So you may have seen a flurry of activity from me on the wiki, adding counts and descriptions of elements to all the 4D and 5D tapertopes.
I quickly realised I'd need a convention for naming these, as there are many different ways to curve n-surfaces.
I think I have a way of enumerating and naming the curved elements of tapertopes now, which I'll elaborate on later, but suffice to say a side effect of this is to create a set of not only all tapertopes and their curved elements, but all toratopes too, since all toratopes are elements of higher dimensional tapertopes (I think). So I figured, why stop there - I've been looking for a way to reunite the tapertopes, toratopes and bracketopes ever since they were first split up.
This brought me to the question of curved elements of bracketopes - specifically those of crinds and tegums since those are the ones we don't yet have.
The standard 3D crind, or circle{square,line}, has four curved faces which I call lunes. circle{triangle,line} has 3 lunes, and in general circle{n-gon,line} has n lunes. I'd like to refer to all these lunes as the same object, regardless of how many sides the crind has, much like how the iscosceles triangles of n-spindles are all called triangles even though they have different angles.
With that in mind, my question is - how does one identify different types of lune in higher-dimensional crinds? And do unique elements exist in higher-dimensional tegums? The 3D bicone, for example, only repeats curved surfaces already known - specifically, it has two curved cone-surfaces, whereas the cone only has one, but they are the same object. Does this continue into higher dimensions, or are there higher-dimensional tegums with objects not seen in their cone/pyramid equivalents?