The Theory of Stretchy Thingys

[jehovajah]

Ausdehnungslehre continued:


Now–this Harmony allows me to relate to all sorts of things: permits that I would close the gate onto a whole new field of Analysis which I submit here so that  it might lead to important Results! Yet, back then, this idea again remained quiet for a whole long time, as my responsibilities in other circles of professional activities drew me back down from there! And also the strange result initially deeply troubled me, because for this new method of products to work  with the rest of the usual rules of mathematical multiplication, and in particular for the relationship with addition to be retained on standby, One but only had to accept the interchange of the factors of the product switched the sign value of the product from+ to – and vice versa !
 
One  may commute the factors, but only if the foresigns  are "turning around" an equal amount ( + into  – transformed and so "turned around")

Footnote: compare J. G. Grasmmanns Raumlehre part 2  p. 164 and that of the Trigonometry p.10

[jehovajah]

Ausdehnungslehre continued
A worked exercise that I later had to take on the Theory of the Ebb and Flow of tides lead me to the Work and Ideas of La Grange his Mecanique Analytique, and those ideas brought back these ideas of Analysis. All the developments established in that work I used this new Principle of Analysis to establish again in the most simplest way! Often the Calculation was as little as one tenth of that in that original in the way it was lead through, and the result just fell out!

This encouraged me to apply this analytical method to the Difficult Theory of the Ebb and Flow of tides; there were many kinds of new handles to define and develop in order to suit the Analysis to the task! And specifically I was lead, by the label specific to the Exponential magnitudes relating to the swivelling round( of things) , to the Analysis of the angle/ arc and the trigonometric functions  analysis, etc, etc... And I had the Joy to see that the newly established extended method lead to highly simple fully symmetric Formulae, rather than the unsymmetrical and convoluted ones which are the basis of this Theory! And these formulae not only contained the old convoluted and unsymmetrical ones but also The method of development of the Formulae went to the  side ( of the page) of  the handle or label.

In the practice of advancing stepwise formulae from one formula to another things  not only go so easily, leading to the least wordy formulae, the development leads to greater symbolic handles , new expressions and the development of specific laws every time; but also the stepwise advance of formulae seems to lead to its own parallel going, consisting in labels, guided Tour .  Dark and obscure ideas shroud the development , and the Spirit of the idea is killed, by such an aforementioned practice. The introduction of arbitrary coordinates that do not relate to the things essential natures but obscures them shows this death dealing formula development!

In practice I could not only every formula which arose in the process of development clothe with the lightest of words, and I expressed then a new law( rule) in this way every time, but also every stepwise advance from one formula to another formula relentlesdly seemed like only the symbolic expression of a parallel going, guided tour consisting in labels or handles! Compared to this, Other usual Methods I showed that they, through the introduction of arbitrary Coordinates, those which had nothing to do with the essential idea of the thing in hand, the  idea shroud in darkness, and the calculation do in a mechanical (lifeless )formula development , the basic concept of which is definitely not presented, and which around that basic concept is formula development that is lifeless!


Footnotes
The nearest validation is found below

Compare with La Place Celestial Mechaniques book IV

[jehovajah]

Commentary

I think I might have got the gist of what Grassmann did in his the law of the third Strecke phase!
However I do not know if I can communicate it clearly!

As you may know Bill Clifford quoted a paraphrase of the scripture which said unless you are like a little child you cannot enter the kingdom of God, because the gate is so low!

In my wordy and voluminous thrashings about I have passed by this gate many times, but could not open it! And yet I was outside the gate and sometimes inside the gate and other times on either dude of it simultaneously!

I am raving like a lunatic, so forgive me. However vectors do not help you understand Grassmanns Analytical Praxis! They are a product of someone else's mind grafted onto Grassmanns straight forward but rigid even autistic thinking!

Everything has to be disassembled into pieces and constructed in the synthetic process precisely as Grassmann describes. It is tempting to leap ahead and think you are on the same path, but you are not likely to be. There are many many similar paths but they do not all end up where Grassmann goes. In particular, Grassmann builds new connections new paths but they rely upon his original path.

Are all these other paths wrong? Not necessarily, but they are not Grassmanns path and may not lead to his conclusions!

There are fixed conventions in geometry and trigonometry. No one really knows when they started or if they were ever really discoursed. It just seems that they were imposed as standard conventional form. These conventions of the plane are not discussed just enforced. All lines are drawn left to right , horizontally in the plane or page or slate. In some countries this is reversed and lines are drawn right to left horizontally. Even left and right are conventions of reference frames of a personal nature. A good proportion of us have difficulty with our personal left and right, and that extends to apprehending another persons left and right!

