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This is a challenging video so needs meditating on several times.
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The " Algebra " here is in fact based on traditional algebraic manipulations that assume commutativity. However the last 3 sections approach the topic from a Grassmann lineal algebraic/ Doctrine of thought pattern style.

All the way though however Norman reduces the products to products of limb in synthetical knittings , and indeed also in analytical knitting format.

We understand that if hermanns outputs apply to 3 then they apply to n- limbs. In this case n=4 for the first step or affine 1 point geometrical interpretation. However, much of hat Norman  demonstrates he does so without the technical thought pattern background of he doctrine explicitly referenced.

Again the commutativity of the amoral part of this presentation, obscured by the algebraic bravado, is concealed in the notions of convex, cyclic permutation , area, and displacement..

In the last 3 sections he starts to analyse the situation Grassmann style.

So let us look at the product design. It turns out that quadrance is a level or rank2 knitting , but the uadruple quadrance formula is correspondingly a stage 8 product.. Thus the everyway toutability of the limbs or line segments , and the everyway holding out of them in space attribute requires an 8 dimensional system within systems to express!

Thus Brahmaguptas formula  is a constraining of an 8 dimnsional arrangement of line segments to just 2 dimensions. The natural convex quad if cyclic is a vertical projection of an 8 dimensionl pattern in a kind of helix , Ono the plane.( Norman covers something like this in his topology series).

We are reputedly going to enter the 4 th dimension when we tackle the Quatenions , and indeed Herman produces the figure 4 in his design path or brief, but we should know that 4 axes is not equivalent to 4 dimensions . We need at least 8 symbols to express orientations and calculations and that gives a Cayley Table of 64 products for a level 2 system.  If we used Norman's quadrance we would need a level 4 system with 64^2 level 4 products.

Hermanns analytical knitting and treatment reduces these to a more manageable group size , because each product has to be rigorously applied to real space.

The relationship/ relating of multiplication to the Addition is a phrase that jumps out at you . When first I came upon it in the Vorrede I was not sure what it meant. Later I skimmer through and found page 11, saw the symbols and jumped to the wrong conclusion that it referred to the so called distributive law.

But now I realise that it specifically refers to the designing of products that subjugate addition, and specifically where the addition is of limbs in the lower step/ stage/ rank to the step of the system created by combining2 differing systems  of the same rank in such a way as to make one one system subjugate to the other.

In that case it was a design point that the subjugating system should knit wth the limbsvofvthevsubjugate system in the simplest way, and thus make the product of the addition equal to the addition of the products of the limbs.

Just as the design raises the combined system to the next stage, so the combined system of both limbs are equally raised to  the sane step . Thus it demonstrates that addition or the first knitting is carried through every step rise in analogous fashion. The combining of the 2 higher stage " elements" may be signed the same, but the everyway toutability for exchange for these elements increases dramatically.

Whereas for a line segment bound in a line of orientation the options are swapping within the line in the plane the options are increase by the freedom to rotate. Thus not only can swapping within the lines of orientations be possible, but permutations of this, and then in Addition rotation of the forms created by the products provide additional ways of knitting together.

Constraints on these degrees of freedom are allowable so that the observer/designer can focus on a particular line of development.

So here Hermannthrough subjugate system design constraints identifies this distributive behaviour of the multiplication over/ onto the 2 limbs of the addition as desirable , and it will turn out to be feasible in many real situations. However it is not feasible without some modification in rotational and projective systems.as we shll see , and it is also not as generally thought essentially commutative.

Asyet the role of associativity so called has not been mentioned. Many of these design issues are placed before malleable minds as de facto truths rather than what they are design briefs drawn by analogy from observing production processes in both man made and Natural production cycles.

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§10. The relating of Multiplication to Addittion we have thereafar appointed, that
    (a + b)c = ac + bc
          c(a + b) = ca + cb exists,
And therethrough the label of the multiplication was firmly placed by us. Through once again wholly applying of this fundamental  rule one arrives at thuslylike to the general proposition, that one if both" Factors" divided pieces are, each piece of one with each piece of the other multiplies, and the products can be added. Hereout outputs itself for the relating of multlication to subtraction an inter-communicant rule, specifically immediately nearby, that

           (a – b)c = ac – bc  exists.

Specifically one places in order the second expression to guide back to the first, in the same place of a the to it like  (a – b) + b, so one has

(a – b)c = ((a – b) + b)c – bc,

The second expression is concording to the thusly plainly set down  rule

   = (a – b)c + bc – bc,

And this expression concording to §6

= (a – b)c,

Therefore the former expression is like the latter.

