I construct and examine the properties of Lie and Clifford algebras which are used to describe certain types of particles. These algebras are then related to the traditional theory of division algebras. Quaternions are applied to these algebras and their properties are exploited to model physical properties of particles. The Lie algebra so(3) is shown to be isomorphic to su(2) and can be used to represent the Pauli spin matrices. The relationships between the three Lie Algebras are shown geometrically using combinatorics. Finally, the weak force is related to the remarkable fact that so(4) = su(2)+su(2). The weak force is naturally described by restricting to one of the two copies of su(2), one of which is left-handed, the other of which is left-handed.