Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

The difference between numerally derivable and numerically derivable is the difference between numbers and numerals. I do not assent to numbers as Mathematicl objects formal, defined or otherwise. To me they are symbols of names applied by a sequence to real Metrons covering a real magnitude, enabling us by commensuration to "quantify" a magnitude.

That means we compare smaller against larger by a factoring or division process. The comparison causes us to describe or express what we discover, and this is the meaning of the Greek Logos, and thus the meaning of the word ratio.

On this expression we build all our Rational Thought patterns .

The great exploration is how these rational thought patterns apply generally to many differing circumstances, and allow us to develop an expertise that is transferable as an artform or Skill Set!

[jehovajah]

 PThe Place of Hamiltons Quaternions in the Doctrine of Extending/ Extensive Magnitude

By  H. Grassmann.   In Stettin
--------------------------------
Page 376

I want to name this artform of multiplication  Middler/ Middle-man/Go-Between, and indeed "important matter-like" to name  its aid,  because it builds a representation : the middle Step/ rank/ stage( how  itself comes to be shown thuslike), between both the most important artforms of the multiplication, which I  have named the " spreading out" and the " colliding into" .

The spreading out multiplication has specifically,  the two groups (2) and (3), toward constraining likenings( equations)
And the colliding in  multiplication  has the two groups (1) and (2)

I have the spreading out product of two line segments a and  b with [ ab], and the colliding in product of the same line segments with [ a | b] besigned, and will be in this offprint-handling  under  ab, (without the sharp brackets) continuously and consistently the Middler product of line segments  a and  b everyway  standing.

Then the middler product output-esults itself thuslike, that the Middler product  ab of two line segments let themselves present  in the thought pattern

  ab = [ a | b].  +  [ a ^ b]

Where and   are constant and immediately nearby, arbitrary, each still not null .

Out of the constraining likenings( equations) (2) output-results itself:

, that it  gives
for the Middler product of two line segments   from one another independent Monadic products ,
from which products
one kind( types lay to ground(as a basis) products for the colliding product [a | b]  ,
and  the others () lay to ground( as a basis) products for the spreading out product ab .

In Space, where the account tally of the  independent from one another line segments three wears / deports, therefore exists,  therefore the tally mark of the products of the monads, upon which the Middler multiplication is guided back  , is  like 4.

The constraining likenings ( equations) of the Middler  Multiplication become then
(a),,
(b)

But the essential centring  activity   in (a) is  like the account tally of the monads , and one can therehere those onto these guide back .

Thusly, then, the Monads of the product therefore remain, even if one still takes fromafarto (here) the "in (b) lying to ground ( as a basis)" Tally mark Monad  , these same monads remain how the originating entities remain.

This simple relating disappears( every way speeds off), by considering the fields of higher steps/ranks/stages, thus that the Middler multiplication in the doctrine of extending/ extensive magnitude , which fields of whimsical Steps behandles, no simple assigning holds by.

I  constrict myself therehere upon the space and take to my side for scrutiny, that the 3 ( as a basis) to ground laid monads , 3 like  length, to one another vertically dropped line segments are, whose lengths 1 deports.

Now I have in the Doctrine of extending magnitude(&2,50,51) concordingly demonstrated, that the constraint likenings( equations)  of the spreading out Multiplication, still remain by standing , even if one guides into place for the originating monads whimsically other entities, and (&2,330 ff.), that if a normal everyway unioning builds a representation of, that brands: they upon one other vertically dropped stand and the lengths 1 have,  

that brands [| ] = 1  exists, the constraint likenings( equations)  of the colliding in multiplication therefore then still remain standing by, if one sets place to the originating monads the monads of one whimsical normal everyway unioning.

[jehovajah]

Commentary on page 376

This page nd it's preamble on page 375 sets out the construction and design of a new product.

What is this product about?

Well of course it is to demonstrate that quaternion products are derivable from the method snd propositions of the Doctrine of extending magnitudes. However this presentation is to an audience which has studied quaternions . To someone who has not so studied the obvious question is why develop the quaternions in particular?

One must avoid the jingoism that the Doctrine discovered Quaternions, because that is not what Hermann is suing. The quaternions are a type of extrnding magnitude. As such the Doctine should be able to construct them. In addition the doctrine should give fundamental insight into what kind of magnitudes they are.

It is hard to over emphasise how deeply empirical this treatment Is. Clearly 33 years of continued exploration and refinement, research and discussion articles and memorandums etc have passed by. And so the three likenings are as a result of empirically testing product designs. As a consequence he found by trial and error that all knittings are constrained by those 3 constraining groups in various ways.

Thus it is an expectation that they should constrain the quaternions. However there is no obvious immediate way to see how.

The Middler product is pure invention and imagination, but it is the imagination of an expert in these matters.

One must be very careful. The thought pattern is designed to reflect standard "mathmatical" terminology but these forms of thought are firstly an expression of the designers goal and design principle.

The expression for the Middler product looks algebraic but is not yet, it is merely an overview of where Hemann would like to go.

We now see how he goes from this expression to a concrete design of the Middler product.

First is the fundamental distinctive products: the spreading out and the colliding in. These are 2 different types of projective products: the parallel or affine projection, and the projective projection. Over time Hermann has come to realise that both projections capture the third projection which is rotational projection.

