Affine and Projective Planes (and the Skew Fields Underneath Them)

<p>In this paper, we study affine and projective planes, and their algebraic foundations. Following the early sections of Emil Artin's book Geometric Algebra, we develop a theory of affine and projective planes. His text develops a correspondence between affine planes and skew fields. The notation and terminology used in Artin's text is somewhat dated, and so we use more modern algebraic constructs and ideas to make his ideas more legible to a modern reader.</p> <p>By establishing a correspondence between skew fields and affine or projective planes, we can move from a synthetic setting of axiomatic models of planes, to a constructive model of geometry. Basing geometry on algebraic structures allows us to appeal to ideas from algebra to discuss properties of geometry.</p>