We do not seem to have a problem with forward and backward or up and down, other key aspects of our personal reference frames.

Back to the page and by extension the mental plane! The other fundamental orientation for a line or a line segment on the page is called vertical. This mental page vertical gets transposed into our 3 dimensional apprehension of height and depth, and yet these are entirely different experiences!

Finally on the page/plane/ mental page we can draw line segments at any orientation between the horizontal and vertical by several methods. To draw a direct line segment one needs a straight edge. The manufacture of these straight edges is of considerable importance!

When Newton begins his Principia he does so in the context of the dance between Mechanics and Geometry, each contributes to and impacts on the other. They firm a dancing pair, an iterating couple, by which we hone down our ideal forms to the greatest exactitudes until we become exhausted! We never achieve perfection we endlessly approach a mental or formal ideal.

Where does this ideal come from? Ah, both Plato and Socrates invented a game to play on that question! Poor Arustotle, however took it seriously and attempted to state what was logically true as opposed to what was logically not true. I make no bones about it: Aristotle went a little crazy in this regard!

So, returning to the page, we can use a manufactured straight edge or we can use a rigid property of fixedness or fixity namely the circle! In fact we use a combination of both in the plane which makes all our Geometry actually based on the spiral!( now there's a surprise!). One other convention: we specify our orientation relative to the horizontal which means we have 2 measures of orientation that are usually called supplementary.

This theme of couples runs all the way through geometry, or as I prefer Spaciometry. It is only in trigonometry where we distinguish them, and then promptly attempt to forget the other supplementary facts and details!.

We need all this conventional set up before we can even begin to talk about a point in a plane!
In fact the point, the line, the circle, the plane are all concepts that we have induced in our mental experience from mechanical interaction experiences. These we have re-applied to those mechanical experiences, iterating to abstracted mental ideals over time and experience.

What is a blank page or a clear plane is in fact a pregnant space waiting for us to pour out our mental conceptions from the full page or plane in our " minds". It is how we apprehend these mental conventions that Grassmann deals with in his masterpiece. It is the method or Praxis or Art that he discovered that he imparts in the Ausdehnungslehre. And it relies now on 3 arbitrary points and the 3 Strecken that join them in the context of all these prior conventions!.

When Hamilton invented a mathesis for complex numbers, he went step by step from one point moment  to couples of point moments. But he got stuck at 3 point moments because he was looking for something else. He was looking for rotation. Grassmann was not looking for anything in particular, but he was sensitive to patterns and interested in labels. He noticed the simplest things like a child. And the rigour 3 points placed on all notation was ignored by everyone because they were busy looking elsewhere!

[jehovajah]

Commentary

So starting with the first 2 points A,B he draws the horizontal Strecke AB . The concept of contra is introduced by a simple observation : retaining the points as fixed. He could have started with the horizontal Strecke BA.

Only the child knows the magic of what it has just done!

The statement AB = –BA introduces the contra sign as a negative sign, but the child knows it as a rotation symbol. Unkehren is to sweep round in a great arc and go in the opposing or contra orientation . That is to change direction.

Now draw the second Strecke BC . We could have drawn the second Strecke as AC but tht does not " flow"

The flow shows up in the statement AB + BC = AC

This is label poetry. .

The child delights in it!

This says something magical about the plane and 2 flowing Strecken in the plane: they create poetry. This is where the child starts from, this is the child's Förderung.

Now we come to trigonometry. Serious adult business! We do not want all the fuss about all the points and all the labelling. We adults want it ll neat and tidy out of the way.

So we start with the rectangle. Break it into all it's pieces(Stücke) and name them all and label them all. The vertices , the edges or sides, the diagonals, the area of the shape, the products of the sides or edges which are adjacent. Use lower case for the sides so we do not have to keep writing vertices or the points.

But then the bemused child sees an old friend, a reemerging of the poetry
 a + b = e the label for the diagonal
–a + d = f the label for the second diagonal
And look d + a = –f

Switching the adjacent sides in the sum and the contra on the a creates a contra on the diagonal f!
But multiplication is boring !
ad or ab or bc or ba or cb or cd or dc or da they are all the allowable products
So what about de or ed?
Now we can write  ed = ( a + b)d and thst is ab + bd
ab is the allowed product but what is bd? It is not allowed
So the product of the parallelogram is the allowed product of the rectangle  wit a disallowed product.