The three likening groups are a reflection of these 3 elemental projections. (1) is that projection that is the same irrespective of order. By that is meant we can set the output result to a fixed evaluation beforehand, defining invariance. (2) is the projection in which cyclic interchange reverses the output result in an alternating pattern. It corresponds to all cyclic parallelograms as a special but fundamental group. The only interchange ( cyclic) that remains invariant is the identity interchange or rather the identical element projects itself in an identical manner no matter what orientation and direction of the entity, if those are fixed beforehand. Again the second set of likenings captures Rotation , but in a static, staccato manner. The 2 differing elements usually form a system in which one is a priori rotated relative to the other in a fixed orientation . This rotation is sadly then discounted in all subsequent and dependent synthesis and application . Thus when Cardano and Tartallia uncovered the anomalous result they were in a deep quandary , but intuitively Bombelli realised this was just a symbol of a geometric difference on the flat plain.

His carpenters ruler showed him that a bend of 90° was indicated, and he attached the to his ordinary notation for Arithmoi and called them Adjugate numerals. Later he followed Brahmagupta and set down the poetic or aphoristic rules for products, an extension of product design beyond that which Brahmagupta introduced. He also named the conjugate pairs in this notation!

Product design was then stuck for about 300 years until Hermann Grassmann formalised its process in a heuristic and empirical methodology. The intervening centuries were not fruitless because Hermann gathered and feasted on the strange berries produced by it. This he utilised to write his conception of how all these meanderings actually form a kind of consistent expertise!

(3) was the strangest and most beautiful projection of all. The cyclic polygons were captured by this projection in which not only were parallelograms fundamental to the design but literally any convex polygon regular or not! This type of projection combined projectile and rotational projection, but not only in a fixed staccato manner, but also in a flexible and free manner which later was able to become a fluid and dynamic manner for dynamic projective rotational systems.

This is one of the most beautiful thought patterns of Natural philosophers, first theoretically described by Cotes and DeMoivre collaborating under the supervision of Newton, whose Multinomial theory, including the binomial series expansion was so far advanced that even today he goes beyond most " mathematical" minded explorers. On the strength of it DeMoivre was let into the Royal Society for astonishing solutions to Multinomial equations! The joke was that Newton had published these results in a tract tht nobody else read or if they had they certainly had not understood!

DeMoivres devotion to Newton did not remove him from the vicissitudes of life and historical movements, but it did provide us with a remarkable insight into the depth of thought Newton had shared with his acolytes, and how there was too much for even his great powers of calculation fully explore. Thus devotees like DeMoivre did groundbreaking and extending research under Newtons direction.

Few realise the centuries long effort to metricate the circle by the chief to diameter ratio of the ancient greeks, nd how the Persians carried this forward based on the Indian methods which used the half chord and the half diameter. This came to be called the Sinus or " cove/ pocket" ratio, as a mistranslation of the Hindu for " limb" ! This great work produced many interpolation formulae of greater and greater complexity. These became the basis of a difference calculation which was the underpinning of much of Newtons concept of Fluxions and fluents. These insights along with many geometrical innovations are published in his works, but are buried by the amount and subtlety of them.

DeMoivre because of his devotion went carefully over every scribble, and for that was rewarded with a lifelong friendship with Newton and great intellectual assistance. Thus when he glimpsed the relationship between the sines and the Pythagorean triangle in the unit circle he was in a position to use it almost mystically to solve "impossible" equations by looking up the factors in the published sine tables . However more importantly for Cotes De Moivre was able to share his insight so that Cotes was able to present a solution to the Rhumb line problem vital for navigating on a globe in a ship with only a pole star, a sextant nd a lodestone compass!

Cotes and DeMoivre then collaborated under Newtons supervision to set out the fabulous Cotes-DeMoivre theorem for the roots of unity, and then subsequently Cotes revealed the Napierian logarithmic version of the Cotes-Euler equation. Euler set out the exponential version some 70 years later!

This fascinating story is retold here because it usually is treated in the context of the history of polynomilals especially infinite series. However it represents a substantial advance in product design in the guise of polynomial equations. Once you realise this you can grasp the generality of the product design process Hermann is laying out here. (3) represents the product design based on the quadratic roots of unity, that is the degree 4 roots of unity.

I have digressed enough, but realise that is not a degree 2 root but a degree 4. It is based on a quarter arc and relates thus to square forms, and all squares are cyclic. Thus this is now the standard representation of a quarter arc rotation, and fractional sum combinations in fact give us any arc rotation in radians nowadays. This constraint is the most highly rotationl of them all, it has turned out.

Hermanns research had indicated that the spreading out product was constrained by (2) and(3), thus this represents the most highly rotational constraints he had yet uncovered.

He also had determined that the colliding in product was constrained by (1) and (2). This was therefore the most directionally fixed constraints of his method. Thus his expression , the Middler product design brief is really statin that some fraction of fixed orientation and direction has to be combined with some fraction of incredible rotatateability!

[jehovajah]

Commentary page 376 continued

Having determined the design mix he naturally would pick(2) first! It combines a fixed identity element with a cyclical rotateability.

The application of the binomial expansion to determine the independent elements is a crucial design stage.

This is perhaps the most difficult conceptual element even though it is relatively clear how to calculate. The question is Why?

Why go into combinatorics?

I suppose the design of a product, 2 things written in a string or sequence is the easiest link. When you make that link you are no longer in the real world. You are fully in the formal mental world of sequence patterning. In that world however one must be very careful not to be constrained by the surface on which one writes the symbols down in a sequence!

A sequence essentially is a copy of a dynamic referencing of points in space. Thus almost inevitably sequences have to be assigned to directions in space, and this is the ignored link between combinatorics and permutations and spatial reference frames.

Because Herman starts with the line segment in his construction of Geometry, all sequences are naturally directional in space and on the plane . This was a huge conceptual advance on his colleagues who were directed toward the point as the starting concept.

