Quaternions and Finite Projective Planes

Projective planes were discovered by Renaissance artists who needed to depict tiled floors on canvas. Quaternions, discovered in the nineteenth century, were used by physicists to represent rotations in three dimensions, which do not commute with one another, In the early twentieth century, mathematicians discovered that quaternions could also be used as coordinates in projective planes where certain theorems of Euclidean geometry fail and the rules of ordinary algebra do not apply to coordinates. This course focuses on an article published at the start of the twentieth century by American mathematicians Oswald Veblen and Joseph Wedderburn that constructs a type of finite plane that violates some of the axioms of geometry and requires quaternions as coordinates. We also focus on another article, published at the dawn of the computer era by the great American geometer Marshall Hall, that describes an exhaustive search, with the aid of a primitive computer, for all finite planes of order 9. We replicate and extend the results in these articles using the R scripting language, in the process delving into group theory, finite fields, quaternions, and finite geometry. A secondary goal is to compare the transition from classical to modern in mathematics with similar transitions in music, art, and poetry.