Suddenly the child sees the parallelogram has a simple product and it is the rectangle if disallowed products are ignored or set to 0

All parallelograms on the same base are equal!

But what about de which is da + db?

db is not allowed but da is
Is da the same as ad?

The poetry says no.they are contra!
The adult says poppycock!
The child curious goes back to peek. The labes drop away to reveal the Strecken and their points. The multiplication drops away to reveal the construction of a parallelogram compared with the construction of a rectangle.

In both the construction flows only one way! To interchange the constructed elements requires introducing contra construction processes.. These contra processes are not cancelled, thus they carry forward and make the whole process simply contra!
It is intuitively simple, but the adult is deeply troubled by it.
The child rejoices in the poetic harmony.

How to construct a parallelogram  ABCD( cyclic vertices) given AB is a and BC is b and CD is c  and parallel to AB and DA is d and parallel to b
Using a b as the product instructions that is construct the parallelogram using a and b only.
Draw a horizontally. Using a compass swing an arc radius b. mark off the required arc using C . Now bisect the arc to construct a rhombus , extend the side of the rhombus through C to meet D by marking off D  measured as c from C. Join D to A and check it measures as d
Thus ba as the product instructions: draw b horizontally.( already a major diagram shift! ) . Using a compass swing an arc radius a ( where is the centre for this arc? It cannot be at C as this violates the convention established. It cannot be at B because that violates the convention  a as AB ! To proceed we are forced to use not a but –a) . Correcting swing an arc radius -a. Mark of the required arc using A. Bisect the arc to construct a rhombus, extend the side of the rhombus through A to meet D by marking off Dmeasured as -d( because we are forced to use the contra convention) from A. Join D to C and check it measures as –c.

It is quite clear that ab is the contra of ba, and this is written as ab = –ba.

The question is can I cancel the constructions out?

The answer is no. These are allowed constructions.
I am tempted to say that we can arrange the coefficients to cancel, but the point is that the product is there, and the contra product is there.mthey are not annihilated in the notation, they are equilibriated!

While this may mean little to the child as it grows it will appreciate the value of dynamic and static equilibria in mechanical systems as well as physical ones.

Some say that geometry revealed these set ups in physical reality, but forget to point out the dual nature that runs through every relation we observe!

[jehovajah]

Commentary

I thought that if I let the last meditative transaction sink in I would gain some insight into why it was so fundamental and yet do underwhelming!

And bingo!
I awoke thinking about the Galilean transformation principle and how it could be analysed Grassmann style!
Then I thought of Miles Mathis protestations that Einstein got it wrong in his 1905 papers  regarding the principle of Galilean trsnsformation.

And then finally I realised how fixed Strecken are associated to one dynamic Strecke. But then I realised it was arbitrary which 2  are fixed and then only one might  be fixed. Finally what if no Strecke was fixed!?

That brought up the crucial role of a point and their fundamntal role in our referencing Strecken, and representing annihilation and creation of Strecken constructions.

And finally I realised it all starts with memory, or as Hamilton put it: moments in the progression of time.
But even time is a subjective relativity.

For exapmple a measure of force is a measure of pressure produced with a measure of area that is a measure of pressure producing a bistrecke.
A measure of pressure is by cross ratio a measure of force divided by a bistrecke for area.
Now a measure of force is a constant scalar product of a Strecke of acceleration; and a Strecke of acceleration is another constant scalar product of a Strecke of differential velocity; and subsequently a Strecke of differential velocity is a constant scalar product of differential displacement of position in sequential order Strecke.
And finally differential displacement Strecke are a representation of a lineal combination of unit basis or primitive Strecke with differential scalar coefficients which are functions of sequence progression or implicit iterated sequence or inductive sequence functions.

Thus underpinning any mathematical model of our experiential continuum is an arbitrary or generalised selection of unit Strecke that are accepted as a basis or primitive set of Strecke.

But what if those basis Strecke are dynamic? That can only mean: what if they rotate relative to each other interdependently?

[jehovajah]

Commentary
Grassmanns analytical method opens my eyes to one thing in particular: our arithmetical models are totally formal, and in our heads! Consequently the measurent schemes(metrics ) can not answer the question,"What is matter?" The models enable us to model behaviours only through applied iteration, carefully conceived.