Now a series of unique coincidences occur making(2) the natural choice. The first is space is usually described by 3 dimensions and (2) Combinatorially has 3 independent sequence patterns. Thus those 3 patterns match the usual concept of an axis with bothe positive and negative evaluations.
Next the identity behaves the same in each of those axia directions. Thus we can set property (a) to represent the axes and property(b) to represent the projection result.

This confluence just does not occur in higher dimnsional stages or ranks and thus no simple rule or application or this product design presents itself except in the case wherein is set to 3. This 3 is arbitrary, but it is chosen because we have arbitrarily chosen 3 axes to measure space in.

The consequence of chosding 3 is that one now has 4 independent types, but the last type is not really independent . It occurs uniquely to each direction and is collineal in each unique direction. The combinatorial binomial series expansion suggests the 4 th type should be set to a unit monad in each direction .

Doing so Hermann points back to previous results he has published, does not alter the basic structure of the system, but the projection , a vertical one, has a consequence for the projection product.

Do now all these constraints are In place , we now have to check if we get the quaternion product results using this constrained product.

I note that Eigenthümlich is used by Hermann regarding the product likenings (a) not (b). This helps me clarify a vague conception I have of this label. I derived a sense of it from Augen and Tun. Augen is outwardly oriented or looking , thus Eigen is inwardly or more specifically centrally oriented. Tun in older German is written as Thuen or Thun from which I link to " thum " or " tum"  in Eigentum. Now this may be etymological nonsense so please let me know if it is.

In any case the label describes a centrally situated or oriented process, like a camp fire with everyone sitting round staring into it! Which aspect of that scene is identified by Hermann was difficult to determine, but going off the concept of an Eigenvalue I related that to the central diagonal of a matrix. However here Hermann relates it firmly to the off centre entries in a matrix, that is the ones which are symmetrical and commutative or anti commutative around the central diagonal.

These ones I found very useful when designing the Newtonian triple product, because I could centre them about an origin for the axes if they were anticommutative. Thus I concord with the idea of a centrally acting arrangement, all axes emanating from a centre to which they centripetally return as the numerals reduce in the sequence.

Here Hermann introduces the notion of a Normal unioning of every direction or orientation. Again he describes it as a projection onto each other . Each line segment considered as a projection project vertically onto the other line segments. Thus it is known that there is only 1 such mutually orthogonal arrangement for any given orientation. That is called a mutually orthogonal axial arrangement.

However as there are an endless crowd of possible given orientations there are also an endless crowd of such mutually orthogonal axes. That is the real message, because each of these is independent of any other unless all 3 are given, thus fixed!

Hermann restricts himself to this endless crowd of possibilities while also carefully explaining that the unit line segments can be thought of as 1 without reference to direction without altering the system output.

While this is true regarding the output, I suspect that they play an important role in real space by configuring the actual output orientation, but that is at this stage a difficult concept to explain. Suffice to say we get identical results for every orientation so to determine the precise orientation we need these unit line segments to be identified not just evaluated as 1.

When I actually designed the Newtonian triple product I had to identify each element with an axial ray direction. By doing this I eliminated the need for a calculation/ evaluation axis , but I did have to evaluate the coefficients( the account tallies) as un dimensional sums and products. Thus unlike things if they had a product output could have their coefficients producted( multiplied as factors of this new output).

I have discussed how e•e is defined differently to e^e , but in my opinion that destroys the symmetry of  the projection products. I understand that they represent 2 entirely different focuses on the parallelogram pieces, but that is an internal mental difference not necessarily a physical one. In our geometry that construction line is a mental structure we impose to help us ratio and thus reason properly. The actual physical projection( say a shadow cast) does not fly on that construction and subsequently many physical event may not either! I am thinking of Synchotron radiation which by our assumption is not an expected outcome, but by retaining the projection process would be expected even if the order of magnitude is off our scale!

[e |e] =1 has not been properly understood by mathematicians as a likening not an assignment. In addition this colliding product is more general than the dot product which specifies only vertical projections, whereas the colliding in product can represent collisions of any orientation.

This becomes very important when we move to a symbolic formula for normal. Then we can by using the cosine of integer multiples of an angle define related but different normal representations that coincide with the Cotes DeMoivre roots of unity.

[jehovajah]

In my conception [e|e] should be set to 1e and [e^e] should be set to 2e we then get the coefficients as 1/3 rd and 2/3 rd  for the Middler product.
 
It has also dawned on me that the spreading out product and the colliding in product are not the fundamental product processes! I now understand from the general doctrine of thought patterns that the fundsmental primitive multiplications or knittings are the subjugating knittings, post and Pre subjugating knittings!
 The two Main or important products rely on the two primitive products or multiplications. The discussion identifies thes knittings as multiplications only when the defining rules are satisfied or rather the general design guidelines are satisfied.  Clearly I have skipped over much fundsmental groundbreaking work that has established the 2 main products from these primitive multiplications or subjugating knittings.

Thus [e|e] and [e^e] are not in primitive format, as they do not show the primitive multiplications that underpin them. The primitive product forms are the constraining likenings( equations) and thus we see that [e|e] is a structure that is made from (1) and (2) and the best structure or format we have at our disposal is a  Cayley - Grassmann table or a " product" matrix . But in that case (1) and (2) contradict at the off diagonal elements, unless we accept 2 formats for [e|e] or one simple primitive ee.

What that shows is the constraints act on the circumstances. If you have only one element the multiplication of it subjugated by itself makes no sense !

If you have a second step system then one subjugates Pre and post the other . (1) does not mean commutativity but what is effectively the same , that is Pre and post subjugation give the like result.

(2) means that in certain circumstances (1) does not hold except as Pre and post subjugation of the same entity , but in a step 2 system this now does make sense.

