.
Fortunately his sons Hermann and Robert ingested his ideas.

While it does place so called Geometry at the heart of So called Mathematics alongside Arithmetic and symbolic Arithmetic ( Algebra) it highlighted logical assumptions that were pure arbitrary wishful thinking and not grounded in spatial constraints. The most innocuous of these was the concept of multiplication!  Justus just could not give a logical ground for it, he could only point to a geometrical figure! This was because he held on to the oncept of commutativity.

It was Hermann who accepted that commutativity is not logically justified in general, and applies only to special situations in which processes of construction are truly independent of starting position, or thr symmetry of the product means the process is not uniquely definable !

So product design is crucially a thing of expertise in which every calcultivar process in the product has to be demonstrated as giving the designed outputs. Fortunately the method of induction or inductive reasoning means structures can be built in a logically defensible way and fundamental behaviours can be demonstrated as an outcome of a prior process.

Thus it is not always simple to say " let us assume " and then fail to demonstrate the outcomes forbthevprocesses based on those assumptions.

These demonstrations can be very tedious initially, so it is fortunate for those of us with Los boredom thresholds that ou forebears were not distracted by TV's radios and games, but often delighted in tedium for the small rewards it gave!

Evenso, the more romantic would rush ahead with great visions of what was to come hoping that those who would plod through thr detail would justify their exuberant forecasts.
Hermann presented himself as  a plodder providing a " ringside" or stadium seat for all those interested in the drama but unable or unwilling to engage in the necessary arduous trainings! His 1844 print revels in this metaphor, whilst at the same time profusely apologising for any error he might commit in the process due to the part time nature of his studies!

For all it's imperfections it is and was a masterpiece, and recognised as such by Hmilton, perhaps the only other Natural Philosopher naturally inclined to that way of thinking! We may dismiss St Vainant as a plagiarist attempting to claim primacy for France, even though he may very well have been studying the same ideas from the greats of his time like LaGrange,Euler, the Bernoulli's and of course Newton. The tedium is what distinguishes Hermanns work from any other. He has lived through these boring demonstrations and somehow managed to develop a huge enthusiasm based on them! Only a very few would thereby be drawn to sit at his feet as a student and Hamilton was one of those ! Bill Clifford was another, but few others! Most would take his ideas and fly their own jets with them!

So what does the composition i*5 depict, or refer to or lable?

http://bpeers.com/blog/?itemid=1001
The power series

http://bpeers.com/blog/?itemid=1001
I chose W00t because he goes through the expected Demonstration of the quaternion as an exponential and logarithmic form.
This is Mathemythics!
The Thought Pattern is: there must be a way of showing that this is consistent with standard arithmetical processes especially differentiation ones.
The student and the teacher are left in awe snd somewhat bemused for lack of understanding.
The whole demonstration begins with product deign!  Thus it is not a natural consequence of everyday arithmetic, it is a designed use of standard arithmetical forms but the forms are analogies. They are analogies by design. And it is because they are analogies that we call the process symbolic  arithmetic aka Algebra!

In designing these analogies the combinatorial structures are important , but the internal semantics are different. So we replace the arithmetical operations by combinatorial and compositional ones.

Algebra since Hamilton Boole and Grassman et al has pushed this formal analogy to the limit.

The label for the exponential form is just that. e^ix is a label or function form . Somewhat we put in its argument is what we express as defined.

The design ofnthebquatenions product rules and addition rules ( composition and combination) are dealtbwithnextensivly by Grassmann in his paper on " where to find Hamiltons Quaternions in the Ausdehnungslehre"! It is a very sophisticated read of a very sophisticated thinker ( Hamilton) . Hermann dispels the Mathmythics by clearly demonstrating the constructed nature of these combinatorial forms.

The geometrical implementation/ motivation/ interpretation is then rather obvious . Where the obscurantism lies is in conforming the processes to the expectations and intuitions of arithmetical mathematicians!

For most Mathematicians it is precisely this " mind fluff!" of symbols that confuse 95% of the rest of humanity that is such a drug! We love to tortuously work through the twist and turns of any demonstration to get a result! But it has to be the "correct" result! It has to Bethesda right answer.

Hermann and other Algebraists of his time recognised the constructive psychology of mathematical behaviour and gave an exposition of it.
Math,sticians I'd not like this, it was too revealing of what they were actually doing! But philosophers were saying: don't be Luddites! You can simplify and improve and extend your expertise by understanding what you are actually doing!!!

The exponential label came after the logarithmic process was clarified by Napier and Briggs and Bruges(?). Mercator worked out he Logarithmic series which Newton Wallis et al confirmed. These were difference processes for interpolation something Mathematicians had been engaged in for centuries in calculating the Sine or Chord tables for different arc segmnts/ angles.  The establishing of a serviceable labelling convention was part of the so called Function revolution. Mathematicians got so bogged down in detail they lost sight of what they were doing psychologically and philosophically, and to a certain extent theologically.

If you read Newton, Euler and Grassmann you find a clear philosophicl,and psychological adjustment to what they were about. As Berkely pointed out it was as nebulous as any religious discusdions or definition or debate of their time! Those who pursued this kind of constructive thought patterning usually had some expertise to apply it with. Thus a master craftsman might develop these kinds of notions over years of studying nd applying his/ her trade.

Newtons trade was Mechanics Astrology and Alchemy, all of which he applied throughout his lifetime.
So dear reader do not get the impression that quaternions are strange otherworldly entities, as Hmilton poetically inclined to believe. Realise that Grassmann explained it as a constructed analogous method or system for labelling and expressing ordinary spatial experince of dynmic entities like moving and shaping a block of wood !
Of course they have more sophisticated applications beyond spatial even temporal experinces, but you need to grasp Grassmanns analysis of extensive and intensive magnitudes to appreciate how a line segment and a step rise can encode those distinctions.

There is one point about commutativity that is prominent in W00ts  explanation. The issue is not about arithmetic or counting numbers. These are compositional expressions and in general commutativity of compositions has to be defined or constructed . Nothing can be assumed .

If we define the Composition of two exponential quaternions we have to be aware of th conjugate something Hamilton was not initially clear about in his original presentation of the rules of composition of ijk. Indeed he was not clear about a lot of things regarding i j k !  For that matter Grassmann struggled in his early work on rotation trying to grasp the way to cope with the immense freedom line segments or rays had in the Raum!

The solution is that i,j,k specify planes of circular arc that are orthogonal. This was a olution based on Justus Grassmans work on specifying a system for points planes etc in the Raum.