What matter is, is the subject area of alchemy or modern chemistry and physics. But Alchemy cannot say what matter is but what ratios of distinguished matter appear constant or conserved.

Our conservation rules and sense are how we can move between scales, testing all the time. At a certain scale our conservation rules break down. At that scale we change alchemical description and say the fragments we measure or identify in other measuring schemes like interferometry , are new elements or particles. The smaller the scale the more fragments we identify.

But we know our mechanical concepts can be scaled down ad infinitum. So what prevents us from multiplying fragments ad infinitum?

Our conservation rules guide us to look for systems where they hold , and those scales then become the new fragment scales.

At the quantum level, then,  we are looking for conservation rules that hold. That then defines the scales of the new fragments.

How do we even measure on those sorts of scales?

Again we cannot, so we resorted to statistical methods. This does not tell us about particles at all, but about collections of statistical measures. If we get excellent agreement on these statistical samples, either as probability distributions or as statistical curves we infer that the measures bear some relation to an underlying assumption of fragment size.
This is modern quantum particle physics.

However, these measures are equally applicable to fluid elements, spatial density ratios, invisible energy density regions, Fractal regional boundaries etc. in other words the mathematics does not identify any substance. We assume, impart or infer substantive properties from sensory signal inputs processed by our proprioceptive mesh.

Our experience of reality is our reality, and it is beyond our powers to step outside of it in any sense or way at all. We simply reprocess the data to establish a particular sensory map. How useful that map is is how much behavioural control it gives us over local reality.

[jehovajah]

Commentary
One of the very real difficulties that defeated Newton in developing his fluid mechanics was the sheer complexity of the reference models.
While Descartrs system of coordinates were of a generalised nature, and it was Wallis who first placed them as rigid cross hairs, it still was difficult for Newton to relate roationally mobile local reference frames to a larger external reference frame with all the freedoms of movement in 3 dimensions. Hamilton's development of Quaternions made that possible for the first time, but it was fiendishly difficult to get an intuitive feel of what the axis were doing.

Even today in quantum physics theorists will not accept mere rotation as a description of these calculated state changes.

However, Grassmanns method makes it natural and intuitive to think of these local transformations as relative rotations, even though they can also be described as reflection in crossed lines.

It is better to include these rotational behaviours in the model, alongside the differential or integral calculus scalars for the Strecken.  The result is that rather than describing fields as spherical potential scalars, we need to describe them as vorticular Strecken scalar complexes. Using this model, it is easier to model fluid element behaviours as well as fluid dynamics as a whole.

While the curl of a vector field is an attempt to model this rotational behaviour it is too rigid to deal flexibly with vortex shedding , energy propagation along or between " reflective or refractive wave guides" etc. in particular the phenomena of reflection refraction and diffraction do not seem to be easily connected to the curl and or div of a vector field with integral or differential calculus scalars.

There is a better way is what Grassmann is saying.

[jehovajah]

Commentary.
It struck me this morning, that the train of thought inspired by Grassmanns thinking style only deals with the fundmental primitive or basis Strecken of any model. , that is to say that the importance of Grassmanns model is not just the basis Strecken in Lineal combination, but in fact that he could multiply any pair of properly configured Strecken!

The work he does on multiplication I have not yet studied because I wanted to get his Vorrede right. But already , after my experience with vectors and Clifford algebras I can see that many have lost the simplicity of his approach, as discussed here.

Yes Grassmann goes on to redact and to extend his method into more and more difficult analytical subjects, that is subjects that rely on symbolic arithmetic to collate and coordinate the theoretical system. And yes he does create new terminology and even new products, but what he comes back to is the essential simplicity of his 2 major initial insight: the representational sum, and it's use in defining the general geometric product.

The representational sum can support affine and projective geometries, and we have seen how it can support a dynamic mechanics. The general geometric product extends the method into the enclosures of the Strecken products.

What I mean by that is : when one learns Geometry the line and point are usually demoted to the role of boundary and boundary measure or parameter. The things that excite geometers are the forms or shapes starting with the triangle.These forms are the first intuitions of space and what it might be.

However , as I have pointed out, these forms are totally formal and in our heads, so cannot answer the question. However these forms are abstracted by us from alchemical substance relationships, that is matter behaves in ways that gives rise to these forms approximately. Rather than go down that arrogant route suffice to say that we abstract from reality these abstract ideal foms by a process of fractal distillation!  In other words, over time and experience we iterate by design to the ideal forms in our heads.