I came across this phenomenon while writing out factor squares. Of course using numbers obscured things, but I actually realised that we should write 2 factor squares for each product because we did not specify which factor was operating on which! But because the "answer" was the same I rationalised this away as needless rigour!

How can the colliding in product be constrained by these 2 mutually exclusive constraints?

The only solution is a dialectic one, and this is in keeping with all we have learned about Hermann and Hegel.  The central diagonal is the only solution for the colliding in product, and that is how the dot product was conceived!.

However, more fundamentally, the dot product does not exist unless the 2 constraining likenings are in operation simultaneously.  That means that we have 2 subjugating systems and within either one a cyclical rotation of the labelling is occurring at the same time.

Strange as that sounds we can find it in a dynamic situation!  Crudely if we have a tank track as one system and the terrain as another system, the tank track rotates independently of the terrain but the tank is positioned by the terrain, and where the tank track moves the tank is a simultaneous combination of these 2 subjugating but independent systems! The colliding in product is where the 2 systems have to agree.

Once again the example highlights the fundamental rotational dynamics underpinning this analytical and synthetical Method.

[jehovajah]

So now we consider the primitive subjugating products for the spreading out product (2) and (3).

Again we have a contradicting set of circumstances . That implies a general independence or a notional " orthogonality" between the two constraining likenings. Strictly the word orthogonal means like the quarter knee. Directly we have used the right triangle, but indirectly we use the independence of these orientations for directions in a spatial reference frame.

We are taught that space is 3 dimensional , but that does not mean we can only reference it by 3 mutually orthogonal line segments. We can in fact reference it by any 3 independent line segments . But the important independence here is an independence like the orthogonal one. This likeness consists in independent planes!

So when we think about 3 mutually orthogonal line segments, by which many of us define 3 d, and subsequently wallis 3 orthogonal axes we deliberately ignore the planes by which this remarkable phenomenon requires to instantiate!

The quality or property of independence is thus in the planes , not the line segments within the planes . And for planes to be independent they have to be " embedded" in 3 d space . But what does 3d now refer to? To a generality or whole crowd of independent plane orientations taken 3 at a time! In the mix of these independent triples the orthogonal set of planes would be lost, but for our love of its special characteristics, the mutuality og the right angle.

However this characteristic is not unique in space as it exists in any given orientation! We only make it unique by factoring out these unique characteristics into an equivalence class. We say technically up to a homemorphism to describe this factoring process.

It is this factoring process that obscures the full rotational transformability by orientation change within this equivalent class.

This is very revealing, because space is literally filled with all these orientation tripos , quadruples and on and on once again 3 independently oriented planes are not special, because not only can there be any number of triples but even more planes can be evoked.

So what does (3) express?

It expresses an arrangement of multiple squares that connect so as to form a closed loop at at least one set of connected edges. Think of a necklace of planes as a limiting case! The planes may be connected as a closed cylinder or parallelepiped.

ee as a primitive product is a parallel projection where the pos-t or pre-subjugation is indeterminate or equally applicable. In addition, because this is occurring in a single system, it actually transforms from knitting by subjugation to direct synthesis. The result either way is an extension in the primitive direction.

In (2) this is put to little use but it is in fact a scaling up centrifugally. Nothing but "volume" changes.
In (3) however the planes arranged in this way represent a spiralling rotational wave dynamically . Instantaneously the volume changes  by a circular expansion or growth.

The spiralling rotational dynamic is by name a trochoidal one, rather than a circular one, but it is a closed loop hence the sum to 0.

Once again the two can only be combined if the two constraints can be met and that again is the sum of the squares arranged in a loop the perimeter of a shape like a triangle.

Such a shape is not normally associated with a rotation, but in fact it is the simplest rotation we can describe and its 3 points lie on a specific circle the circumcircle.

If the sum of these squares is 0 then either they are all0 or some are negative. To be negative , which means at least  one is directed opposite to a principal direction or the strange behaviour of imaginary values  must be in evidence.

We could of course speak in terms of complex numbers, but in reality we are  looking at rotations by a half or quartet turn.

[jehovajah]

Norman revisits rotation nd dilation( trochoidal spiralling)

<a href="http://www.youtube.com/v/sqlugOYr&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/sqlugOYr&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/sqlugOYr-_Q&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/sqlugOYr-_Q&rel=1&fs=1&hd=1</a>

[jehovajah]

I am now happy to accept  Normns presentation of affine geometry  at the beginning of his WildlinAlg series and in his Universal Hyperbolic Geometry.

The problems I have had in understanding hermanns work has been down to defective mathematical training, which Norman has consistently and persistently set straight. The Wallis-DesCartes coordinate Geometry developed by Fermat , Descartes and Wallis , drawing on earlier work particularly Bombelli, Viete, Harriot and others, was a reintroduction of the Pythagorean mosaic reliefs into the metrication of space..

The processes of doing this and the philosophical motives for doing this are based on a few ideal Monads , Monas being the concept of a unit. That concept of a unit is a formal one, an analogy that applies generally and severally. It allows us to start at whatever level we choose. This is very fractal and economical. That means we can master an infinite regress by thoroughly understanding a level or two.

I thusly accept that Hermanns system is based on a grid like and projective structure, which artistically would look like parallel lines. Or circles tessellating the plane, or even spirals , and these are Hermanns primitive products from which he constructs the spreading out and colliding in products..


It is an historical misfortune that rotation became discounted and then rediscovered and presented as a complex number product, and then mystified for centuries!

As Norman points out, rotation or projection separately mislead the proper intuition of dynamism . They must be indissolubly combined as a trochoidal spiral  behaviour , that is a rotation with a dilation.

