The Theory of Stretchy Thingys

[jehovajah]

Commentary
I often disagree with other mathematicians on fundamentals. In this case I feel that rotation is a fundamental transformation that then defines all other. Consequently it action has a big job to do!

When I first coined the term Spaciometry in the fractal Maths foundation thread I recognised the difficulty and thus defined rotation as an experience: it is an experience of the boundary of a quantity or form.

A magnitude is an experience of sensory excitation, so I would not define rotation as an experience of an experience foundation ally. We get a better apprehension when we define foundational concepts in as many sensory modalities as pertinent, and in every case quantity or quanta invokes bounded ness and boundaries. Such concepts are necessarily subjective, but they are basic and have easy referrents so can be communicated usually accurately.

Rotation as an experience is thus a magnitude which can be further defined adjectively. So part rotation means we see only part or feel only part or receive expchoes from only part of a forms boundary. Total rotation means we experience all of its boundary.

One of the realisations that makes the definition so tricky is the reliance on the circle or sphere. It is such a fascinating form that no wonder we model rotation on it, but it is in fact an ideal created by pragmatism. I have meditated greatly on Laz Plaths trochoid app which applies circles to the notion of arbitrary rotation, and as far as curves go trochoids or roullettes are a more satisfactory notion of rotation!

The complexity of rotation is well illustrated in any of his trochoidal presentations.

The notion of a point, a pragmatic seemeioon often is left out of the notion of rotation being considered as the centre. However experience of material points, reflection and translation indicates that we cannot remove points from the transformation. Points necessarily itate also.

Rotating about a point is generally replaced by rotating about an axis. And it is this axial concept that removes the centre from consideration. The work of Grassmann on Schwenkunglehre while not complete allows one to see rotation in terms of cyclically traversing a boundary . Whatever the path of thst boundary it is clear that to end up where you started from is a rotation! The rotation of a point as it traverses a boundary is transmitted to the tangent concept. The line segment concepts of Grassman and the ancient Greeks enable us to sum the rotations of points by summing line segments in the form AB + BC =AC , an absolutely fundamental formula for the physical world!

Thus the concepts of rotation do require this infinite geometric series approach, which can then be simplified to logarithmic addition. It is just not usual to have this pointed out in such a clear geometrical form!

As Grassmann states" whoever searches for these adopted concepts is unlikely to ever find them ! "  . He believed he had been given an insight that was from God or his Muse!

[jehovajah]

Ausdehnungslehre 1844 continued Ausdehnungslehre VorredeYet I am to me, regardless of all trouble related on the form , still conscious of  the great unfulfillment of it
True I have the complete Thing worked through multiple times in different forms, sometimes in Euclidean  form Expositions and propositions for best possible rigour, another time in a melded development form with best possible overview guides, at other times both with one another interleaved, in relation to which the given overviews precede  and then let the Euclidean form follow on.

Truly I have been of it completely aware, that by the repeated  processing around it many an improved form would step from here , that is of a bit, more rigorous, of a  bit, clearer guided overview given. But from the pervading Conviction, that I could  hope for no   full satisfaction  , and the plain fact,  the truth compared to it, i must the presentation still always only leave wanting, I myself concluded with the form that seemed best to me at the time, with the form which steps forward  arising from here.

A special reason  for hindsight leniency also thereby i hoped to find, that to me the time for processing these was by virtue of my appointed duties only most extremely  and meagerly and piece wise apportioned, and also to me my post did not present the opportunity the lively Freshness to extract,through memoranda out of the field of this expertise, or also of  related content,  which like a life giving breath,must blow  through the whole work, even if it should seem like a lifegiving arm stretched out to the organic body of wisdoms.
Yet even if also a professional duty, which such memoranda out of  the field of this expertise should become my singular work, as my  wish goal and deep yearning stand to me before my eyes,yet still I believe the processing of this expertise will not until the achievement  of  the goal be allowed to be postponed,  markedly since I can hope through the processing of this volume  by myself the way to the goal to be able to ride.

Stettin 28 June 1844

[jehovajah]

Commentary
I only just realised that when I was in a British school in Germany we studied the history of these times! Not only the Napoleonic wars but more cogently the  period of Bismarcks rearmament of Prussia and his industrial reforms! Of course we never studied the intellectual and cultural history of the times at the secondary level.

http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Laplace.html

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lagrange.html

The French Renaissance in Science and. Mathematics came through ome dark and troubled times in Rurope, mostly relaying to the Spanish kings seeking to regain control from the Islamic empire. But the Islamic empire was the one bright light in the gloom and dismal darkness of plague ridden Western Europe.

