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

Commentary
Normsn's discussion of Affine Geometry explains Barycentric Coordinates simply. But you see immediately the disconnect . The parallel lines are discarded and replaced by point notation! This is precisely what Möbius did, but what Hermann chose to keep!
Further Hermann regarded the parallel Strecken as a product of points!

The reference to coordinates is lost in the computation.

A point in Cartesin geometry is given by 2 lines called axes and 2 parallel lines that intersect. In Barycentric coordinates we use two parallel lines that do not intersect as axes! We thn use a common line that intersects both as our line of interest . To find any point on that line of interest we use ordinates on the axes. This then enables us to draw a line that intersects the line of interest in a unique point!

Barycentric coordinates therefore give us a unique point. By varying the ordinates on the axis in a proportion that remains as a fraction sum equivalent to 1, we can traverse the intersecting line through all points on the line of interest.

If the proportion sum exceeds 1 or is less than one then a point of intersection on the line of focus does not exist. Using negative or harmonic proportions we can extend the line of focus.

Now by changing the points of the line of focus we can change the position of this point in the plane. This is the Source  of Bezier curves .

In 3 dimensions we trace out positions of points in a surface in space.

These coordinates become very general algorithms used in animation or vector graphic applications  to animate or deform space .

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

Thanks Kram1032.

If and when you have time , I would welcome your input into the thread, particularly your version of the translation.

The etymology of the word Schwer, takes me to many ideas, and so I consider them in the commentary. For example besides heavy Schwer has the sense of difficult  or dense.  The idea of a complex point or a dense point derived from nodal connections to all other points in a system suggests itself from the term Schwerpunkt.

Newtons practice of reducing a body to a poit mass is precisely the notion of Schwerpunkt.

But there is another connection. Newton attempted to introduce an explanation of his Fluxions method on the basis of a notion of the moment of a rectangle. This derivation recounted in the foreword to the Principia gave rise to many misunderstandings. Yet it is almost precisely the Barycentric method!

aX + bY,  for a rectangle with sides X,Y, with variations a,b

Moment
This formulation of the derivative as a Barycentric equation is more precisely Hermanns notation than Möbius, because the capital letters refer to the Strecken of the rectangle not the points. Yet it is clear that Möbius acknowledged the contribution of the  Strecken of the parallelogram to determining the Barycentric coordinates of the centre.

It is therefore significant that this point plays such a vital role in the analytical methods of Möbius and Newton and Grassmann.

Further we see how it underpins Lagrange's mastery of the multivatiable calculus.

I sense there is more to this point than meets the eye!

Commentary

There is a direct connection between the Schwerpunkt and the so called complex point.

The easiest way to realise this is to use the Barycentric calculus to trisect an arc  of a circle. Trisecting an angle is supposed to be impossible by straight edge and compass. Bur Archimedes demonstrated it was possible. He is often accused by some purists of using Neusis to achieve the result, but in fact he probably used a form of the Barycentric calculus.

For a symmetrical Barycentric Bezier Casteljau system setting t = 1/3 gives a trisection of the arc, and so the required points for the trisection of an angle.

A ruler can be constructed using a straight edge alone let alone having a compass pair available!

The Barycentric notation conceals a reference frame based on pairs of arallel axes. The affine geometry is a coordinate Grometry that makes this clear. Using this geometrical reference frame together with a compass enables oe to establish a point at 1/3 rd circular arc.

But this was demonstrated to be an impossible construction for a so called real value. Gauss of course showed that a complex valued solution was possible. The difficulty was knowing what that meant!

It became fashionable to talk about Argand diagrams and eventually the complex plane, where the solution could be directly constructed,. However few realised that the Barycentric calculus gave a real solution to this problem, mostly because they were not looking for a frame of reference beyond the Cartesian as established by Wallis.

anti trisection bias


Jim Loy

Thus the direct relationship between the Schwerpunkt and the complex point either as a polar coordinate or a complex plane coordinate was missed. The direct connection to vectors was also hard to accept, but over time it has tended to come together, through he transformation rules.

Later we will see how Hermann derives the as the constant required to describe Strecken associated with the circle in the plane uder an exponential trig relationship.

I'd really like to give more feedback on all your work here. I just have to say, your collections are rather overwhelming. It's impossible to keep up with it all.
And often it's a bit hard to see what your point actually is. You seem to assume fairly high standards and you take all these topics in a deeply philosophical manner.

