## Response from Prof. B.

You are quite right that the animation does make a big difference in trying to see how these objects look in four-space. I find it helpful to imagine a polyhedral torus filling in sixteen of the faces of a hypercube in perspective. In the "cube-within-a-cube" model, it seems all right to use the four vertical squares of the smallest cube and the four vertical squares of the largest cube, with their free quadrilateral rims connected by "flanges". Once you see that, you can imagine what happens as this object begins to rotate in four-space. Try this exercise with the picture of the rotating wire-frame model in the book. You can even trace the picture and shade in some of the faces to help you with the visualization. Then you can try to subdivide once and see a torus with 64 faces as a slight distortion of the 16-face example in the hypercube.

Once again, it is analogy that helps to clarify things. Think of a pair of circles of latitude on the orinary sphere in three-space as they are projected stereographically into the plane. If we approximate one of these circles by a spherical quadrilateral, then the image below will have some distortion in it. In any case, if we look only at the images of the four points, then they can be connected to give an approximation of the image of the circle. Then subdividing further, we approximate the circle in the plane by an octagon that is a "swollen up" version of the square, and so on. Does that help? It is not just an analogy--it represents a slice of what is happening in the higher-dimensional situation.

With respect to the "depth" of shadows, a Flatlander would see the image of a circle under stereographic projection as an ordinary curve in the plane, although we know that it is produced as a shadow of something up in three-space. Just how that shadow is rendered in the plane will affect the illusion of it being something substantial. In a similar way we might see something wraithlike in three-space, as insubstantial as a bubble or perhaps something akin to a "spherical smoke ring". We might not think of it as a "hard surface" but we could still appreciate its shape, I believe. Comments?