How to Explain Affine Point Geometry

Ramon Gonzalez Calvet, Institut Pere Calders, Spain

Hermann Grassmann based his extension theory on Möbius’ barycentric calculus [1]. According to Grassmann, a line is the exterior product of two points, a plane is the exterior product of three points, and an extension having what we now call $n$ dimension is the exterior product of $n+1$ points [2]. Also, the exterior product of a line and a point is a plane, and the exterior product of two non-intersecting lines is the affine space. How should we explain this Grassmann’s point geometry to our pupils instead of the usual vector geometry? The algebraic way to teach it [3] is by using barycentric and homogeneous coordinates [4]. It will be shown that they have many advantages such as the natural introduction to projective geometry and duality which become trivial when they are understood by means of pencils of lines and sheaves of planes.

[1] A. F. Möbius, Der Barycentrische Calcul (Leipzig, 1827). Facsimile edition of Georg Olms Verlag (Hildesheim, 1976).

[2] H. Grassmann, Extension Theory (2000), in the series History of Mathematics vol. 19, American Mathematical Society and London Mathematical Society, p. 138. Translation of the 2nd edition of Die Ausdehnungslehre (Berlin, 1862) by Lloyd C. Kannenberg.

[3] R González Calvet, Treatise of Plane Geometry through Geometric Algebra, (Cerdanyola del Vallès, 2007) p. 43.

[4] R. González Calvet, El álgebra geométrica del espacio y tiempo, (2011-), p. 64.