# Chapter Five comments

## Leo Mariani

This chapter is definitely the most interesting one so far. Is it true that the Greeks only constructed figures using just the straight edge and the compass? Did they have other tools at their disposal or did they consider only these tools to be perfect? Many concepts that I was not aware of have been introduced in this chapter. Though it seems that I should have been aware of them having taken high school geometry. I find it quite fascinating that it is impossible to trisect a 120š angle. I wish that you could have included a more formal mathematical proof in your book, but I guess it was not allowed. The progression from regular polygons to regular polyhedra to regular polytopes is done wonderfully. The explanations for the existence of these figures are also very easy to grasp. I had never heard of the concept of a dual before, and I am somewhat intrigued by them. It seems that the regular tetrahedron is the perfect shape as it is self-dual, and in four dimensions the self- duality of the twenty-four cell also has beautiful symmetry though it is somewhat more difficult to see. Does this pattern continue to higher dimensions? Is there always a self-dual figure in higher dimensions? What is the relationship between these figures? The fold-outs are by far the most practical part of your book as they are easy to construct and provide such excellent practice in visualization techniques. I am glad that you finally made a more formal connection to art though I think that this could have been explored further. Perhaps you do this in a later chapter?