By reversing the iterative process we can " sum" or aggregate or integrate iteratively these foms into models of more natural forms , foms found in nature.. We have to reverse our arrogant taxonomical thinking to do this, which is why fractal geometry is such a subject for the people, the artisan, the artist, the builder and the mechanic, the engineer etc. and the little child can contribute as well and even lead us to new innovative thinking.

So now Grasmmann gives us a way to represent these forms as products of Strecken. Again, we need to avoid artogant assumptions .

Many who have read my earlier writings might remember my friends the spiders. They are the ndimenioners! it is a curious experience but once we get past about 6 dimensions in the plane we can no longer construct from these nets 3 dimensional solids that are closed. Thus multiple dimensions in the plane remain as planar forms. Hence spider webs which typically have between 20 and 30 dimensions are flat!

You see dimensions do not have to be orthogonal or mutually orthogonal! This one bit of classical nonsense has impaired understanding of the space in which we live!

We are so used to saying we live in 3 d we neglect to apprehend that is not a truth! It is a shorthand for a representational system based on orthogonal standards. These standards initiated by Wallis are one of many, and are not always convenient, for measurement, construction, engineering or technological process, or even Alchemical process..

Crystallographers and crystallography certainly would struggle with just 3 dimensions!. Grassmann made a point of stating this notion of 3 dimensions was a severe limitation to understanding. It turns out that the real numbers also supports this limitation in thinking!

Many physicists assume they need the real numbers to describe reality. But increasingly, quantum mechanics is showing that we can only really apprehend the natural numbers and their construct the rational numbers, if we need number at all!

As you may know I do not believe in mathematics as a subject or numbers as its primary objects. Here is why!
<a href="http://www.youtube.com/v/w-I6XTVZXww&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/w-I6XTVZXww&rel=1&fs=1&hd=1</a>
This is nonsense. But that is just my opinion, I hasten to add.

Fortunately for me I have had opportunity to apprehend the Arithmoi and Shunya, both are blessings beyond measure!

[jehovajah]

The motion of  a fixed local point when the local basis Strecke are gyres or Twistors of fixed radius

[Sockratease]

...Many physicists assume they need the real numbers to describe reality. But increasingly, quantum mechanics is showing that we can only really apprehend the natural numbers and their construct the rational numbers, if we need number at all!

As you may know I do not believe in mathematics as a subject or numbers as its primary objects. Here is why!
[silly video]
This is nonsense. But that is just my opinion, I hasten to add....

Things like this are why I refuse to believe infinity exists as anything other than a mathematical construct.  It simply does not work in the real world.

I get so many arguments from everywhere when I say such things that I prefer to just keep quiet about the fact that I hold the same opinion of Zero - it's another useful concept for math but it has no place in reality.

[jehovajah]

Thankyou and welcome to the tHread Sockratease.

I hope you will feel able to share many other points regarding the foundations of mathematics here!

Arguments are not the way forward. Disagreements are acceptable and welcome. If you or anyone wants to put an alternative view that is welcome, but ad Hominem or disrespect as tactics will not be tolerated.

I once made a comment on YouTube which requested a collaboration with prof Wildberger on research into Grassmann. It attracted the most virulent comment from another viewer that I reported it as spam! I was happy to see YouTube immediately took it down!

So why am I going on about these standards of civility? Because I want you and anyone else to feel safe to voice your opinion in and on this thread and know you will not get embroiled in some argument!

The video may appear to be a silly argument. I think it is a confused one. I am certain that most of us can see where the argument leaves the normal experience of additive process or aggregation. The question is, why are these men smiling?

Non verbally it is because we knw they are Ill at ease with what they are being forced to say! The reality is that they cannot explain it, all they can beg us to do is accept the process.
I have a choice, and I say no! Others may say yes.

This kind of dividing occurs all the time. We are divided into groups who accept certain things as standard. It is when one group attempts to coerce another group to conform that hostilities break out.

The summation of alternative opposites is classically undecideable if a process is endless. We found this out when our computers went into their first infinite loop!

So why decide on a 1/2?

You have heard of probability. This is where the half summation comes from. What the presenters probably do not know or fail to tell us is that the summation is no longer an arithmetic one, but a symbolic representational one.
The confusion lies in the lack of explicit definitions of the symbols on the page. Because of this lack of clear definition everything else is confused. The answer itself is also meaningless within the standard probability theory. Negative probabilities are not usually defined, although there is a great philosophical debate about their possible usefulness and interpretation!

