This chapter dealt with coordinate geometry and the extension of two and three dimensional coordinate geometry into the fourth and higher dimensions.  The question that continued to run through my head throughout this chapter was regarding the computer representation of the coordinates.  How does the computer "understand" coordinates in higher dimensions?  I would assume that with two dimensional geometry the computer can recreate axes and then plot them according to number of pixels away from the origins.  How does it do this with three or four or five dimensional geometry?  The picture on page 162 of the unit hypercube in four dimensional coordinate space makes sense.  There are four axes, each of which go in a different direction on a plane.  If we were to lift this off the page and create a model, we would be creating a three dimensional representation of it, with the axes going in different directions in space.  If it could be lifted again into the fourth dimension, I suppose the axes would once again diverge in four directions in hyperspace.  What is the difference between four directions in hyperspace and four dimensions in threespace?