# Maths - Geometric Algebra - Categories

We have already seen that Geometric Algebra takes a vector algebra of dimension 'n' and generates an algebra of dimension n2.

So the properties of the basis vectors largely determine the properties of the entire algebra. Unless otherwise stated the algebras will be based on conventional vectors which square to positive numbers, however it is possible to base the algebra on vectors which square to negative numbers or zero.

So we can define a geometric algebra by its signature of the form: Gp,q,r

where:

• p = number of basis vectors which square to a positive number
• q = number of basis vectors which square to a negative number
• r = number of basis vectors which square to zero

We can take subsets of these algebras, denoted as follows:

• G+ = take only even grades of the algebra (0=scalar,2=bivector...)
• G- = take only odd grades of the algebra (1=vector,3-trivector...)

Here are some examples:

Geometric Algebra based on Designation
2D mulitvector based on standard vector geometry G2,0,0
3D mulitvector based on standard vector geometry G3,0,0
4D mulitvector based on vectors which square to positive G4,0,0
complex G0,1,0
dual G0,0,1
double G1,0,0
quaternions G+3,0,0
space-time algebra (STA) G1,3,0

So can we generate a geometric algebra from any other algebra? Can we generate a geometric algebra from another geometric algebra?

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.      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.      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.