Geometric Algebra, Geometric Calculus

[Roquen]

All I've done is reformed, say (x²-y²-z²) = (x+y+z)(x-y-z)+2yz to avoid the possibility of catastrophic cancellation and specify the direction of the line through the origin with respect the angle doubling is occurring.  The linked equations are ignoring the translation part.

[kram1032]

yeah, ignoring the translation part is what I meant by doing it in one iteration

really nice work, now that there are examples of it all in the other thread ;P

[jehovajah]

Excellent introduction to GA Kram.

I had a difficulty with the kind of mathematical Houdini approach of others, but you set it out for the most part sensibly and in a motivating way.

The Algebraic manipultions is where most non mathematicians get lost. Then the 1 to 1 isomorphisms( one thing in one notation is attached to only one thing in another notation, so we can say those things identify with each other, and the behaviours are the same for one notation as it is for the other notation) is the next place where people get lost.
Finally the lack of specificity in the term vector  also loses people.

Honestly, each of these areas of loss and confusion I experienced reading your article, but because I knew what I was looking for I could see how you had tried to minimise these crisis points.

I learn something new each time I look at Grassmanns work, and I still think we over complicate it unnecessarily. The real problem is overcoming decades of bad teaching practices to be able to grasp the naturalness of these arithmetical concepts.

Nobody explains why the signature of a system is -1,0,1 because no one understands Grassmans dynamic model. I would highly recommend Norman Wildberger for all things Grasdmann even if he still gets some of it wrong, he is mostly right as far as the Grassmann method of Analysis is concerned. I have learned a lot through his clear exposition.

Grassmann defines Strecke as Die Begräntzte Linie, but also in the same context as the symbol for many discrete and continuous magnitudes in dynamic change. In this way he takes what is trigonometry of the parallelogram and the Triangle deconstructed into 3 points and 4 points with their associated stretch between them, into the subtleties of our interacting with space.

But what few grasp is that through Lagrange and his Father Justus, Grassmann was giving a version of Newtonian Analytics. And Newton was giving us his deep meditation on Pythagorean Mechanics and Astrological thought, as promulgated by the Platonic Academies.

Of those associated with the Euclidean branch of Plato's academy in Alexandria 2 stand out: Archimedes and Apollonius. But it is the work of Eudoxus that most clearly introduces the Pythagorean lineal algebra in the Logos Analogos format.

Making sense of how we interact with space before believing results of Algebraic manipultions is crucial to our sanity and sense of identity.

[jehovajah]

This is link to the Cayley contribution to Grassmans analytical method.

http://rsta.royalsocietypublishing.org/content/356/1740/1123.full.pdf

It takes a familiar Clifford algebra approach, and is too technical for my tastes, but if you have followed Norman and his video on area and volume in his WildLinAlg series you will recognise the point being put across.

How unfortunate that Mathematicians feel they have to bamboozle there audience with utter tripe while attempting to make a valuable contribution to methodology!

[Roquen]

Years ago I skimmed the complete works of Cayley (available online)...it's quite interesting the range of topics and how various fields ended contributing to the development of linear algebra.  While I'm babbling about old math papers I found it kind of interesting to see that "flame-wars" existed (at least) in the 19th century...they just did it in journals.

"What are you doing in there?"
"Writing an article...someone in Philosophical Magazine is wrong."

[kram1032]

"What are you doing in there?"
"Writing an article...someone in Philosophical Magazine is wrong."

[Roquen]

I figured someone would get the reference.

[jehovajah]

 
Can you imagine, if they had the communication back then?  Fast connections are not always a good thing! Lol!

[Nahee_Enterprises]

    Years ago I skimmed the complete works of Cayley (available online)... it's quite interesting the range of
    topics and how various fields ended contributing to the development of linear algebra.   While I'm babbling
    about old math papers I found it kind of interesting to see that "flame-wars" existed (at least) in the 19th
    century... they just did it in journals.
    "What are you doing in there?"
    "Writing an article...someone in Philosophical Magazine is wrong."

            http://imgs.xkcd.com/comics/duty_calls.png

    I figured someone would get the reference.

Yes, a very good reference.     
 

[Nahee_Enterprises]

   
    Can you imagine, if they had the communication back then?   Fast connections are not always a good thing!  Lol!

Just wait and see what will happen when technology is able to directly link the brain to the computer for transferring thoughts immediately as they are thought.   
 

[Dinkydau]

I figured someone would get the reference.

Haha

[Roquen]

Alan Macdonald has an updated survey paper:

http://faculty.luther.edu/~macdonal/GA&GC.pdf

[kram1032]

nice, thanks

[kram1032]

If I had to guess, this probably isn't quite legal, but I found this on google scholar:
Guide to Geometric Algebra in Practice - Leo Dorst, Daniel Fontijne and Stephen Mann (pdf)
Normally, you can only get a preview of it on Google books. It might not be up for long.

[jehovajah]

Good catch kram1032.
Sometimes they do make books available for free download, but springer are normally very tight!
Lol!