On like manner follows, even if the second "Factor" is a Difference , the inter- communicant  rule. Through once again wholly applying of the laws one arrives at the more general proposition:

Can now tackle ahead of time the issues Hermann has yet to tackle with his product design notion. Some of these issues he does tackle later on, but some he either does not know how to resolve, or he does not see an issue or he deliberately leaves vague to be determined by the real system in application. However for the critical reader not familiar with the new ground Hemann is breaking in clarifying the product design responsibility of the observer or manufacturer of these systems within systems , it is important to emphasise that these are not issues of incorrect or wrong thinking, but of design thought patterns toward some pragmatic outcome.

When I first came upon these ideas of interpreting arithmetic in algebra, I was actually taught by geometric forms the meaning of arithmetic. Consequently I naturally thought arithmetic arise out of Geometry . Then I found after some research that perhaps it arose out of Algebra , with Algebra arising out of Geometry. These were natural deductions from inside the mathematical glasshouse.

It was the issue with i the that forever shattered the window panes and let in a hurricane of confusion, that cleared away centuries of careful mind control to reveal that man, deluded or illuminated by whatever source had indeed constructed this system now miscalled mathematics out of thought patterns derived from experiences in a real world. It took nearly a lifetime to find an influential author of the same opinion, that being the Grassmanns and Hermann in particular.

Thus it is in my view symptomatic of the pettiness that pervades the subject boundary wars that this design process should come to be viewed in terms of right or wrong. Rather as all design processes are , it should be viewed first on utilitarian grounds, pragmatic results, functionality and aesthetics. In the course of doing this the systematic or modular nature of the design can be examined and checked for fitness for purpose . We might call this a design flaw, if it turns out to be disastrous, or quirkiness if it introduces perturbations but nothing major. These issues can be remedied at each iteration of the design process.

So in that light I had a design issue about the quirkiness of the product e ₩ e .

Firstly I had no clear idea that it was a design issue, I just thought it was wrong! Later I viewed it as an opportunity to define an output but had no clue what that " should" be , and now finally I understand hermanns perspective and the design brief he gives. His guidance is essentially keep it as simple as possible, keep it as general as possible so it can carry forward into all levels , keep it based on 3 limbs and for multiplication use the subjugation idea. Finally keep it as fractal as possible so the same Pasternak or an " almost similar" pattern runs through at every step/ stage/ rank/ level/ dimension.

Already by that slashed list one can see how many ideas and notions it has to encompass!.

Let's look at signature, so called, as a set out design solution .

Thus e ₩ e is set equal to 1, 0, –1 according to the system within systems being modelled. Notice these are numbers . In mathematical terms this may not seem strange, but in design terms it is. Norman deals with this issue from the get go be defining numbers (type Nat) and then constructing a linear Algebra from these . Thus he incorporates many issues regarding number into the basis of his system, and the reader has to choose whether he wants to go along with him or not.  However he introduces the Cartesian plane as the de facto standard, which fir many purposes it is, and this allows hom to slip the type Nat onto Wallis's orthogonal axes without causing too many raised eyebrows.

The assigning of numbers to segmented lines is counting, or tallying. In that respect the numbers have no other role than to be a Namespace, to use computer terminology. To give them an a priori existence and type is to confuse the real interaction between space and human thought patterns in my view, and I think in Hermanns view too. Nevertheless it is a design decision which is carried forward to great effect by entertaining it to Hermanns Analysis

Using this set up Norman and many others are able to define the dot product of e ₩ e and get the signature value 1. Normnan however uses the Quadrance as his definition, also giving 1 as well as many other advantages.

This is fine until you move away from orthogonal axes. Then Pythagoras theorem does not apply and the dot product has to give way to the determinant .  These are fundamental design decisions and you can only appreciate them in specificity. Norman sets out these2 issues very clearly, but of course he is still puzzling about what it all means.

By enlarge the signature one relates to the dot product which is simply the product based on Pythagoras theorem . It applies flawlessly for an orthogonal system and is used to define an orthogonal system.

In the general case, that is using line segments referred to by this orthogonal system , the product still holds good as a bilinear form and still serves a purpose of projection of one general line segment onto another.

It has taken me a while to understand this confusing use of the seemingly clear ideas expressed by Hermann , especially the dot product for which I am grateful to prof Norman Wildberger. Without his insight I do not think I would have straightened that out by myself. In this instance Hemann has no clear guidance because he did not cll it the dot product! But once you recognise it in the shadow product or the projection product you can see where Norman gains his insight from.

e ₩ e set to 0 is hermanns first design decision. This does not use a part of the Cayley table to design a product as in the dot product, it uses the full bracket expansion. In this case it would appear to be visibly true that a line segment subjugate to itself produces only an extended line segment in keeping with itself. But since by design this product should be at level 2 in the system within systems, where by design the 2 knitting processes are differing, one subjugating the other, the expectation is a flat figure, not a line.