I think that, while there is nothing wrong with a philosophical stance, and sometimes it can even be helpful, if you take it too far, it is more confusing than anything else. And I fear you are taking things too far at times.

(* Hestenes, after all, began as philosopher interested in truth and then switched to math because he thought that that's where a good, rigorous notion of truth lies, and that eventually lead to the very nice, clean nature of Geometric Algebra)

Can you, perhaps, start at the beginning, fully explain your points not only on a philosophical level but also on a mathematical one and then slowly (slowly being the keyword) build the explanation of your views from there? - There is nothing wrong with hinting at eventual philosophical or mathematical implications, of course. But don't get derailed. Just give one or two sentences of future outlook and come back to the ground. Take your time to lay out more of the basics, then more people will be able to understand your points.

The other thing you are doing is putting a lot of value into the precise history of how things developed. I agree that this, also, can be both interesting and enlightening. Especially, in hindsight and through views and tools informed by our modern understanding of math, we probably can significantly improve some old historic flaws if we understand how they came to be in the first place.
Once again a good example for this is Geometric Algebra which is just a big combination of historic ideas that initially were perceived to be very different approaches to the same problem.
Another, as far as I can tell, seems to be Homotopy Type Theory, combining three completely separate notions into one coherent one that gives a formulation of the foundation of maths on a level that promises to be both the most rigorous and the most intuitive one yet.

In that way, history does matter. However, while there certainly are historical oversights that may or may not have slowed down progress for years and years(**), for the most part, new developments are there for good reasons, well understood and straight forward to apply and, unless you want to point out such a historically grown problem, there is often little use in going back into the past and solving some problem that nowadays is hard not to solve in less than five steps that might once have taken twenty or more steps of deep insight, just because the theory behind the techniques wasn't all too well understood yet.

(** All the various formulations of geometric spaces as used most commonly in physics today and also as taught in schools and universities, vector-, matrix-, tensor-, spinor-, twistor-, or quaternion formulations, are a confusing mess that often seems highly arbitrary and the notation hides a lot of the inherent internal structure of a given problem. Something that can be avoided with the grand unification of geometric spaces found in Geometric Algebra which makes working with all those things crystal clear and uniform. Had the people who once decided this already known that, they surely would have put more value into the teaching of Clifford- rather than Vector Algebra and we might live in a different world today. That's where historic retrospection is important.)


And while I'm at it: "Strecke" simply means line segment. In some rare cases it might mean "distance" but really, "distance" is just the length of a line segment, so this still indirectly refers to a line segment. - That's another thing. It would be advantageous if you kept your explanations to the common words of the language you use to write your explanations in. As far as I know (and I might be wrong) it just so happens that either the largest or the second largest group of people in here are from a German speaking country (the other one, from what I gathered, seemingly being from English nations, probably mostly from the US), but not everybody will understand a German word if there is a perfectly good English word to be used instead.

One of the rather rare exceptions to this is "Eigen-" which is a German term commonly used in Math and Physics and refers to something that is special to a particular mathematical object and, in some sense, gives that object its most natural description. The English word for that would be "Own-" but historically, "Eigen-" got used instead.

However, for things like "Schwerpunkt" or "Strecke" or other such words, there are perfectly good English words that you can use just as well or, since you are using English to write all your posts, even better.

Commentary

The Berührungspunkte is the next idea mentioned by Hemann as his discovery.

<a href="http://www.youtube.com/v/CizogTmSju4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/CizogTmSju4&rel=1&fs=1&hd=1</a>

http://www.ifi.unicamp.br/~assis/Archimedes-2nd-edition.pdf

http://www.cs.uiuc.edu/class/sp06/cs418/notes/10-MoreSplines.pdf

I think he has made a point that has been overlooked. The research into the barycentres, based around Möbius Barycentric calculus has revealed some taken for granted assumptions, not the least of which is the static barycentres of a system!

The law of levers is used in some calculations of the orbits of planets, and it's validity and valid use is not often checked by those who teach it.

The biggest discovery for me is the impact it has on rotational motion ! Because orbital motion is ellipsoidal it is clear that the barycentres cannot be stationary! A dynamic barycentres is not discussed very much anywhere, although it is known.

The barycentres of a system is not the same as the centre of symmetry, because the barycentres for a dynamic system is dynamic. Thus it is ok to calculate a fixed barycentres for a fixed mass, but it is misleading to calculate it for a dynamic system without mentioning it is dynamic.

