Arrow Spaces: A Unified Algebraic Approach to Euclidean Geometry and Inner Product Spaces

Abstract

Given a postulated set of points, an algebraic system of axioms is proposed for an ``arrow space". An arrow is defined to be an ordered set of two points (T,H), named respectively Tail and Head. The set of arrows is an arrow space. The arrow space is axiomatically endowed with an arrow space ``pre-inner product" which is analogous to the inner product of an inner product vector space over R. Using this arrow space pre-inner product, various properties of the arrow space are derived and contrasted with the properties of a vector space over R. The axioms of a vector space and its associated inner product are derived as theorems that follow from the axioms of an arrow space since vectors are rigorously shown to be equivalence classes of arrows. With arrow space's tools, Hilbert's axioms of Euclidean plane geometry follow as theorems in arrow spaces. Applications of using an arrow space to solve geometric problems in affine geometry are provided. Examples are provided to equip complex Hilbert spaces with a structure analogous to the structure of Euclidean geometry and trigonometry.