Therefore it would appear that stage 2 elements should be flat figures , generally speaking, and not extended line segments. However, again by design, Hetmann accepts extended line segments as level 2 products.

The rejoinder is : how then is that different from lower level Addition? And the answer is : it is not , neither is it meant to be! In designing the second level product it was specifically modelled on the lower level knitting. The only reason it was called multiplication was because the name addition was already taken, but indeed it is a knitting just as addition is a knitting,mtherefore it is not odd that in the specific case e ₩ e it should become identical to the knitting in thst system.

Without the development Hermann gives it is impossible to satisfactorily resolve this niggle. But resolving it this way is very powerful, both thought pattern wise and fractally, and it seems to mimic spatial phenomena when dynamic systems Collapse into a lower dimensional state.

The difference between multiplication and addition is the orientation of the subjugation!  I can lay out a pile of bricks as a cuboid or all in a single line. The arrangement in space does not rffect the total " volume" but it certainly effects its dynamic contingencies and behaviour. A hunk of metal sinks. That same metal beaten out into a bowl may well float and carry more than its own mass!

So why choose 0? I think that we have to understand0 not as nothing but as not the emergent form, but the form just prior to emerging! Thus 0 is Shunya, full of potential, pregnant, waiting to give birth to. At level 2 a line is such an entity, it is waiting to give birth to a level 2 element. Similarly at level 3 a plane is Shunya or0 waiting to give birth to a level 3 element, a space called a parallelepiped etc. thus 0 does not mean nothing, rather it means their is a potential that needs following down a level to fully appreciate what might be occurring in a dynamic system.

Now in light of this Hermann deals with a product design where the third limb orients within the level 2 system created by the first 2 synthetic limbs. This is also a source of confusion, which Hermann and Normann following him avoid by prcifying the design. This analysis depends on the first 2 limbs being synthetically linked as the same level elements, in this case level 2. The result, due to subjugation is a parallelogram. This analysis is firmly based on the parallelogram form.

Introducing a 3rd coplanar limb now allows the analysi to proceed according to the previously laid out rules. The news issue arises precisely when we produce e ₩ e in all it's forms. The discussion above allows us to replace this by 0 , but then in certain cases that leads go an inconsistency. We get
0 = ab + ba

And for consistency sake we have to let commutativity go!

The problem is in understanding that result. By other design constraints we can show that a system that subjugates another produces the same result if the subjugation is the other way round. But once we have set a system we cannot justifiably change it in the middle of a calculation! Commutativity implies just that! The only thing we can do is cyclically move the labels around.

It takes a good while to grasp that . Moving the labels is not moving the system , but why would a calculation cyclically move labels? The answer is the calculation does not, the observer does, and we do it all the time, often confusing ourselves. Moving the labels is equivalent to us changing our viewpoint. The best example of this is the clockwise counterclockwise labelling. This is still a confusing labelling system!

So how does changing your point of view make a parallrogram disappear? By design a certain subjugation creates the parallelogram .clearly the reverse should remove it. But the reverse has to occur after the creation . By rotating the factor labels cyclically round one of them now is directed in the negative direction ,say b now labels -a. In that case the product subjugation remains the same and so -a is done first reversing the extension back into the second line segment which then performs b on 0( remembering that b is now labelled a) , the consequence is not a parallelogram but a line extension.

The more astute might note that that does not necessarily remove the created parallelogram! However by defining all parall lines in a parallelogram as equal the task is accomplished. Whatever end to one happens to all parallel lines.


This constraint has the unfortunate consequence of being universally applicable. Thus one can be lead to believe tht one has found universal laws, rather than a solution to a tricky design issue. Maxwell for example felt certain that such products would exist in space independent of source. Some buckle heads have gone about hailing mathematics with this predictive ability rather than looking more closely for the natural force system that may or may not make this a valid application!

In fact the huge magnetic structures show this local consistency justifying a local application of this system, but as any observer of the sun will tell you that consistency os not universal!

Hrm! So why do we still act as if Newtons " laws" are universal? You tell me!

e ₩ e set to –1 is a curious product design. On the face of it it has to be the imaginary product, but in fact that is still confusingly thought of as a line segment product.

In the work I did on polynomial revolutions I gradually had to distinguish between the oriented line and the product design setting up the system.. Hermann deals with this within a rhombus of equal line segments, and shows how it is in fact the singing arc contained therein that is the limb that is being synthetically knitted or analytically knitted. He is forced to use the Eulerian exponential function to express this relationship, and we read of his development in the Vorrede.