The Relation between the Schwerpunkt and the Berührungspunkte is an. Indication of this dynamic nature. Hermanns insight not only meant he could determine it for " static" systems, but also for dynamic ones. This is a much more general idea than the principle of levers ! It reveals something about the transmission of force in space and action at a distance.

When it was first determined that a centre of mass can lie outside a body, some light bulbs should have switched on. Instead, the focus was on making the rotation uses look like the translation ones, and obscuring this physical difference.  Certain shapes can only be moved translatioally if a force or pressure applies equally to points or areas around the barycentres , otherwise rotation results!

The concept of angular momentum is flawed by this kind of analysis, without caveats regarding the Barycentre. Discounting the Barycentre has made the gyroscope into a modern mystery, whereas including it not only explains the gyroscopic behaviour but reveals how "empty" space can influence the dynamics of a system.

Newton's third law, the third stage in setting up his compete dynamic equilibrium or dynamic inertial frame, obscures a lot of internal physical interaction, that is body force interactions . We now commonly use a fluid dynamic model for body force interactions, but we use strain and stress parameters, both surface and onto to model these interactions. The Barycentric calculations are not to be found, because they are simply not included.

Yet, consider as a strain propagates through a body the Barycentre of that body also is dynamically altered. This effect means that for a given body , wave propagation of strain will induce rotational oscillation!

The Berührungspunkte with the Schwerpunkt by definition mean this is an inherent property of space, no matter how dense or vacuous! This is action at a distance, and it applies to any measurable magnitude. Thus any energy field, fluctuating will have a dynamic Barycentre that also oscillates. The significance is, that in such fields rotation will be induced as a matter of course for some fractal regional systems.

Typically a point in geometry has a value of position only. But physically, especially in Newtons point masses, motive must be attributed. The one motive that is ignored in a physical point mass is rotation, but it must also be included to make complete sense of rotational dynamics.  Berührungspunkte and the Schwerpunkt rotate, and thus the null vector hides the rotation of the barycentric point.

For a Schwerpunkt outside a physical mass, I may concede that point has no physical rotation, but in fact I do not know if that is the case, although it seems likely.

The doubt arises only because the transmission of any strain wave into a space outside the mass will inevitably obey the principles for that spatial density. It may thus severely dampen the incident strain wave resulting in null rotation. On the other hand it may increase the transmission of the strain wave resulting in higher rotation, if the density of the space is greater! In such a case it would take more time to speed up the rotation of. A denser medium, as we know.

Hallo kram1032,

I have the same problems as you following Jehovajahs thoughts. I think its important to have him write his idears down.
He has given a lot of excellent and understandable background information in the form of YouTupe Videos and papers.

Specialy the excellent lessons of:

• Norman Wildberger
• Leonhard Susskind
• DrPhysiks

At the end of last year I spent much time to see the videos which gave me a deep view in physics and mathematics.
I discovered that Jehovajah was writing about mathematics and physiks I am personaly very deep interested in.
I also have the problem to understand all his idears. So I started to write down my own thoughts.
By doing so many shapes of Jehovajahs thoughts become visible.

For me the the main problem is, that I do not have enough time to work on the subject.

Hermann

From the german Wikipedia I learned that Hermann Grassmann wrote two main versions of his "Ausdehnungslehre"
One  "Ausdehnungslehre von 1844" and a second Version form (1861 - 1862?)
Both seems to be completly different.
Also from these to basic books there seems to be different versions.
http://de.wikipedia.org/wiki/Hermann_Gra%C3%9Fmann

Lloyd Kannenberg had written an englisch translation of Ausdehnungslehre
http://www.amazon.com/New-Branch-Mathematics-Ausdehnungslehre-Other/dp/0812692764#reader_0812692764

Wouldn't this be a starting point to combare the german and the english text?

When I look at the following text of the 1844 Ausdehnungslehre I discover that the first Pages of "Vorrede" are missing. With this scan.
https://archive.org/details/dielinealeausde00grasgoog

The same problem with the following scan:
http://www.uni-potsdam.de/u/philosophie/grassmann/Werke/Hermann/Ausdehnungslehre_1844.pdf

The "Vorrede" in this scan seems to be complete. Know days a book has a "Vorwort" interesting how language changes over 150 years,
But it is the 1877 issue
https://archive.org/stream/dieausdehnungsl00grasgoog#page/n7/mode/2up

This I think this is the Kannenberg translation.
http://books.google.de/books?id=yeGPeaPVLKoC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

Hermann
P.S The word "Ausdehnungslehre" I would translate to "Expansion Theorie", I don't know if this translation meets the spirit of the book.