Consequently many strange constraints become explicable, not as imaginary but as real behaviours that break the symmetry of our formal concepts.

The constraint that the quadrance of each of 3 squares is 0 is consistently associated with a normal unioning in which the dot product is defined on the primitive products. The three product groupings as likening groups are mutually exclusive but speak of Pre and post subjugate products of primitives, related specifically to the grid systems , the systems within systems. They have no meaning apart from such real geometrical systems that fulfill the criteria.

My mistake was to think that the innere and Aussere products were the fundamental primitive!

The fundamental primitives arise only in the context of a mosaic or grid artform . The fundamental primitive products are the spatial elements in those basic systems. And from these we build the product designs for the spreading out and the colliding in and now the Middler products or Multiplications.

In the groups (1),(2), (3) the (3) constraint is the so called dot product.(1) is Pre and post subjugate multiplication set as giving a fixed result. (2) is the so called anti commutative multiplication with dilation , but in this case it is the half turn, not the quarter turn definition.

This video on Möbius band indicates how the use of rotation was associated with the complex number transformations . Even though Gauss at the time was dubious about the Metaphysics of complex numbers it was widely explored and experimented with. Hermanns contradicting or impossible constraints represent these experimental conditions. The explorations represent the dialectical resolution of these constraints .  The dot product being like 0 covers this "orientation " issue. The concept of a line segment thus has to become generalised to include arc segments! Clearly straight line segments won't get back to 0.

Once again the Shunya symbol here is relative to the step level thus 0₂ and such a 0 may describe a loop in 2 dimensions  as a sum. While ee is ostensibly a straight line segment , if it is a semi circular arc we get a complete loop which is identifiable as 0

Thus the complex root i is an analogue for a quarter arc or a semicircular arc, depending on the proportion perimeter : corresponding traverse on diameter.

The Elementargróßen I now believe are these fundmental system primitives on which everything Hermann designs is based. In a sense the second chapter of the Doctrine of extensive Magnitude should logically be the first, but in that case the actual realisation of his concepts would be reversed and with it some impetus to study his ideas, because coordinate geometry was Reasonably well established and his distinctions would have not been understood at all!
•• a very recent translation ( 27/03/15) found in the Ausdehnungslehre thread changes some of the thoughts expressed here.

[jehovajah]

Now let us consider the solution for the constraining equations that Hamilton chose, and the one Hermann chose .

Herman chose (2) and set them out as (a ). And (b). However these do not look like i,j,k. We will see Hermann has to do some more designing to reveal the properties of the rotation in a comparable form. Actually the difference is crucial . Hermann is using a half turn to construct his product.

Hamilton is using (3), that is the dot product. The issue is that there is no way 3 line segment "planes" can be arranged to give 0₂ without the line segments lying in the same plane. You actually need 4 planes! And if you have 4 planes they cannot be mutually orthogonal. So the independence of the planes is the crucial constraint.

Effectively what Hamilton did is to ignore the conjugate of one of the factors and treat it as if it were identical to the factor!  The result is that he forced out the quaternions by brute force using e²₁ + e²₂ = –e²₃. This makes the anti commutativity reside in the design which forces a conjugate rotation to be assigned to a the same symbol as its conjugate.

However Hermanns choice is the 3 planes conical rotations with interspaced or simultaneous dilationl. The rotations are cyclical around the parallelogram planes, the dilutions occur at the orientations of the parallelograms. Because he uses only 3 planes the planes act like a tetrahedral structure about the origin.

However, since the rotation is " cellular" , that is within the plane form of the Parallrlogram, the three parallelograms have to be " stitched" together to allow for a general rotation in all space orientations. But Hermann carefully chooses the square as the parallelograms and so constrains his rotations by the edges of a cube. With dilation at the edges.

The anti commutativity in this design is thus due to the 2 elements cycling , but only one ditches sign at each stage of the cycle. Effectively this semicircular rotation is designed to act as a quarter arc rotation, within the square planes due to the geometry. Thus the result that the angle of rotation is always twice what is expected, is built into the design from the get go, and we must always remember to use 1/2 angles to specify a general rotation and a reciprocal to counteract the dilation.

Hamiltons approach was to work through the endless coordinate transformations. Because of these inter communicant general relationships. Hermann was able to work directly, but carefully from the line segments . It is known that Fter reading the Ausdehnungdlehre, Hamilton strove to rework his approach from the "coordinate free" point of view, that Hermann appeared to have, but in fact Hermann relied on coordinates just as much as Hamilton, but his approach was only to tackle the coordinates at the last possible stage. His general propositions allowed him to reduce the calculation load by 90%!

However the mental labour and meditation as we shall see is not reduced , but it's rigour is in fact increased.

[jehovajah]

The description of rotation has had a long history, but the crucial impetus has always been the determination of a final position given an initial position. That determination has been made by one sort of calculation after another based on Data, that is pure measurement by some agreed monad.

Monas is the idea that we can count in ones , to enumerate, and evaluate a magnitude. The next step is to Record these counte of monads. The step after that is to mull over the records to find patterns of synthesis and or analysis, and then to find patterns of factors as the whole is analytically divided by monads of differing greatness and least ness. .

Greatness and least ness finds its apprehension in the notion of quantity of Magnitude, while magnitude remains an experience of intensiveness or extensivenesses.

The use of small pebbles or stones to keep track lead to the notion of the enterprise being a calculation ( calculus being a small pebble), and this tautology becomes further obscured by language changes until we have multiple iterations of the same basic idea, each " almost self similar" each a tautological fractal.

