END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

How do we reference curved lines( and ultimately surfaces)?

The answer Grassmann style was in fact given millennia ago by Apollonius! We call it polar coordinate geometry , but Apollonius stated that one of the initial extensive magnitudes must be a circular arc segment. The other can be anything else ,but he and we choose the direct line segment families as the second for the plane .

In the 3 directional space, the third stage we can choose a third element as a circle, a curve or a straight line. However if we want to specify cylindrical, spherical, toroidal spaces we have to specify more than 3 elements. What that means is that we are already in more than 3 dimensional space when we do. Because we do not understand the notion of the basis elements we mislead ourselves about the dimensionality of space.

Now it is possible to use trig line segments instead of an arc segment. But to declare a trig line segment as a one element family is clearly misleading ourselves. The sine ratio depends on a 2nd stage system to be defined. Thus we are de limiting a first stage system by a second stage system, making the behaviour of the whole a 3 step process reduced to the second stage system whose characteristic is a 2 step process!

By using an arc segment we restore the 2 step process to this second stage system. As we go into space the sphere maintains the characteristic process fie a third stage system while the trig line segments multiply the steps to get to the same result. Increasing the steps in this way is akin to increasing the directions required to attain a point, and thus that is an increase in dimensions required to attain a point in that basis.

What I am referring to here is otherwise called degrees of freedom. To use trig line segments we require more degrees of freedom in our thought pattern. We do not require more extensive elements in the basis, but we have to use the other degrees of freedom referenced by that basis, even to lay out the calculation on paper.

This calculative axis or axes played an important role in Hamilton realising the Quaternions as a rotational system . He could not mentally grasp how to do it until he allowed one of his degrees of freedom or axes to be purely for evaluating the calculations . This is why ijk=–1, the 4 th axis is used to evaluate the other 3 axes( as products of rotations).

It is a strange thought pattern which we must avoid, because Hermann has given us a better one. These products can be described as arc segment extensions where I,j,k are arc segment extensive magnitudes.

Returning to Apollonius, using one of the initial elements as the circular arc segment in the plane he was able to exposit the conic section curves. Norman Wildberger, inspired by both Hermann and Apollonius et al. has produced a wonderful series called Universal hyperbolic geometry which I can recommend to you.

Commentary
Well I nicknamed it the Schwenkungslehre, but Hermann refers to it as the Schwenkungsbewegung!

The second stage system can also naturally model the behaviours of the circle!

Thus the notion of a system varying within a system that also varies, both free to vary differingly, gives rise to 2 " step rises" above the elementary initials. That is there are 2 stage 2 cognisances Hermann can draw attention to by his system within a system varying structure.

Be warned: this finishes the "elementary" treatment! After this second exposition it get hard!

Commentary
The quest for precision has been buried under the spurious creation of the real number line concept.

Engineers and scientists require precise enumerations. This arises from the need to know the trigonometric ratios precisely for navigation and astronomical purposes. Later, as engineers and chemists determined other physical ratios , the need for precision grew. Precision engineering and exact Sciences became names synonymous with the most skilled application of our discernment and knowledge across a wide range of deliverable services and industries, but few realise that this was the result of a millennia long campaign to calculate the trigonometric and latterly the logarithmic cannons or tables.

The military application to ballistics and also to air flight and eventually space flight are also cases in point. Without Astronomical trigonometry none of these things would have been possible.

It is this creating cognisance, and very much also creative cognisance, that is trigonometry that Hermann introduces here in this second natural 2nd stage and 3rd stage and eventually nth- stage systems. These are all systems within systems within systems...n times or n statements.

These. Are the rank arrays that Hermann clearly imagined, but did not notate by matrix notation, because he did not invent that notation,Cayley and others did. But they only invented the notation, Hermann conceived the content that needed the notation. That is not to say he is the prima facia inventor or creator, he clearly drew together the work of his contemporaries in a way no one else did before him, and definitely before St,Vainant who either came to similar conclusions or more likely plagiarised what Hermann had written seeing that no one was responding to the ideas contained within his first volume.