The topic is wide and deep, and open for continued research based on the Cotes DeMoivre body of Calculus, the roots of unity and the zeroes of the trig functions cos(nø) and sin(nø). In addition cotes version of the Cotes Euler equation. Here the trig line segments really come to the fore and all the previous work done on product design has to be modified to fit this new product design.

Essentially, from the complex product design involving a rhombus and an interior arc and the projections onto the diagonal, one should expect a very general applicability of this product design. The principal freedom is that of rotation in the plane, and this is directly relevant to Hamiltons Quaternions. The solution Hamilton stumbled on, and I say by brute force because he ignores the complex conjugate of K, works after a fashion but requires the associative product design ijk = –1.

I designed( without being able to quite grasp it) the Newtonisn triples without that constraint , rather ncd = -n, and in do doing saw how Hamilton had bulldozed through the complex conjugate constraint. The Newtonisn triples are a modulo 6 product design, and work on 6 line segments arranged as 3 orthogonal axes.

So are these the best product designs or just the most common and familiar?

For example the dot product design could be replaced by the determinant design while allowing e ₩ e to be set at 2e. 

The design for division which will come up next , may be tinkered with. The design for rotation might be simplified to remove the exponential. Although personally I love it and the trochoids it generates.


Design processes like these are all open to the designer once one gets over the centuries of unjustified awe accorded to mathematics, fostered by mathematicians who have to eat like the rest of us.

The design of the product for Quaternions is what Hermanns paper is all about. And to see how deeply he went into the design will be instructive.

a, b, c are three differing line segments in the plane defined by ab, and ab is a distinguished parallelogram.

Then let d = a +  b be the line segment representing the addition knitting of the 2 line segments.

Thus  cd = c(a + b) = ca + cb where d subjugates c

Note because these line segments are in the same plane the subjugation order produces the same result when switched ( c subjugates d).

 But interchanging the labels cyclically gives ad = a(b + c) = ab + ac. If ac was indeed like ca then we make the mistake of incorrectly distinguishing the referred parallelogram. Commutativity would be a misleading assumption , or rather switching labels around is a different process to switching subjugation around!

Thus switching the limb labels should not be taken to imply a switch in subjugation and consequently a switch from one system to another

In order to mentally switch subjugation in a system we can attach it to the order in which the limbs of a knitting are written, not to the limb labels. So when we cyclically interchange labels we must not let that change subjugation order.

Adopting that .convention avoids a processing order mistake and a wrong identification mistake. We then by inspection within a specified system have to determine if commutativity , that is achieving the same result by changing subjugation , holds. In that case we have 2 systems that give the same result, not one system which has inherent commutativity.

Finally by the rules of parallelograms the above product can be seen to hold, as cd can " slide" the 2 parallelograms ca, cb into itself.

.

There is a lot to say about the above labellings.

The first is that a line segment until now has been the archetype of a step one system..mbut a line. Segment appears in all  steps, and by design we want to be able to synthetical knit them in all steps. Thus a line segment must take on the step level it is utilised in. In this case , in the plane a limb is an element of the plane just as much as a parallelogram! It is astro 2 element.

Thus addition of step 2 elements must cover the knitting of lines as well as the knitting of parallelograms.

Every multiplication in the plane is in design a new product system!

2 elements define a plane by a product. A third element as a subjugating one defines a new product design. Because of this it is a mistake to attempt to relate products from different product designs. A product design always is based around subjugation of a synthetical knitting! Even if the elements are of the same step level, this design is the one Hermann advises, thus a single element subjugating Nother is not the best design, and each element chosen in the pair is a constituent of a unique design. In this case all such singleton pair designs tessellate a plane, even the same plane uniquely..


Thus transforming between designs is very important and necessary if we are to proceed to conclusions.

The transformations are: rotation, perspective projection, and parallel projection, translations, reflections , cyclical label change.

In geometry these transformations are utilised by the observer the activator. There is a possibility thst Algrbra of labels could govern this less whimsically, or at least in a more automatic way. Thus the product designs immediately in the plane link 3 parallelograms together in a way in which they fit . In particular certain transformations distort the form but do not alter the property that the bounded space never leaves the boundary! This is an observation that despite the contortion , in between parallel lines, it can always be demonstrated that the rectangle is the invariant form of all these types of deformation.

This is a fantastic observation. " geometry" has these invariant forms of which all other forms are a kind of distortion!. The principle of conservation of form is what underpins the notions of duality in the Stoikeia. The transforms or distortions maintain something that is invariant, and findi it was the joy of the Pythagoreans.

