This thesis introduces IC4D, a means of visualizing 4 dimensions, and a tool for interactively exploring 4-dimensional data. IC4D is a computer program that allows a user to freely rotate a virtual 4-dimensional object, in 4-dimensional space, while observing its continuously transforming 2-dimensional shadow. In this way, the user readily develops an intuitive grasp of the 4-dimensional structure. The 4-dimensional objects explored in this paper are examples taken from mathematics and physics: The Klein bottle embedded in 4-dimensional space; the Complex Gamma Function; full phase space flows of the famous Henon-Heiles Hamiltonian; and sections of the phase space orbits of a ball bouncing between a flat wall and a rippled wall. IC4D is being developed on NeXT workstations for computers running the NEXTSTEP operating system.