I've thought of a way to make our ideas more rigorous using equivalence classes.

If you're not familiar with these, read up on equivalence relations and equivalence classes on Wikipedia. It's an extremely useful concept in mathematics that isn't taught early enough in my opinion. A simple example is the equivalence relation "has the same birthday" on the set of people. This says that two people are "equivalent" if they have the same birthday. It partitions the set of all people into 365 equivalence classes, each of which contains millions of people who have the same birthday. Each person is in exactly one equivalence class.

My basic idea is that our toratope notation can be used to describe an equivalence class. For example, "(II)I" is the equivalence class of cylinders. Any cylinder, whether a tall thin solid cylinder, or a short wide hollow cylinder, or a pair of parallel circles, is an element of the set (II)I. Similarly, any tiger, solid or hollow, and for any values of the three radii, is a member of the set ((II)(II)). Two sets will be "equivalent" if they are in the same toratope class. We could come up with a more rigorous definition of this equivalence.

I think this will make it much easier to talk about toratopes without getting bogged down too early in a discussion of frames and parameters, so it will make our paper more interesting. We can add something on frames and parameters later.