I enjoyed the discussions of generalized dimensional shadows. I in particular would have been interested to look at shadows of some of the trickier objects. All of the higher dimensional objects considered in the book were basically polytopes. This tied in neatly with the discussion of counting edges, faces and the like, but (as usual) left more questions for me than answers.

In particular, I would like to consider some shadows of 4-d objects with curved hypersurfaces. At this point it might get a little tough to talk about shadows without a clarification. In B3D, the shadows are of wireframe models; we see through the faces and bodies and whatnot, and the only things which cast shadows are edges. In thinking of curved shadows, since it is sometimes tricky to talk about edges, let's just assume that all the objects are "filled in" in an intuitive sense, and that all of the stuff inside casts a shadow as well as the edges.

The first such object that came to my mind was a hypersphere, and I was a bit disappointed to realize that it didn't cast a very interesting shadow. No matter how the sun shines on it, its 3-d shadow is a sphere and its interior. So I started thinking about hyperellipsoids, since they popped to mind next. A 3-d ellipsoid, when light shines parallel to one of its axes, will have a shadow similar to one of the three circles or ellipses that define it. But when light shines at weird angles, one gets unusual results. I don't know any name for the figure one gets, but it looks like something of a lopsided ellipse, or an ellipse that has had some mildly interesting stretch performed on it.

So I would imagine possible 3-d shadows of the 4-d hyperellipsoid would be the four ellipsoids that define it, if the angle is just so, but more likely, something of a lopsided ellipsoid.

The entire shadow concept seems like an interesting one mathematically, and in other ways. Since obviously two different figures could have the same shadow, I'm sure there are all sorts of wonderful theorems one could prove about what figures have related shadows under certain circumstances.

I was also glad that this chapter started on a philosophical note, with the shadow allegory from The Republic. The idea that the reality of everyday experience is just a shadow of a higher reality is a venerable one in both Eastern and Western belief systems. It is also interesting to combine that thought with the fact that different objects may cast the same shadow. I habitually dismiss claims that "things may not really be as they appear to be" as inherently illogical, but this is one way to keep a somewhat rational viewpoint of the world and still consistently believe that human beings are somehow manifestations of a "beyond," something which many religions assert.

### Exercises

Find two distinct three dimensional objects (notional or physical) which can cast similar shadows under the right circumstances, and have at least two such shadows. For example, a right hexagonal prism of height two and edge length one and a cube of edge length one share as possible shadows a square and a regular hexagon.

Take an object into the sun, and while slowly rotating it, look at the shadow. For most objects, at certain points in the rotation, the shadow will suddenly change; a side appears out of nowhere, an edge that had been curved turns flat. Identify some of these points, and convince yourself of a decent reason for the shadow to suddenly change.