## Response from Prof. B.

A prism determined by a regular polygon in the plane is obtained by translating the polygon in a direction perpendicular to its plane and connecting corresponding points. In general, the sides of the prism will be rectangles, and if you translate by a distance equal to the length of each of the sides of the original polygon, the sides will all be squares, giving an example of what is known as a "semi-regular" polyhedron, where all faces are regular and all vertices look alike, in a very strong sense.

To get an antiprism, start with an n-gon as before, and translate, but now rotate the top n-gon by 180/n degrees, so we get an upper polygon where each vertex is not above a vertex of the lower one but "in between" two of them. Connecting each vertex of the upper polygon to the two vertices closest to it on the bottom gives a figure with 2n isosceles triangles on its sides, with n of them attached to the sides of the upper n-gon and n attached to the lower. The effect is that of a toy drum, with lacings going from the top to the bottom. I wish I could insert a picture a this point.

Your comment about the square kit in Lineland is a good one. The original kit had four segments end to end, and that can be built in Lineland all right, but as you point out, you can't even begin to carry out the instructions because you can't superimpose two segments in that one-dimensional world.

The picture of the 600-cell is actually wrong. There are more edges that should have been included in the picture, and I will bring a picture in to show what it will look like in the new edition. The reason that it looks like a circle is that there are so many sides!

Visualizing the regular figures in n-space is very difficult, and at a certain point we have to resort to algebraic analogies, using coordinates when they finally appear in Chapter 8.

No one would contradict the statement that a circle is regular, but at the same time it is not a regular **polygon**. *Regularity* in this case involves some kind of symmetry, and the circle and its spherical analogues have much more symmetry than even the most regular of polygons or polyhedra. We'll have better ways of describing this as we go along.

I too like the Max Bill progression. It's a powerful visual image. I should talk about my visit to his studio just a few years ago.

Good work but your entry seemed to stop a bit abruptly?