It was well known and common practice for French mpnobility to plagiarise and use their wealth and status to increase their intellectual standing. But Hermann was on guard for this and successfully defended his primacy against St.Vainant, which is unusual. But Clearly St. Vainant could not carry it off as the inventor , Hermann had put too much detail in his work for anyone but him to be familiar with all it's content!

So this second aspect of the doctrine of extending magnitudes is trigonometry Grassmann style, but it combined the incredible works of Euler,Lagrange,Gauss,Laplace and Newton, De Moivre and Cotes,Fourier into a Phenomenon! Out of this Fouriers students developed the Fourier series and the Fourier transform, clearly independently of Hermann, but not embedded in such a rich Doctrine of extensive magnitude as Hermanns. In fact it is the exponential form that Hermann elucidated that is the real beauty of the Fourier Analytical method. While Fourier was defending his assertions Hermann was creating the systematic framework of systems within systems... for its exploitation.

http://en.m.wikipedia.org/wiki/Joseph_Fourier#The_Analytic_Theory_of_Heat

<a href="http://www.youtube.com/v/xYPw2gY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/xYPw2gY&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/xYPw2gY_3PI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/xYPw2gY_3PI&rel=1&fs=1&hd=1</a>

Ausdehnungslehre 1844
Induction
D. The Thought Pattern of the presentation.

The centrally active Thing of the philosophical method is: that it advances in contradicting statements, and thus reaches from the general to the specific .

The mathematical method on the other hand  advances from the simplest labels to the combined set ones, and thus  wins new and more general labels through a tying together of the specific label.

Therefore, while there (placed where it is) the Overview  highlights beforehand over the Whole work, and the Development plainly consists in the any time everyway pulling apart and limbshaping of the whole , 

here, the "on one anothe"r binding of the specific entity henceforward dominates, 

and each "in itself ending" developing rank array only builds together again a representation of  a  limb for the following  everyway binding, 

and this difference of the method lies in the label entity., because the originating  entity in the philosophy is plainly the monad  of the idea , the specific system  is plainly the derived entity,. On the other hand in the mathematic the specific system is plainly the originating entity, on the other hand the idea is plainly the last entity ,the aspired to!: 

wherethrough (all of the preceding) the "in opposing established" Advancing is given shape.

Describing the Products and Summation in the Ausdehnungslehre



Beware that you do not perceive the difference that placing geometry on the real expertises basis makes a fundamental difference to what operations are allowed to represent geometrical relations!.

I struggled for a great while with how ab could be analogous to–ba if they are all true extensive magnitudes, not just formal thought set ones. In fact I did not even know the difference!

To be a " true" extensive magnitude the extending must come out of the basis element, thus a extended to á must be one continuous or contiguous whole.. If the extension is governed by a different rule, that is a second direction of varying, it is crucial that it remains contiguous. This is where the rule of placing vectors tip to tail comes from.

So now I can understand "auseinander treten", and also " aussere", it means that the extensions must be continuous or contiguous and out of the basis element.

In which case how can you switch them round? Either you must rotate them cyclically, while they remain contiguous, or you must make the extension the basis element, in which case the first extension must extend in the backward direction to its original direction. . Hence in both cases we get a " negative" or contra extension to the original.

Well, you might say , if the original a must extend backwards, surely the b must also be " flowing" backwards to support this extension?  That is an important physical observation which Hermann was not thinking of at the time, but can easily accommodate. Herman was starting with the b direction as originally defined or established( a formal modelling requirement) and extending that basis element according to the original a direction( rather orientation). He is thus obliged to extend in the reverse,backward or negative direction.

It is also clear why Hermann separated Length and Direction as rules to be attributed to a line segment , stuck to them by glue!. His analytical method makes all these distinctions clear, but as he says in the Vorrede, they can be analysed further to improve the application.

There is only so much a single man can do!

Describing the Products and Summation in the Ausdehnungslehre

The summation sign also now cAn be given its exact meaning. It is the fundamental system process of combination or composition. The extensive magnitude rules are constructed into systems, called ranks or steps or even stages. Each stage is ruled by some simple law or set of laws. So the order in which these rules are obeyed is important.. Using the sequence of combination starting left and moving right as a process instruction, the extensions can be carried out in contiguous or continuous order.

