The concept of a shadow is that of a 3-D projection onto a 2-D surface, however the region between the object and the surface is also in shadow. Can this volume also be considered as part of the shadow? What if the shadow volume were attached to the original object? Taking the object and its projection as a whole would give a dual dimensional object with one component fixed in 3 space with another component, the volume-shadow, that could vary in 3 space with regard to the position of the light source. This new object, I think could be an analog to a 4-D object.
Imagine a pyramid with the sun casting light down on it. On the flat desert around the pyramid a shadow is projected in the shape of a triangle with one side of it "attached" to the base of the pyramid. The tip of the shadow can be thought of as a point in four space and if you connect the apex of the pyramid as well as each of the vertices in the base to the tip of the shadow you get something resembling a 4-pyramid in 3 space.
I found the application of higher dimensional geometries in practical research in the chapter to be very interesting. For humans to find correlations and useful information in tables of numbers is next to impossible. The best way to understand data from a wide range of variables is to visualize it. Up until recently data for the most part has been visualized in 2 dimensions, with an x and y coordinate representing two different variables. By viewing the graph a relationship is hopefully found between the two. However, for a lot of processes that are of interest there are many more that two variables at work. A squiggly line in two dimensions could turn out to be a smooth spiral in three. With the aid of computers multi-variable data can be modeled as never before, however I feel there are severe limitations. After 3 variables are added what's next? Color could be used, and then animation could be incorporated, but then after that those trying to find relationships are in the same predicament as before. Ultimately slicing has to occur in order to comprehend higher dimensional data series, but that leaves holding other variables constant to try and extract information. Graphs can only go so far in simplifying complicated information.