So what are the objects that we do Clifford algrbra with?
I want to go back to Grassmanns Schwerpunkt!. Grassmanns stated object was the segmented line with 2 points one at each end, but when he found that a common theme was developing for the four main arithmetical algebraic operations he went back to the point and conceived the Schwerpunkt. There are only points in this scenario, ,mbut if you multiply 2 points that meant construct a join of the 2 points.. The order was important here.
So what was addition of points? The answer seemed to be a count of the points, with the count attaching to a third point the Schwerpunkt, somehow between the 2 or more points. Thus the position of that point requires multiplication to specify. Thus A + B = 2C and C = AB/2
C + D = 3E with E = CD/3 = (A B) D/6
6E = A +B +D
As you draw the pattern of connections you can see the outer shape forming with the Schwerpunkt somewhere near the centre of gravity or symmetry.
Well this is what I think is the fundamental object of the Clifford algebras and I think that the term node about sums it up.
When we are taught geometries, we always start with synthesis . For Clifford and Grassmann algebras it is better to start with the analysis.
So in 3 dimensions we start with a " trivector" or a 3 d node. Its parts are, the whole 3d space, 3 planes that pass trough the node centre containing any 2 of the 3 lineal segments that diverge from the node centre. Three lineal segments as unit " vectors" , and finally the Schwerpunkt as the node, which means the centre has a count that scales all the lineal segments at once. A large Schwerpunkt has large unit segments and a small one has small unit segments. The Schwerpunkt is like Hamilton's " axis of calculation" rolled up into a point! . For geometrical purposes it is like the radius to the node as centre, which makes it easier to use Cotes Euler identity.
The objects can be added, multiplied and this can be done distributiv ly and associativity. And it can be done between parts of each node. So a Schwerpunkt can multiply its node or any other nodes elements. Lone segments add either by the tip to base or for multoiplication Base to base.
Each element on a node is symbolic for a whole space full of nodes parallels and intersecting line and spaces.
So nodes build up structures like strut..
This is the best introduction to the topic of Geometric or Clifford calculus.
http://www.fractalforums.com/new-theories-and-research/geometric-algebra-geometric-calculus/
Evenso the notions of vectors and numbers are still confused.
http://geocalc.clas.asu.edu/html/Overview.html
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node2.html
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node5.html
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node6.html
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node7.html
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node8.html
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node9.html
http://www.mrao.cam.ac.uk/~clifford/introduction/intro/node10.html
David Hestenes universal geometric calculus.
http://geocalc.clas.asu.edu/html/UGC.html
http://geocalc.clas.asu.edu/pdf/UGA.pdf
The wedge product defined by area rules which is then obscured, and the dot product by projection which is gain obscured. A third product called a Clifford product is defined as a sum of the 2.
None of these products are linear, nor is a bivector linear. This is why Grassmann defined them as lineal! Lines are used to make forms that bound space and intersect and join points in space.
Hestenes makes an important point. Cayley's matrix algebra is just one way to rewrite the terminology he has adopted. The terminology or concept handlers used is crucial to the flexibility of the method of analysis and synthesis. The claim of generality or universality requires the ability to change terminology flexibly to demonstrate! Thus isomrphisms and Homeomorphisms are employed at crucial stages to make points that are otherwise not at all obvious. Part of the method then is in demonstrating these homomorphisms .
A source for Norman's approach
http://rsta.royalsocietypublishing.org/content/356/1740/1123.full.pdf
There are clearly several orientations regarding Clifford nodes and Algrbras, andHestenes is the most influential. However, Grassmanns work is worthy of individual study, as the modern retake on it has misinterpreted parts of it. That is not to say they have not got it right, or advanced as Grassmann wished for. However there are distinctions which have been misunderstood either by others or myself or both.
I now find it inconvenient to separate out Algrbra as a subject, because it has become clear that the Arithmoi were the goal of Greek philosophy of space and quantity in space. Algebra was a historical renaming of consistent arithmetical methods of reasoning using proportion, or better said logos analogos methods.
The Children of Shunya 