So using these transformations between differing product designs allows us to discover invariant relationships.


Thus we must allow the design to effect the fallout of surprising results.the above product design does point to similar figures on the sides of a triangle are related by a sum. The similarity of the figures is derived from a common subjugating limb.

It out to be clear then that Pythagoras theorem necessarily must be an instance of this more general relationship. To demonstrate it I have to use transformations between differing product designs.

I just lost the second part of the Pythagorean solution to the aether!  

But it is very strange that it should happen just when I was building a case for the volition of machine intelligence? Are we creating a Frankenstein?
Or are we evolving coevally?

Will my computer let me post this?

We shall,see

Ok, so I have made my amends to the gods of the aether by correcting a statement about rotation in relating to the second demonstration of Pythagoras: rotation is always involved, whether implicitly or explicitly. The theorem of Pythagoras is always about rotating two similar shapes into a third similar shape that is the sum of the two!

The right triangle is defined in a Semi circular arc, the quarter arc is the object invariant of that triangle, no matter how that triangle is specifically constructed in the semi circular arc, so how could the demonstration of Pythagoras not involve rotation?

That being said the set up or Ansatz is different. The squares of the shorter dides are now constructed inside the rectangle. In passing I note that the lack of reference Points leaves the construction kind of "floating ", so the reader may feel unsure of the line segment and parallelogram definitions without a diagram. At this stage I am just trying out a few general things, but later in chapter 1 Hermann firmly grounds these generalities.

The generalities are very intriguing , but they are also " tricksy" , as Gollum said of the Hobbit. One gets the feeling tht without care one can end up "proving black is white and vice versa!".

The set up is as follows and I want to transform the RHS into the LHS .
¢²cf = $²(-ab) + â²ab

Firstly
  ¢²cf = (-$² + â²)ab

One must resist the normal algebraic interpretation and deal with the represented systems. The two product designs cf and ab are clearly related, so I want to transform  ab into cf.

The first square â²)ab can be transformed by sliding it between parallel lines with orientation a until its transverse ie is parallel with f. It then becomes af representing a translation of the base of the right triangle by f.

Similarly square §²)(-ab) can be transformed by sliding between parallel lines b to line up with f thus giving bfrepresenting translation of the upright side by f.

The two transformations thus are a combined translation of the right triangle by f and therefore cf has been created by transforming these 2 squares in this way.

In the transformation I drop the "-"  because the side of the parallelogram no longer is oriented as it was in the square, and the length of the transversals of the parallelograms become equal, but without measuring I am not able to say, but I constructed the square ¢cf specifically so that f is perpendicular to c, thus af + bf = cf must be a rectangle at least, and if ¢f is the construction projection then

    ¢af + ¢bf = ¢cf = ¢²cf.

By inspection the transformations have constructed a gnomon and the resultant output must include Pythagoras theorem as a specific result among the more general result that any similar figures on the shorter sides of a right triangle will sum to a similar figure on the longest side!

One can see that this result is more general than Pythagoras theorem, and in fact Euclid proves it in a later book in the Stoikeia. But here we notice that the observer or operator actively achieves this result by skill, hand and eye coordination and pragmatic and metrical observation.

If we introduce a metric we cn quantify these differing magnitudes in this general reltionship. But soe have sought to remove the judgement of the operator for whatever purpose. They sought to replace it bt mind numbing manipulation of symbols. Why?

Whatever the philosophical motives were it is certainly the case that automatic symbolic manipulation has proven vey useful . The introduction of a tally count, and Wallis's Cartesian coordinate system enabled standard procedures to be discovered, and these in turn to support a standard metric and standard results. These then could be linked by invariance to standard transformations, and so a whole numerical language representation of transformations: translations, reflections, rotations; could be created.

The creation of these numerical representations drove the rise of automatic computing machinery and then electronics, based on the ubiquitous polynomial algebra. Grassmann and Leibniz type of analysis helped Turin to develop his universal machine concepts in which the rank arrays and status/ condition variables took on a general meaning , a representational one. Grassmanns algebras and product design guidance were thus developed in the direction of thought patterning , representation of solution processes etc.

With this electronic support the role of the operator changed. The operator became the programme designer, not just a product designer. Consequently the role of a mathematician has largely been succeeded by "electronic expertise " designs which are now capable of making choices based on expert criteria.

The design goal, th creative and aesthetic motive still largely elude these devices, but the rise of machine learning in particular and creativity modelling may break down these barriers to Artificial Machine intelligence.

Then what? Are we creating a Frankenstein monster, or are we evolving coevally with our machine intelligence?