The + ign is a natural mathematical candidate for this combination or composition. However, in this use it does not mean aggregate from every direction, it means aggregate in sequence order from left to right.. Again aggregation is the wrong concept, it actually means extend continuously in the order of the combination obeying the rules governing the initial elements.

Within this fundamental practice other rules are installed to ensure the arithmetic operations of the Tally marks  remains in the familiar mode.

Of course the natural meaning of = changes to be consistent with labelling and assignments,

The requirement of Equality or duality imposes certain conditions on the summation statements. Thus If a,b are differing initial extensive elements in a 2nd rank / step system, then the system of a is primary, the System of b is secondary. That is the extensive families are in an order. a + b as an extensive Process  is different to b +a in which the primacy of the families of extensive magnitudes are switched.

But if we identify a point by our second stage system, that is in space, it really does not matter if our system changes in any way. The point in space is the fixed thing.  So I could change the primacy order, the initial elements and so forth and should not expect the 3d point to change because of that. The geometry is now on the real expertise basis not the formal one.

So using that point in 3d space , the "real" space , I can ground my formal systems of extensive magnitudes, or the gathered( summed) totality as an n- step/ rank/ level system, (or some say " bind"  it) to my empirical situation. Now all directions, absolute lengths , continuous and contiguous extensions are defined from this real 3d point.

a has to be specified by this point or a point derived by this system in a system. This means that I have grounded my system and applied it by essentially establishing a cage or framework of parallel lines , planes and spaces , all specifiable by unique points in real space relative to this initial fixed point in real space.

This initial real point is called the origin or O. It is NOT a point in the system. It is a grounding point in real space. The confusion arises because this clarification is never made.

When your teacher starts by saying pick an arbitrary point on the blackboard as an origin, he or she should also say that this is a real point, it is now the centre of our universe by a formal decree!

Of course what they do say is that this point is completely arbitrary, but that is not the same as distinguishing between real space and the formal model.. For example , in real space the horizontal axial family is actually running in the curved surface of the earth, thus it is an extremely large arc segment with an extremely large radius. It is not a straight line segment. Consequently our model universe is already at odds with real space!

Now consider some other assumptions we make: because the reference cage is the same at every point in the formal system based on the one real point we assume that the rule established in a local frame applies everywhere. Thus we unjustifiably assume universality of our formulae. It is our job to justify that assertion any and every time we reach a point in real space that is modelled in our formal systems.

In fact Newton was not so deceived. By establishing a spherical geometrical model of space, based on a real local point, he in fact took pains to find out several real points on earth where he could ground his model. He and his collaborators chose several points after much sifting of information held within the trading empire. He also established points for astronomical observation of comets, to establish that point itself as a real point in his system.

By measuring tides, apsides of the moon, comets, pendulum swings etc he established several real points on which to found his reference framework, and thereby to carry out his calculations to see if they resulted in observable phenomena. Of course they did, and this is why his contemporaries eventually gifted his formulae with the divine Title of Laws of nature!

However, we should not be so ready to grant such a description. Indeed we are justifiably impressed by his Astrological Principles, but few have ever read even beyond a couple of pages! The principles themselves demand that we empirically justify at each point their application. Thus as we have ventured into space and back again we have learned a thing or two. They are not laws of Nature!

Now we use so called Einsteinian relativity, which is a kind of joke, because it is in fact Newtonian, and clarified over time by LaGrange, LaPlace  and Euler as Well as Cotes, and Gauss. However it is in fact Hermanns systematic method that underpins all modern physics including Einsteins. And we still have the responsibility to verify and adjust our reference frames at each real point in space!

We now have a lander called Philae on a comet. It's job is to verify our formal systems from that dynamic point in space.

If we ground our reference frame to a specific point in real space then we can identify a specifically and b specifically, and all the family of extensive magnitudes they each belong to as systems, and then we can referent a crowd of other families as systems by these initial 2 elements. The sum represents the continuous/ contiguous process required to do this, and we can start with a specific identifiable a continuously joined to a specific idrntifisbleb. However if I switch the order then the idrntified a runs into a problem. It cannot extend out of the b except backwards. So a + b does not equate to b+ a.

