Maths - Grassmann Algebra

Grassmann Algebra starts with a vector space (or more generally a module) of dimension 'n' and from it generates a vector space 'A' of dimension 2n or, another way to think about it, the vector space 'A' is made up of a number of smaller dimensional vector spaces.

The elements in Grassman Algebra consists of a compound element made up from parts of different 'grades':

Grade Description
0 Real Numbers (or whatever the base element over which the vectors are based)
1 Vectors (or more generally modules) with a given dimension 'n'.
>1 Higher order bases 'generated' from the above bases by the multiplication rules which we will describe.

Addition in Grassmann algebras is always defined by adding corresponding terms, this is relatively simple in that, it does not involve interaction between the various components.

So in order to define a Grassmann Algebra we must define multiplication rules. We require the algebra operations to be 'closed' and finite, that is, when two elements of the algebra are combined the result is another element of the algebra. For example a complex number multiplied by another complex number will always produce another complex number. Grassmann and Clifford Algebra has a number of sub-algebras within one big algebra, however the sub-algebras are not closed within themselves only the overall algebra is closed. Apart from the case where grade=0 which is just the real numbers all clifford algebras have a subset which are vectors, but multiplying a vector by another vector produces something that is not a vector, we can think of this as generating other types.

The number and dimension of the vector spaces have the structure of the binomial triangle. For any dimension 'n' of the vector space that we start with the vector spaces are given by the following sequence:

n                                   2n=
0                 1                 1
1               1 1               1+1
2             1 2 1             1+2+1
3           1 3 3 1           1+3+3+1
4         1 4 6 4 1         1+4+6+4+1
5       1 5 10 10 5 1       1+5+10+10+5+1
6     1 6 15 20 15 6 1     1+6+15+20+15+6+1
7   1 7 21 35 35 21 7 1   1+7+21+35+35+21+7+1
8 1 8 28 56 70 56 28 8 1 1+8+28+56+70+56+28+8+1
                                       

The dimension of the kth element is given by:

n!/(n-k)! k!


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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 

flag flag flag flag flag flag New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation.

 

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