PRELIMINARY STUDY ON THE BASIS OF GEOMETRIC ALGEBRA FROM A NEW PERSPECTIVE
几何代数基础新视角下的初步探讨

A problem about the basis of geometric algebra in this paper is posed: what kind of algebraic structures may be set up on the geometry given transformation groups?Firstly it is proven that it is impossible to define an operation which is invariant under distance-preserving transformations among points of Euclidean plane so that Euclidean planar point set can form an abelian group under the defined operation. Further discussions show that only by expanding Euclidean geometry into particle geometry could the commutative and associative operation which is invariant under distance-preserving transformations be established, and this operation can only be the addition operation in the particle geometry. If we hope to make it an abelian group under the operation we have to introduce vectors. At last, we conclude with a discussion of the significance of the results and put forward a number of issues.