These are interesting times, with many connections. Grassmann was in a time when despite his provincial status great opportunity for advancement by merit existed.  Gauss, Möbius and Euler were out on the Ruropean limb, as was Newton, but from east to west Europe similar changes were taking place with communication becoming faster and more reliable, and printing more democratic!

[jehovajah]

Good news. I have now gained access to the article :
Der Ort der Hamilton'schen Qaternionen in der Ausdehnungslehre.

I will start a thread to translate that after this one!

I do not propose to translate the whole Ausdehnunglhre here on this forum, because I do not think that is necessary, or completely relevant.
I chose the Vorrede for several reasons, the main one being it is a fundamental introduction to Grassmanns thinking, and to his interpretation or promotion of complex or imaginary arithmetics. Similarly the short article on Quaternions will be instructive.

I will of course continue to " durchbearbeit" the Ausdehnunslehren in my blog site
 Jehovajah.wordpress.com.

As you know I love this forum and have had most of my major insights while posting here. Also, I have said it before: this forum is the most important forum on fractal imaging on the web in my opinion.

Long may that continue.

[jehovajah]

Commentary
 Having seemingly started out in a bold fashion, declaring great clarity and inight in what some might view as arrogance, Grassmann finishes with a tale of Woes, anxieties and dissatisfactions!

Last few sections have been discussing his struggle to find time, peace and muse to properly process and edit the work he was publishing. Even up to the last minute he was making modifications, worried whether he was rigorous enough, whether his overviews were good enough. But in addition he admits to be struggling with known imperfections in his work.

As a publisher and editor he wants his book to sell. It must then appeal to a target audience. It seems he was undecided who to edit and format the text for? In addition to that the applications of the method were so numerous and deep that he felt hopelessly inadequate. But he was a fighter. This first volume for. All its shortcomings and compromised decisions, compromised by lack of quality time and opportunity, he hoped in hindsight to have demonstrated its truth and importance, and thus to find a way to progress to the completion of his plan

Although he has no direct appeal for help, indirectly he begs for it

What follows gives him nearly 15 years to perfect his system almost singlehandedly, and the help in the processing, editing and presentation of his work by his brother Robert. For at least 2 years they laboured to redact the 1844 texts and the additional texts Grassmann had set apart for the second volume,

He had 17 years to perfect his system , because no one in Prussia was interested enough to buy his book.eventully he burned most of his first print run!

[jehovajah]

Commentary

<a href="http://www.youtube.com/v/2t61M-3jniQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/2t61M-3jniQ&rel=1&fs=1&hd=1</a>

This video relates how we think and frame "law", and draw theories and apply them

[jehovajah]

<a href="http://www.youtube.com/v/ZYodF7ikj5g&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ZYodF7ikj5g&rel=1&fs=1&hd=1</a>

This is a remarkable insight into the nature of geometry of the line and the circle! I have taken and recommended Normans course on universal hyperbolic geometry, but this is the best description of duality I have ever seen him give. To me it represents the deepest insight into nature of geometry, and as yet I cannot trace its equal in Grassmanns original work.

The explanation of bilinearity is at its clearest here, but his thinking I would distinguish as bilinearity!

Notice how he highlights the circular radial values of 1,0,-1.  These values pervade the Clifford Algebra of today, but I am sure that Hermann used only 0 to characterise his products, deriving 1 or -1 as particular constraints when designing applicable algebras. I will know more when I translate the article on Hamiltons Quaternions..

Perpendicularity is a key concept but you have to let it go into the general dual form Norman reveals if you want to understand how your subjective mind processes space! Things which are locally perpendicular rvertheless seem to meet at a distance beyond our horizons or on our horizons. We know this is not anywhere near infinite space, yet our reasoning makes little discernible distinction between this vanishing point on the horizon and some mystical point at " infinity ".  In fact the vanishing point behaves like any conceptual point at infinity.

We have to acknowledge our limitations even at the level of subjective processing. We cannot really imagine " nothing" neither can we imagine " infinity". All we imagine is discernment disappearing outside of the sphere of our view. Beyond that, as they say " there be monsters!". However Aristotle decided to populate it with mythical entities as inversions of our own selves. Religions populate it with good and bad entities . As an adult human being you are free to choose!

[jehovajah]

Commentary.

I have reshaped the final translation as is my wont! Over time I will probably redact and review the whole translation in this thread but for now I have done what seems the best to me at this time.

Precisely the spirit of Hemanns closing words of his Vorrede.

