A unified mathematical language for physics and engineering in the 21st century

The late 18th and 19th centuries were times of great mathematical progress, Many new mathematical systems and languages were introduced by some of the millennium's greatest mathematicians. Amongst these were the algebras of C lifford and Grassmann. While these algebras caused considerable interest at the time, they were largely abandoned with the introduction of what people saw as a more straightforward and more generally applicable algebra: the v ector algebra of Gibbs. This was effectively the end of the search for a un ifying mathematical language and the beginning of a proliferation of novel algebraic systems, created as and when they were needed; for example, spino r algebra, matrix and tensor algebra, differential forms, etc. In this paper we will chart the resurgence of the algebras of Clifford and Grassmann in the form of a framework known as geometric algebra (GA). Geome tric algebra was pioneered in the mid-1960s by the American physicist and m athematician, David Hestenes. It has taken the best part of 40 years but th ere are signs that his claim that GA is the universal language for physics and mathematics is now beginning to take a very real form. Throughout the w orld there are an increasing number of groups who apply GA to a range of pr oblems from many scientific fields. While providing an immensely powerful m athematical framework in which the most advanced concepts of quantum mechan ics, relativity, electromagnetism, etc., can be expressed, it is claimed th at GA is also simple enough to be taught to schoolchildren! In this paper w e will review the development and recent progress of GA and discuss whether it is indeed the unifying language for the physics and mathematics of the 21st century. The examples we will use for illustration will be taken from a number of areas of physics and engineering.