So we moved from the great Astronomical recordings to the recording of everyday commerce and land acquisition , and the rotational root of our methods of calculus became forgotten, ignored and set aside. But before this a flowering in spherical trigonometry and geometry fruited in many circular , tautological relationships of great beauty and practicality.

In the west the drift toward a rhetorical aphorism called algebra by the Arabs, based on the aphoristic sophistry og the Indian Astrologers gained a foothold and clawed its way up past the artistically relevant arithmetic, the love of surveyors, architects and great artists while commerce made use of its calculation s to establish trading weights measures and norms.

Algebra threw up a curious magnitude , more curious than the non commensurable quantities of magnitude. The result was almost 600 years of confusion. Today we recognise this quantity of magnitude as a symbol of rotation of orientation!

One could say Rotation of direction but that begs the question which direction ? Orientation is a unique experience and we have recorded it by means of our compasses for millennia. We have recorded and marked the changes in position in such detail that the form a circular arc when drawn out in the night sky or on the ground. How could we have forgotten about rotation in our Algebra?

We have the simple protractor, the map compass directions, the polar coordinate system, the vector plane or space system and more all doing the same tautological job of recording rotation of orientation.

And now, through the work of Newton DeMoivre Cotes Euler , and Cauchy and Argand and Wessel  we have the definitive notation Introduced by Gauss, but extended and fully developed by Hermann Grassmann ( without collaboration or prior knowledge of each others work, except where JustusGrassmann may provide a link.).

Utilising the Wallis Descartes coordinate-ordinate system as a basis of fundamental elements, employing the Pythagorean right triangle trigonometries as an analytical and synthetical roadmap or compass direction Hermann sets out the most comprehensive heuristic Method to dialectically combine all these tautologies into a whole. And he uses the unit circle parameterisation to do this.


But there is another way to set coordinates of reference in the perimeter of a circle, and Norman demonstrates this as the fundamental rational parameterisation of the circle. So this provides another route to the goal of a lineal algebra,the rational basis that avoids incommensurables. By using quadrance, the product of these line segments arranged in the plane normalised by the quadrance, but to be conceived later as in a spherical space, he provides an analogous foundational basis for the Systems of the Ausdehnngdlehre. And he provides the rational parameterisation also.
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[jehovajah]

Exponentiation has always been a puzzle for me, until now.

One is simply taught to accept it as some " real" aspect of number rather than a fundamental attribute in space.

The point is that counting is so sublime a song that we have forgotten its power to conduct our thoughts and emotions to other placed and times and magnitudes.

The usual exposition is based on aggregation and it's counterpart, or intercommunicant associate disaggregation. But these are figures of speech. For it is my awareness that eventually perceives and names aggregation, and in so doing immediately defined the intercommunicating disaggregation. It is my minds perceptions that are intercommunicating, conducting my awareness from one aspect to a related aspect of the same perceptible entity.

So my aggregation of disparate things can be recast as my synthesis of analytical resultants, and my counting response is the simplest way I cn interact with the space in hich this all " happens".

The response delivers perceptions of symmetry and relationship, but also imperfect or " almost " similaritie,s of the notion called artios and perisos in Greek.. And with that perception a whole fractal pattern develops based on scale, and level and structural arrangements in space.

Each of these experience we simply respond to by counting, but in our Sumbola, the symbology we use we cn represent these levels by Exponentiation!

By raising the counting symbols above relative to each counting symbol we can represent the different levels at which are thoughts are conducting their enquiries succinctly.

But these levels are based upon the concept of " multiplication"!

But now I say that the common meaning of that word is flawed. I do not repeat my reasons here, but move on to the notions of Hermnn Grasmann who realised that we have much to learn bout pupil able and workable processes to ebpnriches we might like to attach the label " Multiplication".

Based on his fundamental extending magnitude, the line segment, and his propositions for systematic knittings of such arbitrary magnitudes I surmised the difference between a synthetic knitting ( with its intercommunicating analytical knitting) and the immediate next level knitting process relating to these thusly " lower" level knittings . The motion of post snd presubjugation was set out in the manner of these knittings, the thought pattern clearly written own but the applicability yet to be determined clearly.

Because I have skipped many years of laborious rigour on his part I have yet to go through in detail chapter 1 of the 1844 Ausdehnungdlehre, let alone the complete mature work of the 1862 redaction, and many articles, essays and papers , addenda and footnotes that he utilised to continually develop his approach on an empirical basis. So I do not know if his knowledge of the exponent and the method of Exponentiation is based on the same sympathies as mine, yet.

Many have, through the dot product defined e•e =1 as existing., but I define it as e•e =1e. This is because I define the primitive subjugation processes , the primitive product processes as projective subjugation processes, parallel, perspective and circular, etc. consequently ee gives an output result of 2e. However the colliding in product is a fundamental primitive that outputs as a result [e | e = e as existing, and so is equivalent to the dot product in my. Design.

The spreading out product is usually left to be defined in situ but in my design it is based on the 2 e output result. Thusly I realised that e^e is the primitive subjugating projection ee or e²

On the face of it, symbolically we have a square multiplicative process bring equated to a sum recess! We cannot object because in fact multiplications are summations . But we might object in ignorance protesting that yes 2x2 = 2+2 but that is a special case. In fact it is no case at all! Numbers do not dictate the underlying subjugating projection! Orientation of the subjugating projector  does! In this case the orientation dictates that all exponents must obey the synthetical knitting rule. Exponents only take on their " usual role" when the orientations differ. But thrn the labels differ and appear to be simply quadratic or hyperbolic forms ( thought patterns) .

We learn that numerals do not dictate the knitting proceed out come , orientation does.

