A Few Comments on the Clifford Algebra $${C\ell_2}$$ of the Standard Euclidean Plane

Abstract

This paper is a continuation of the author’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 $${C\ell_2}$$ associated with the standard Euclidean plane $${\mathbb{R}^2}$$ . Miquel’s theorem will be given in the algebraic context of the even Clifford algebra $${C\ell^+_2}$$ isomorphic to the real algebra $${\mathbb{C}}$$ . 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 $${\mathbb{C}P^1 = P(\mathbb{C}^2)}$$ 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 → S 3 → S 2 leads to the space of spinors $${\mathbb{C}^2}$$ of the Euclidean space $${\mathbb{R}^3}$$ and to the isomorphism $${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$$ .