The first thing that this chapter did was to fuel my suspicion that the fourth dimension is not an average dimension. Indeed, as this chapter tells us, fundamental analogies may not hold true across all dimensions; the progression of the number of regular n-figures, from an infinite number in the plane, to five in 3-space, to three in 5-space and beyond, would seem to imply that there would be between 3 and 5 regular figures in four-space. But this is not the case. What other eccentricities does the fourth dimension hold? I'm prompted to wonder again if it is not an accident that our universe has a four dimensional space-time.
The discussion on folding up polygons, and the significance of the number which can fit around a single point, brings up the subject of tessellations. To tessellate is to fill a plane with figures, such that those figures fill all available space. In a sense, any painting is a tessellation, albeit with very irregular figures. What's more interesting is shapes which tessellate and are regular, or repeating, and among the most interesting tessellations are those which use only one figure; MC Escher is famous for his work with such planar tessellations. Here's a nifty Mac or PC program which lets you create and animate tesselating figures. You can use tessellations to make nifty "Magic Eye"-type stereograms; here's an example. Tessellations can also be done with certain regular polygons. Unless I'm mistaken, only three regular polygons can tessellate the plane: the triangle, square, and hexagon. It's easy to make a tessellating pattern by reflecting portions of an image onto opposite sides of a square- the background of my wacky fun page is an example. A hexagon would, I imagine, work just as well, though a triangle would not be as suited, because tessellations require the triangle to have alternating orientation. What about tessellating space? Cubes work well, the Rubik's Cube is an example. What about the other regular polyhedra? As far as I can tell, none of them will work, though this assumption comes not from mathematics but from my own attempts to visualize these shapes stacking together. Our explorations in class of subdividing polyhedra suggest that this may be the case, as the cube was the only one which could be divided into its own likeness. Does this suggest that the hypercube is the only 4-hedra which can be cut up into little versions of itself?
