A. N. Whitehead's Geometric Algebra

My interest in A. N. Whitehead's geometric algebra grew out of dissatisfaction with the vector algebra that is taught to students of science and engineering. The unsatisfying aspect of vector algebra is that it attempts to be a geometric algebra that enables one to solve geometric problems in a coordinate-free way, yet it falls short of this goal by being unable to express many of the notions that are needed by a geometric algebra. For example, the vector language does not have a syntax for the join and intersection of linear subspaces. Consequently, it is not straightforward to write down (say) the point at the intersection of a line and a plane in 3-d space. This deficiency means that the theorems of projective geometry, which have important applications in computer vision, for example in camera calibration, cannot be expressed in the language of vector algebra. Another deficiency is that vector algebra is asymmetric with respect to translations and rotations. Translations are easily represented in a coordinate-free way, but there is no syntax for rotations. In order to represent a rotation, one has to introduce a coordinate system and use matrices.

Similar concerns about the deficiencies of vector algebra have been expressed by others. However these authors all suggest, in various ways, that a geometric algebra should be based on a Grassmann algebra of vectors. However, it seems a little bizarre to choose vectors as the basic elements of a geometric algebra because a vector is something which naturally arises in the context of affine space - a space with a notion of parallelism. Instead, I think that the Grassmann algebra should be based on points and hyperplanes instead of vectors. This follows from the nature of geometry as expressed by Felix Klein's Erlangen Programme. Vectors are displacements. As already noted, they arise naturally in an affine space. According to Klein's genealogy of geometries, which is shown below, all the geometries in which space is homogeneous are constructed from projective geometry by specifying certain structures to be absolute (invariant). As the diagram shows, affine geometry is not fundamental from the Kleinian point of view. Instead projective geometry is fundamental, and the elements of projective geometry are points and, dually, hyperplanes. 

                           projective 
            ___________________|__________________ 
           |                   |                  |
         affine            elliptic           hyperbolic
      _____|______
     |            |
 Euclidean    Minkowskian

Interestingly, in 1898, A. N. Whitehead published a book entitled A Treatise on Universal Algebra which contained an exposition of a geometric algebra built from a Grassmann algebra of points instead of vectors. Whitehead's algebra of points naturally represents projective space in accord with Klein's genealogy of geometries. However, in Whitehead's book, points have a privileged status compared to hyperplanes. This is contrary to the principle of duality which holds in projective geometry. According to which, any theorem involving points can be turned into a theorem involving hyperplanes if the points are replaced by hyperplanes and a few words altered in the statement of the theorem. The privileged status of points in Whitehead's theory complicates matters. He finds two products which he names progressive and regressive. However, when one treats points and hyperplanes on an equal footing according to the principle of duality, there is only one product and the theory has an appealing simplicity.

The following book is a detailed exposition of Whitehead's algebra. It has theory and examples using the software. The applications are in computational geometry, computer vision and in the physics of space-time.

Recently, I've found that Whitehead's algebra is a module over a Laurent polynomial ring. There is a fair amount of work to do to update the material to reflect this better understanding. In the interim, an introduction is provided here.

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