I do not expect many to understand what I have just written, because the video is such obvious "non sense ", showing no common sense that it cannot be redeemed. Nevertheless  we humans do not think " mathematically" we think analogously. It is the analogies we use that give us our sense of meaning and contrariwise our sense of nonsense. It is for want of meaningful metaphor that many of our most innovative artisans alienate their constituencies. In such cases they revert to the " magical" paradigm, or the " mystery" paradigm.

There is a place for magic and myth as well as fact, as long as we realise that all 3 are totally human creations.

[jehovajah]

Commentary on Ausdehnungslehre continued.

The deep , conjugate and fundamental relation of Mechanics to Geometry is perhaps not appreciated by theoreticians even today.
 
In the ancient times the basis for geometrical knowledge was the practical and technological skills and experience of Artisans. Education, or rather academic education removed that connection by establishing qualifications based on written responses.  Education especially monastic based education was based on discipleship and apprenticeships. A student had a master and would gain practical and technical skills over time under the masters supervision.

Plato's academy, as a case in point, was discoursive in nature but required, demanded a level of practical expertise in Astronomy and Geometry. As time went on, and particularly when funding became an issue these entry requirements were modified.

Eventually academicians becme so snooty they would act and state that the Tekne or skilled artisan was beneath them socially and intellectually! The greatest historical Student of Mechnics and perhaps everything else was Archimedes. But I have heard that some accused him of " cheating" when he trisected the angle by neusis! That is he carefully moved a straight edge into position so that he could mark off the measurement required to trisect an angle. Later he demonstrated how a spiral could achieve the same exact measure.

Trigonometry is about applying metrics to space. It is called trigonometry because it relies on the trigonos . This is not the triangle ( ie the trilateral figure) but the actual 3 rotations at the corners of a trilateral.. However, these 3 rotations require a plane and a figure that is a closed " loop" in that plane.

There are many candidates for this role, but the do not arise from geometry! They come directly from Mechanics! Over time the best mrchanicsl archetypes turned out to be very strong rope or cord that could withstand high tension without stretching appreciably. But rigid objects of any description were also used.

It is perhaps excuseable to think that geometers start with a plane. In fact they start with a surface and mechanically produce or scrape out a plane!  The ideal surface and perhaps the easiest to produce is the spherical surface. Rotation around a fixed pole enabled these surfaces to be carved out. Some artisans became so skilled that they could do this by eye nd hand alone!

Returning to the scraped plane , the circle is the next easy fom to create, and the fundamental one. Choosing The Diameter as the greatest measure enabled a taut line to be defined as straight; enabled this to be confirmed by a curious property of a third point on the perimeter of a circle relative to the 2 diameter points. The rotation at that third point if a trilateral was formed by taut cords was always the same. It was called orthos!

Tekne were able to produce and reproduce highly curate models of that rotation! In fact it was so special tht it was used as one of the symbols of divine power for an Egyptian ruler or Pharaoh. It is the origin of our term " Ruler"!

The point is that our reference frames or Strecken are determined and constructed by artisans . The orthogonal reference frame was devised by artisans as a quick way to cut out an appropiately sized block for further preparation or construction..mthe orthogonal mutual rotations are placed into position by construction and measuring processing. At each stage measurements are taken to ensure accurate alignments. These constructs are our fundamental tools of reference and they exist in our heads not in nature.

As we construct using orthogonal frames usually, it is acceptable to use these as our ultimate basis for space, but using these and Grassmnn analytical method we can actually construct generalised reference frames based on Strecken in any rotational relationship.

The issue of dynamically rotating Strecken still requires us to reference these against a fixed fundmental set of basis Strecken and these may as well be orthogonal.  However, the point is, that building atop these fundamental orthogonal set we can have exotic dynamical basis Strecken in lineal combination. These are what are needed to describe fluid dynamics.

Einstein utilised this Grassmann geometric freedom to establish his relativity theory. Others, not understanding the freedom Grassmanns ideas give formally point out the fundamental "Euclidean" basis Strecken! The point is all these basis Strecken are in our heads not in Nature! In any case Ruclid was a consummate astronomer who used spherical trigonometry not plane trigonometry.

