Maths - Clifford Algebra Theory

Clifford Algebra is the most general of this types of algebra. This page discusses the structure and rules of Clifford Algebra and how it can define other algebras like Geometric Algebra.

Clifford Algebra starts with a vector space 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.

theory

Because addition in clifford 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 Clifford Algebra we must define multiplication rules. In most types of algebra operations are 'closed', 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. 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 n=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!

n=2

If we take the case where n=2 then we have a total dimension of 4, that is an element in this algebra can be represented by 4 scalar numbers, 1 for the real, 2 for the vector and 1 for the bivector.

n= 2 case

n=3

If we take the case where n=3 then we have a total dimension of 8, that is an element in this algebra can be represented by 8 scalar numbers, 1 for the real, 3 for the vector, 3 for the bivector and 1 for the trivector.

 

n=3 case


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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 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.

 

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.

 

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