The translation into English would have been more difficult if I were attempting to create a literary masterpiece. Instead I was struggling to understand the mind of a particular man in his times. The form of language, the allusions to biblical and cultural references( Michaelangelo's : breathing life into Adam) and his hubris tempered by trying to advance himself in academia by playing the game, all come through, along with his determination never to give up!

Much of the language, in its convoluted form is very hypnogogic. Thus his wish to be free to carry out his desires burns through the page in this last part, and is a direct appeal to his intended audience for patronage! Unfortunately in his time it fell on deaf ears. Unlike some of the mathmatical greats, LaPlace Leibniz, La Grange, Euler, Steiner( also from Stettin) it never fell right for him. In every sense he had to work to achieve everything he gained in his life, while carrying one of the most precious insights god has given to man, if you will!

Still, from his humble mustard seed the great tree of Modern theoretical and quantum Physics has undoubtedly grown.

It is unfortunate that it has been bent out of shape by the intransigencies of the human heart, but it was ever thus!

[jehovajah]

[jehovajah]

Group theoretic approach that influenced Justus Grassmann.
<a href="http://www.youtube.com/v/VSB8jisn9xI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/VSB8jisn9xI&rel=1&fs=1&hd=1</a>

[jehovajah]

Commentary

The more I translate about Grassmann the more I understand the unique confluence of historical streams meet up in and around him.
https://openlibrary.org/books/OL24340512M/William_Gilbert_of_Colchester_physician_of_London

It is clear that in the history of European mathematics Hermann lived at a crucial epoch, but also crucial in the history of the physical sciences, and the history of actions and nations, of states. But it is publishing that makes the vital link between it all, for Stettin far from being a cultural backwater in the Prussian mire was a minor cultural and print setting community. Situated on the Rhone valley, not far from Leipzig a major print setting town, Stettin enjoyed all the advantages of a dormitory centre to a major european publishing centre.

Thus not only was Grassmann away from academia sufficiently to be independent minded, but also he had access to the major titles of works of worldwide importance. He took full advantage of this nexus to new and old publications and technological advances in printing and became very well read, but independent in the direction of his studies, often taking direction from his father and his brother Robert as well as his uncle who actually brought him up.

De Magnete by Gilbert was published in about 4 editions European wide, one edition of which was published by a printer in Stettin!

[jehovajah]

This is a good explanation of how Grassmanns Ausdehnunglehre assisted the mathmatical thinking of physicists.

<a href="https://www.youtube.com/v/n27_QneBkqE&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/n27_QneBkqE&rel=1&fs=1&hd=1</a>

There are several ways in which this presentation is logically sounder than the Stanford University  one.
The concept of time has always been problematic. Lagrange dismissed time as an observable quantity or magnitude! He stated:" there is only velocity!"

We could say the velocity is made from more primitive concepts of distance displacement and time, but we would be misleading ourselves because we cannot have displacement without time displacement. It is the ratio of these 2 displacements we perceive as speed.

This presentation makes it clear that these perceptionsl primitives are as such relative , and that the fundamental relativity of our thinking, the arbitrary " one" as Hermann put it has to be included as a choosable measurement standard  or Metron in our mathematical descriptions.

[jehovajah]

Just found on Mactutor a partial translation of the 1844 Vorrede

http://www-history.mcs.st-andrews.ac.uk/Extras/Grassmann_1844.html

While this translation is " smoother" it is less informative in crucial ways. It speaks more of the translators mindset than it does of hermanns. However this is an inevitable consequence of translation !

However it has helped me to see my translations, though awkward are not that bad! 

Anyone else is more than welcome to post their own in this thread. In fact I would welcome it.

[jehovajah]

Revisiting the translations of the Einleitung with my newly acquired adjectival skills has proved beneficial. So I am aiming to work through these translations again in that light.

I would encourage any and everybody who feels able to submit a translation to this thread as per the outlined heading recommendations.

The headings clearly identify which page in the Vorrede you are translating and whether it is a commentary or a translation. That is it.

While translation into a language other than English is welcomed it would perhaps best be located in a sister thread in the appropriate language section.

So come on you fractallers, try your hand at a bit of Grass, man! Lol!

[hgjf2]

WoW
You ,JEHOVAJAH,just was made a series of texts and scientifical movie series.
Unfortunethly that those movies, wasn't broadcasted on DISCOVERY SCIENCE TV or SCIENCE TV or other similar TV channel.
Can you request the TV channel "Discovery" to broadcasting your movies from your topics: "Light", "The Theory of Stretchy Things", "Twistor" etc.
Would be cool, and you would can become famous.