This becomes very relevant when the line segments are arcs. The exponents then imply represent the in arc curvature synthetical knitting, and we can naturally use exponents to describe circular extensions!

In kerping with that i as a rotation is a circular quarter arc and thus i² is a semi circular arc extending in the same synthetical way . However ij now becomes apprehensible as a Shunyasutra, a curved parallelogram form in 3 d space, because the orientation of j differs from i.

Similarly I may now combine arc segments of differing curvatures to form a Shunyasutra , again because the subjugating orientations differ.

The power of the motion of a subjugating system determining the notion of multiplication, and to what we attach the label" multiplication" I hope is clear and remarkable. I hope also that exponents now have a clear geometrical interpretation as organising and ordering symbols revealing the fractal level or step of a system.

When we developed and worked on the Mandelbulb discovery we tried all manner of 3d reference frames . It is thus not the exponent of the product design that is crucial, but the orientation of the line segments , or arc segments.

Also we noticed considerable simplifications when we moved from straight line segment reference frames to arc based reference frames.

However the computing power requirements increased exponentially, and this was because our number system becomes compute intenive!

Maybe if we replace the Cartesian arc analogues by actual curve analogues like Bezier curves etc we might reduce the computational load significantly.

While I love the exponential form there must be a less compute intensive version of it.

Maybe Normans projection of the rational parameterisation of the circle is the answer?
e(u:t) is identical to [u²– t²/ u² + t², 2ut/u² + t²]

Compare with
e(it) = (cost,sint)

[jehovajah]

Although Normans special thought patterns have arrived at the above result for the plane , using the projective point or a sloped  extending line  segment emanating from [-1,0], the formula is identical to unit z= z²/|z|².

This is the same as unit z= exp(i2ø).

The main advantage is in the speed of calculation for the computation and the similitude to the exponential  function pattern. The implemented formulary will be different because truncated infinite series will be avoided, but the intuition about rotation as a curve has to be subtexted by the notion of a chord arc structure.

Chord arc structures are very interesting. The desire to " measure" the arc by rolling it out on a straight line is fraught with technical difficulties. The straight chord or secant have interesting relationships with other straight lines in and through the circle, which make certain direct comparisons of arcs absolute and invariant. This these parametrizations are good analogues of position referencing on the circle perimeter.

However the distraction from rotation is palpable. Curved or trochoidal motion is so natural it seems wrong to use anything but the arc pertinent to a curve to describe its behaviour. In fact it is mentally crucial to strive for this because Ampére and Örsted were met with incredulity when they proposed a circular force and acceleration for magnetism! Even today many still find it hard accept such forces and accelerations account for magnetism and gravity.

[jehovajah]

Their is a fundamental difference between measuring and counting, but the 2 processes are indissolubly inter communicant.

Measuring is fundamentally our non verbal response to space and all that is contained therin and constituted from. As such we move and compare whimsically chosen object against arbitrarily chosen object. This is the fundamental act of Ratioing. This is, according to the ancient Greeks, and perhaps sages before them, the real difference between the human and the rest of his animate neighbours. This act, which can be observed spread throughout the animate kingdoms draws forth the distinct Logos, the expression articulated and enunciated  inter communicating with that action.

That logos or expression of a ratio of comparing things is objective evidence of rationality, and is and was thought to be a divine gift to humans.

We now know tht our animate colleagues respond in similar if unexpected ways. Rationality is not a uniquely Humn trait.

However the articulation of the intercommunicating expression of that rationality has lead us to the development of languages. And the fundamental aspect of those languages is the intercommunicant distinct sound that refers to a distinct comparison.

At some stage the ever iterating patterns of spatial patterns were expressed by intercommunicant repeating patterns of distinct sounds. These patterns were are evanescent sequences that recorded patterns of dynamic iterations. Whether these were flows or discrete disjoint events , or rhythmical and poetical dances of motion and movement, fathomable variations in intensity, these became the great song sequences and cycles, which we have in common with the reptilian animates, the birds.

The memorising and passing down of these sequences did not preclude innovation and redaction and reworking. Finally civilised humanity derived an intercommunicant set of pictographs to represent visible objects for which " common" sounds were well known. Gradually these pictographs became organised in the general song cycles and sequences and the aesthetic musical notes and tones guided the phonetic semanticisation of certain pictograms which became organised into syllabaries.

The syllabaries and the song cycles had always been used as standard vocal sequences, and these sequences were always used inter communicantly with objects. Thus " counting" has a coeval beginning with the development of the song cycles, but with the development of the syllabaries definitive counting sequences were formed, well as definitive alphabets.

The progress was never smooth as was to be expected. Abjads and other phonetic syllabric systems are evidenced in the cultures, but it is imperial conquests that established systematic alphabets and counting sequences over wide areas, as well as imperial language norms.

Thus our counting and measuring have developed in an intercommunicant way.rhis means measuring can be seen as a pure matching process which is one to one and onto, and, importantly, projective in space, both external space and internal mental space.

What was not clearly expressed in western renaissance was the projective constraint of the measuring line. This constraint means that if a chord is stretched out , any whimsically chosen objects may be sequenced beside it, and given its own intercommunicant cord segment . The cord thus measures and can be used to record any whimsically chosen sequence of objects.

But once a segment of that cord changes direction it completes the measurement of such a sequence and begins to project that sequence iteratively in the new direction!

Thus we may now iterate or repeat that sequence against the cords new direction and in this way arrang objects in sequences we call arrays or rank arrays.

If that cord again changes direction we can arrang objects against it but they will not be iterations of the previous sequence if that direction remains in the plane.

