"The search for truth is more precious than its possession." - Albert Einstein
The animation above is a stereoscopic illustration of the intersection (shown in thick black lines) of a 4-dimensional cube (hypercube) with a translating 3-dimensional plane (hyperplane). To see the demonstration in 3 dimensions focus on a point about half way between your eyes and the screen (between the two hypercubes). Within a few seconds a perfect 3-dimensional image will appear. There is also a detailed explanation of each of the different sections and their transitions. Also take a look at the fantastic interactive hypercube slicer.
Geometric tomography is concerned with acquiring information about a 3-dimensional object from 2-dimensional information such as projections or sections. Soon there will be much more here. In the mean time you can read the papers below about computing the maximum area cross section and the width of a convex polyhedron. See also the links below.