[jehovajah]

Commentary
Kegan Brill is one of the few artists on the web working intuitively nd accurately with vorticular spaces!
http://www.spiralodyssey.com/Spiral_Odyssey/INTRO.html

After Lazarus Plath he is the next intuitive genius whose work informs my conception of space and it's structural framework and dynamics..
http://www.spiralodyssey.com/Spiral_Odyssey/WORK_IN_PROGRESS.html

The structure of the fundamental or primitive reference frame , the standard orthogonal Strecken is not complete as a wire frame. The structure is best vsalised as a vortex shell. If the orthogonal trecken remain fixed relative to each other we fail to appreciate the shells which govern their relative motions. These shells , expanding or contracting model forces and motions in space and the complex behaviours of matter.

The basis Strecken for dynamic space spin out these shells like the webs of a spider, and necessarily make SpaceMatter fractal and regional at all scales.

However I realise from my above discussions that there is an unfathomsble gap between my formal apprehension of Strecken reference frames , dynamic or otherwise, so that there really is no way of framing an absolute reference frame in any decideable way. It is down to the individual. If they accept an absolute frame they think like Newton. If all frames are formal and personal they think like Einstein.

[jehovajah]

Commentary
It is amazing that Grassmann cut his teeth so to speak on a fluid dynamic problem. While the Scwenkung labels and the exponential handles guided his development of his own analytical labels for the swing, or pendulum like motion of the tidal flows, it was and is apparent that it would generalise to any periodic motion. Bill Clifford felt straight away that it would describe the undulatory motion of waves and in particular light and electromagnetism. Perhaps he was inspired also by Helmholtz and Kelvin, and possibly by Maxwell. Because of his early death we may not have enough documentary evidence to determine that, but we do know bill Clifford was a big fan of Grassmann and Hamilton.

Several references are made of river McCullagh and his rotary description of fluid mechanics . This played a major role on influencing Fitzgerald to redact Maxwells work. But Stokes is said to have demonstrated that angular momentum is not conserved in McCullaghs system. Why this should quite damn the work is still beyond me.
http://hyperphysics.phy-astr.gsu.edu/hbase/conser.html

The belief in conservation laws perhaps was taken to far in this case. Conservation of angular momentum is a clear impossibility in rotating systems. The reason is that angular velocity, as a concept is not mappable to a single valued translational momentum. Instead angular momentum as a transformation to lineal momentum is not a function as far as functions are defined. Thus any narrow , mathematical description of angular momentum is bound to fail.

Rotational momentum, as an analogue of lineal momentum is a nice idea but not physical!

Now I am clearly not talking about the normal definition of angular momentum, because if I was I would be dead wrong! I do not mind being dead wrong if it leads me to a better understanding.

If you consider a disc, for example , or even a stick it's angular momentum is defined fom the concept of torque in comparison with lineal momentum.

The mathematical facilitation is the placing of the mass in connection with the velocity or distinguished from the velocity of an object. However we have to back up to the notion of the velocity of an object.

Here Newton made a justified identity: the velocity of n object is the sum of the velocities of all it's parts. This is what is called a vector sum, by a man who invented geometrical vector Maths before Hamilton and Grassmann and Gibbs! Newton's third aw was a catch all that justified this approach once accepted. Geometrically it was based on Newtons understanding of the sphere nd the circle, as the quintessence of symmetry. Because of this symmetrical apprehension due to the sphere Newton was able to envisage a centre of symmetry collocates with a centre of mass and what Newton called a centre of Gravity.

This vector sum is set to zero at this precise point and an object is demonstrated to balance at this point. Barycentric coordinates apply.

However, if an object rotates about this point the vectors fir rotation are defined variously, but they am mount to instantaneous tangential ones. Or usually ignored nowadays arc or twistor ones!  I named these arc  or curved lines twistors. These do not sim to zero if an object is rotating . For any given circle or spherical surface thes arc vectors will sum to a specific value, but that value is quadratic lily determined or measured. The two measures are the rotational "velocity", an analogy with lineal velocity, and the radius  at which that velocity is measured.

For a real body the notion of angular momentum is tricky to justify!  The mass is distributed through space, and therefore is also a function of the radius at which the "velocity" is measured. The full concept of an analogy to momentum that describes rotation is therefore a double integral  over radius or an area integral or better still a volume integral. My point is I am not sure if it makes sense to discuss an "angular" momentum in this situation, and thus to expect a conservation of angular momentum. However it seems natural to identify a kinetic energy fir the object and to expect that to be conserved.

How dies this relate to the observed behaviour of rotational motion? It relates through density. As the density of the object increases the kinetic energy increases, but if the kinetic energy is constant then one would expect the greater density to impact on the rotational motion reducing it. However, if the density increases via an internal change in volume, because of the cube relation the expectation is that the rotational motion actually increases! This is because despite the density increase the system quantity of matter is constant!