However, if I now make the rule that every parallel line segment will be identified by its extensive magnitude in the system and the point of interest it goes through, I can start b at the point I started a and then continuously follow by a. What I have had to do to maintain continuity of extension is to translate the start of b to the start point of a. The whole connected idea can be represented in a parallelogram.

The points of a parallelogram specify a commutative " process object"  in the system. In that regard we must clarify precisely what we mean by commutativity and associativity and how those ideas might be applied to extensive magnitude summation, and what the consequences are.

It would seem one of the consequences is tessellation of space, or fractally uniform regionalisation.

Commentary
Ansatz is in this section unpacked as "Satz anstände". It is a very general flexible way of referring to any and every type of supporting material. Thus on a map the key box is an ansatze, ant abstract prior to a major work is an Ansatz, any keynote summary or overview is an Ansatz , and any conclusion as a conclusive statement is an Ansatz etc.

Here Hermann presents 2 points of view of his concept of Rigour, which he felt was Mathematics most worrying lack, in his day, and in ours. His first view is that philosophy is a rigorous discipline, requiring everything to be judged against necessity.mbut then he realised that expertise demands the same rigour! Thus his method that he is. Promoting requires everyone and everything to be ayed out in an orderly and rigorous manner.

This was just not being done! He was anxious about this general state of affairs. That was the firs promoted mind set of this section.

The second was a hammer blow to his sensibilities: there was no overview of what Mathematics was as an expertise. Instead it seemed to progress in the most disorganised way! It progressed through blind well. Intentioned enthusiasm, and sudden insights and inspirations. Whatever you might call that process Hermann felt it could not be called professional expertise!

But now Hermann had uncovered an expertise that was rigorous, philosophical in terms of rigour, and possessing a grand overview. Nhowever the received dogma would require him to dismantle it and present it in a hampered, chained and inimicable manner. He felt enslaved by the contemporary dogma which would not allow him to bring forward the discoveries and constructions of a free thinking independent and self chosen process as they were found and how they were meant to be, in his opinion.

This was frustrating and troubling. Mathematicians were slaves , or enslaved, not Masters of their own topic!

However, if the reader would be prepared to take the lordly position, and oversee every development made by Hermann , the slave, then their reward would be a true copy of the expertise he had found, arrived at by freedom of thought , meditation and action as a lord or freeman.

Hermann thus offers to emancipate the reader from slavery to the contemporary lords of mathematics, the professoriate. However this is not a freedom to licentiousness, but to rigour and overview led development as a professional expertise, so that Everyman could stand equally as free and as brothers pursuing the same goal for mathematics, or rather the doctrine of thought patterns.


Historically, this inductive appeal hit home with Bertrand Russel, who consequently made a scathing attack on the board of English Geometers, cloaking it as an attack on Euclid.

Is Hermann right?

He certainly draws on contemporary movements in his time, but also on Hegrls famous master and slave dialectic. But what is very pertinent also is the frre school education he received in his fathers school district. Justus adopted most of the Pestslozzi school principles, which helped to form the mind and character of Jakob Steiner who attended the original founders school in Switzerland, and who was appointed to the high school in Stettin by Justus fir some time before being called to Berlin University.

The Humboldt reforms provided an exciting focus for these primary education reformers who laboured long and hard to change the old system as much as possible, and for the most part had a huge influence on preparing Prussian youth for independent, self actualised, self motivated and self developed thinking. No longer would Prussia hire in foreign expertise, no matter how great, because they would be developing their own, able to take their place on the international stage!

Surprisingly, Klein resisted this movement vehemently, despite being married to Steiners daughter. He set up the Erlangen. Project when he could, but he saw no need to overthrow Academia! In fact he was worried by the geopolitical situation and international prestige. Eventually, even Hermann modified his youthful views and so did Russell.

Ah! It was ever thus! But while it lasted the Prussian Renaissance was a powerful social and international force, and the Grassmanns exploited it better than most.