A Few Comments on the Clifford Algebra of the Standard Euclidean Plane

作者：Angles Pierre^*

来源：Advances in Applied Clifford Algebras, 2012, 22(3): 519-535.

DOI：10.1007/s00006-012-0346-4

• Anharmonic
• Alexandrov compactification
• Cauchy plane
• Clifford algebra
• Clifford mapping
• Clifford-Hopf fibration
• cross ratio
• homography
• Miquel%26apos
• s Theorem
• spinor
• universal property

This paper is a continuation of the author%26apos;s plenary lecture given at ICCA 9 which was held in Weimar at the Bauhaus University, 15-20 July, 2011. We want to study on both the mathematical and the epistemological levels the thought of the brilliant geometer W. K. Clifford by presenting a few comments on the structure of the Clifford algebra associated with the standard Euclidean plane . Miquel%26apos;s theorem will be given in the algebraic context of the even Clifford algebra isomorphic to the real algebra . The proof of this theorem will be based on the cross ratio (the anharmonic ratio) of four complex numbers. It will lead to a group of homographies of the standard projective line which appeared so attractive to W. K. Clifford in his overview of a general theory of anharmonics. In conclusion it will be shown how the classical Clifford-Hopf fibration S (1) -%26gt; S (3) -%26gt; S (2) leads to the space of spinors of the Euclidean space and to the isomorphism .

• 出版日期2012-9