Thus our measuring is intercommunicating with our ordering and sequencing spatial objects in oriented directions in space. Counting is our way of keeping track. It is thus an intercommunicating tally which we use to reference where or how far we have come or gone in a sequence, simple or complex( that is convoluted).

The words we use for counting, the counting name are represent able by numerals, but the counting arrays themselves are called Arithmoi. It is these counting arrays or rather arrays of sequenced, oriented and directed arbitrary objects that we have by slip and error, mistranslation and innovation come to refer to as " numbers".

And then we went crazy and started using the numerals attached to a geometrical line as a definition of a number! And then sets and then some abstract meaningless logic statements and relations.

It all founders however on the notion of multiplication. It is this great iceberg that sinks the Titanic of the modern misconceptions of number.

In any case we do not need to be deceived any longer. We must return to the freedom of the spatial,oriented and translated arrays called the Arithmoi, or mosaics.

We shall see how Hermann evokes these Account Tallies from spatial objects and systematic processes.

[jehovajah]

The Place of Hamiltons Quaternions in the Doctrine of Extending/ Extensive Magnitude

By  H. Grassmann.   In Stettin
--------------------------------
Page 377
There, therefore, by considering this varying of the monads, the constraining likenings of the spreading out multiplication also remain by standing, thusly remains also the Middler product, as set together out of the spreading out product and the colliding in product , by considering this varying of the Normal unioning into one other ( normal unioning) as unvaried .

In the guided to us off-printed handling ( Crelle Bd. 49 S 131  ff.)  I have concordingly demonstrated this  everyway unvarying quality for all entities out of the 3 likenings groups (1,2,3) derivable multiplictions,  also therefore  unrestrainably for the Middler  product.
 
Thereupon, It comes time now to guide back  the 3 products e₁e₂, e₂e₃, e₃e₁  onto the originating Monads,

also this is already fulfilled in the Ausdehnungslehre(&2), where e₁, e₂,e₃ are apprehended as "outwardly completing" entities  from  e₁e₂, e₂e₃, e₃e₁ , and

5)


are set, and where the Stroke | is the sign for the "outwardly completing" entities, and comes to be set out ahead of everything else that e₁, e₂,e₃ build  a representation of Normal Unioning.

Then further comes to be firmly set for a any whimsical  line segment , which out of the originating monads, through the tally marks ( coefficients) is derived , that its outwardly completing entity let be out of the outwardly completing entities of those monads, through the same Tallymarks derived

 therefore

(6)

let be, and it is concordingly demonstrated (&2 37 ff.) that the same Relatings remain bystanding , even if one sets in place of the originating monads the monads of a whimsical other Normal unioning .

With the aid of these  labesl we can now the fundamental likening (4) write in the thought pattern

ab  = [a | b ] + [ab ]

Where and are constant tally marks.

Varying themselves,   and in like everyway holding pattern, to the by considering game, around , thusly varies itself the product only around the same tally mark factor, therefore its character remains concordingly un everyway varied.

We can therehere, without essential varying , one of these tally marks set as 1   . We set = 1. Then  we appoint therethrough, that  each   Middler product should be out of 3 factors  by the rule  of the unionability ( by the Associativity principle) underlying, that brands abc = a( bc) should exist.

This will come to be fulfilled,  if it empowers for the Monas products of the Middler product.

[jehovajah]

Commentary on page 377
I have some LaTex editing to do , but the drift here is clear. A Normal unioning is considered, where Niral here means normal . By Hermanns time the normal Cartesian coordinate system was the Wallis Descartes mutually orthogonal union or joining of 3 axes.

The point that he makes is not remarkable in his time, because it was well known that Descartes did not establish orthogonal axes, his Methid, today called generalised coordinate systems, was simply to establish any pair of fixed lines( both angle and length) as a reference set in the plane. Thus every whimsical notice of that pair would give the same result, but one pair was noticeably easier to work with than any other and that was any pair which were orthogonal to each other . This it became the Norm to use this configuration.

The setting out of the problem in terms of these 2 was pioneered by Descartes and De Fermat, but we have been shielded from these historical facts.

So what was new up to this point? On the face of it very little in character, but in detailed rigorous empirical presentation and deriving of heuristic propositions , I do not think there was ever any other who set it out more clearly, except perhaps St Vainant, who no doubt had a claim to explicating the method of the Great French philosopher, but did so After Hermann and so his claim for primacy was upheld against St. Vainant.

 The more I translate the more I see that Hermann preserves in a kind of time capsule the best practices of his day, but whereas no one else bothered to survey the whole of current mathematics in order to make sense out of it, Hermann did and found a set of principles and derived a set of propositions that would give expertise to any one adopting them.

Thus his hard work was educative in principle. He wished Prussian mathematicians to take there place confidently and competently at the forefront of Mathematical and scientific development. By using his " Keys" to borrow St Vainants word, they would quickly unlock the door to that opportunity. But they would not go so by apeing the best mathematicians of their day. They had to have an intrinsic apprehension of how the best achieved that status!

Hermanns claim: follow my Förderung and you will be essentially there!

The 2 books thus stand as a repository of the best thinking of the best of times and should be studied just for that aspect alone never mind the numerous gems embedded in the material.

What is uniquely Hermanns then?

I would say that the whole effect is Hermanns, and that is greater than the sum of its parts!  I would say, by the time one has ploughed through these pages, visited those thought patterns, that one will emerge considerably enlightened about the process of creating brilliant mathematical and physical and scientific models of processes in space. That is the gift of studying this work.

In a modern sense I see that exposited in Normans work, even though he takes some different starting elements, he has managed to apply many of Hermanns principles, that is the thought patterns of the great Astrologers, philosophers and Mathematicisns of Hermanns time.