The most elementary C*-algebras are the unital abelian ones, and these form the only class to be completely classified. They are the algebras of continuous functions on compact Hausdorff spaces. A different way to describe these algebras is to say that all of their irreducible representations are one-dimensional, and in this way there is a natural generalization: those algebras whose irreducible representations are all of the same finite dimension. These are called the homogeneous C*-algebras, an example of which would be the algebra C([0,1],M_2) of continuous functions on the unit interval with values in the 2x2 matrices. However, they can be much more complicated than this. Using elementary linear algebra, we will introduce an isomorphism invariant for these algebras, and will show how to calculate this using some beautiful theorems from algebraic topology. This talk is designed for a wide audience, and all terms above will be explained.