When an object or a system is periodic the important conservation is that of kinetic energy. The notion of rotational kinetic energy compared against lineal kinetic energy is interesting. For a rotating mass that rotates around an axis , the rotational kinetic energy must be set to the lineal kinetic energy of that mass moving under its own rotation in contact with a non slip surface.
<a href="http://www.youtube.com/v/s_R8d3isJDA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/s_R8d3isJDA&rel=1&fs=1&hd=1</a>

If we now transfer this to a fluid / viscous material one can equate the total kinetic energy of a vortex to the entire system after any vortex shedding has occurred.

The generation or dissipation of vortices is therefore of fundamental importance in describing fluid behaviours, and it is precisely here where Newton ran into difficulty, and stokes makes his non physical diagnosis,

Newton was able to describe how force transmits through a fluid around a shearing cylinder by means of his " calculus" method, but he left out the conservation of rotational kinetic energy. Thus a lot of expected rotational motion did not " appear" precisely because it went into driving shed vortices!

It is apparent that this vortex shedding was not understood or even apprehended. It has taken me a while to apprehend its ubiquitous and vital role in fluid dynamic description.

When I look at Newtons drawing of wave diffraction in water and by analogy on air. He represents it as short lines fanning out. The same kind of thinking carries over into his corpuscular theory of light, but this time he replaces the lines by corpuscles.. So in a fluid a fanning band of lines represented some notion of a wave front, and this epwould be a spreading " hill " or undulation of water. This was his concept of a wave. Later, this hill is almost universally represented by a sine function, this graphical image is burned so deeply into our brainwashed minds that we think it is a wave!

However, newtons corpuscles are perhaps better representations of the fluid behaviour than we are taught to believe.

Corpuscles are alchemical blobs of jelly. Later some regarded them as hard pea like objects or billiard balls. What they were precisely Newton never said. They behaved like projectiles and had a bodily attraction to certain materials, that apparently led them up and diverted them from their straight line course!

Huygens on the other hand looked at the corpuscle as not being a projectile, but a receiver and transmitter of an undulatory force!
<a href="http://www.youtube.com/v/NeaNTbeEelo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/NeaNTbeEelo&rel=1&fs=1&hd=1</a>
This video gives a feel of how circles combine to propagate light spherically. the complexity of the motions hides the fact that these have been deliberately selected. imagine a physical process that selects out the wave phenomena of light:diffraction , refraction , reflection...

Huygens cascade model of light propagation was based on a more accurate understanding of Grimaldi's work. Unlike Newton he understood diffraction to be longitudinal not transverse. His own work on lenses showed him how light inhabits a material, making it glow. Therefore he surmised each corpuscle of matter absorbs and radiates the light "wave" like a pendulum. His model shows light propagating in circles from every corpuscle.  From this model he could isolate all the observed phenomenon simply by defining a kind of sum in a given direction, a vector sum!  
The wave fronts Newton drew as short lines rafting almost in parallel he replaced by circular arcs. Where these arcs crossed more light was generated than where they failed to cross.  I have not read enough of Huygens to apprehend if he regarded this as super positioning of hills or more likely affrefation of pressures in the plenum. In any case his wave front does not correspond to a sine wave , but rather to a spreading firm of curved rhombus. The wave front was made up of these rhombus like regions spreading out as the circles or spheres spread out and I filled by new sources of this rhomboidal form . Thus he had a fractal wave pattern which became more complex the further from the source you went.

That light dimmed the further away you were was not easily explained by Newton's ballistic  corpuscle model.

I mention all this to address the point that a wave is naturally observable as a rolling tube or vortex of water that rolls atop and within larger rolling tubes in a fractal pattern. Thus any physical wave description that does not recognise the rolling tube with an ellipsoidal or rhomboidal form as its fundamental description will mislead. And in addition the conservation of kinetic rotational energy is all thst is necessary to maintain Wave propagation in space matter.
http://www.britannica.com/EBchecked/topic/187240/conservation-of-energy

[jehovajah]

Commentary.

This paper attempts to give a straight forward overview of Grassmanns ideas and developments of them.
http://faculty.luther.edu/~macdonal/GA&GC.pdf
Especially at this point in the translation it is crucial to recognise that much of what gets physicists excited seems to be the development of his Ebb and Flow paper in which he made considerable innovative effort to extend and apply his initial insights.