  Tropical mathematics redefines the rules of arithmetic by replacing addition
with taking a maximum, and by replacing multiplication with addition. After
briefly discussing a tropical version of linear algebra, we study polynomials
build with these new operations. These equations define piecewise-linear
geometric objects called tropical varieties. We explore these tropical
varieties in two and three dimensions, building up discrete tools for studying
them and determining their geometric properties. We then discuss the
relationship between tropical geometry and algebraic geometry, which considers
shapes defined by usual polynomial equations.
