That's definitely more clarifying, thanks wendy!

I should probably point out that the whole C2, C3, C4, C5 is just my little invention, to help describe some common features with how a toratope makes lower-D intercepts. What I noticed over time, is that the number of

inner-most nested circles ( or n-spheres) make the overall size and shape of the array. If there are only two nested, it intercepts low enough as a rectangular array, either vertical or horizontal. It will cut, non-empty, no further down than a rectangle array. If the shape has three nested circles, it makes a cuboid array. If there is four, it makes a tesseract array, etc. So, Cn = n-frame of circle^n prism. I named them the Clifford n-tori, to expand the definition a little. This also means Cn cuts all the way down to an n-cube array of 2^n points. These points are the location of a smaller inflating toratope.

In trying to find a good generalization for this, I made a connection around the nested circles being a solid rotatope prism, whose margin had been inflated by a lower toratope. If we swapped out the nested circles with a single dimension, we end up with the toratope that inflated the margin.

For example:

- Take ((II)(II)), replace 2 nested circles, and we get (II), as S1xC2, cuts to ((I)(I)) : 2x2 array of 4 (II)

- ((II)(II)I), replace 2 circles, and we get (III), as S2xC2, cuts to ((I)(I)I) : 2x2 square of 4 (III)

- ((II)(II)II), replace 2 circles, and we get (IIII), as S3xC2, cuts to ((I)(I)II) : 2x2 square of 4 (IIII)

- (((II)(II))I), replace 2 circles, and we get ((II)I), as T2xC2, cuts to (((I)(I))I) : 2x2 square of 4 ((II)I)

- ((((II)(II))I)I), replace 2 circles, and we get (((II)I)I), as T3xC2, cuts to ((((I)(I))I)I) : 2x2 square of 4 (((II)I)I)

- (((II)I)((II)I)), replace 2 circles, and we get ((II)(II)), as S1xC2xC2, cuts to (((I)I)((I)I)) : 2x2 square of 4 ((II)(II))

- ((II)(II)(II)), replace 3 circles, and we get (III), as S2xC3, cuts to ((I)(I)(I)) : 2x2x2 cube of 8 (III)

- (((II)(II))(II)), replace 3 circles, and we get ((II)I), as T2xC3, cuts to (((I)(I))(I)) : 2x2x2 cube of 8 ((II)I)

- (((II)(II)(II))(II)), replace 4 circles, and we get ((III)I) as S1xS2xC4, cuts to (((I)(I)(I))(I)) : 2x2x2x2 tess array of 16 ((III)I)

- (((II)(II))(II)(II)), replace 4 circles, and we get ((II)II) as S2xS1xC4, cuts to (((I)(I))(I)(I)) : 2x2x2x2 tess array of 16 ((II)II)

- ((((II)(II))(II))(II)), replace 4 circles, and we get (((II)I)I) as T3xC4, cuts to ((((I)(I))(I))(I)) : 2x2x2x2 tess array of 16 (((II)I)I)

and so forth.

By combining this with the " A along B " method of decomposing toratopes, we can get a chain of Sn, Tn and Cn. I saw someone else, and read some examples, of using S1xS2 for a torisphere, and S2xS1 for spheritorus. So, in expanding that idea, I took to chaining the n-spheres, n-tori and Cn together to help describe the build sequence of toratopes.

I came across the Clifford torus definition, here:

http://en.wikipedia.org/wiki/Duocylinder#The_ridge . So, if this is correct, what I did was expand the definition, to include the ridge of a circle^3 prism, as a C3, and a circle^4 prism as C4, etc. If the tiger is an inflated Clifford torus ( duocyl ridge), in which it seems to be, then the tiger cuts down to a square of four circles. Here, I made the connection that a square intercept probably comes from inflating a Clifford torus at some point. In cutting toratopes containing two nested circles, this generalization seems to agree with how it makes intercepts. If cutting a three nested circle toratope, it makes no lower than a cubic array.

This also plays a role in the small shape along large shape breakdown. We've all come understand a torisphere as a small circle stretched along a large sphere, ((III)I), and spheritorus as small sphere along big circle ((II)II). So, what I wanted to do was to broaden this technique with the use of Tn, Cn, and Sn. If describing a (((II)(II))(II)), we can understand it easier with a translation to T2xC3, which tells us it's a small-shape 2-torus ((II)I) stretched over a large-shape 3-frame of circle^3 prism, the C3. What we get in 3D is the cut (((I)(I))(I)) , a 2x2x2 cube array of 8 small tori.

One thing I noticed with discussion in

this post was that the shape (((II)I)((II)I)) is a tiger stretched over a clifford torus, making (((I)I)((I)I)) : four tigers in a square in 4D. It fully cuts down to a 4x4 square array of 16 circles. This 4x4 array can be understood as a small shape C2 stretched over a larger C2, making a 2x2 array of smaller 2x2 arrays. This is why I use S1xC2xC2 to describe the build sequence of (((II)I)((II)I)), starting with the smallest and inflating larger as we go down the chain. And, for a (((II)I)((II)I)((II)I)), it cuts down to (((I))((I))((I))) : a 4x4x4 array of 64 spheres. It's a small-shape 2-sphere inflating a medium shape C3, making triger ((II)(II)(II)) , infl largest shape C3, making a 2x2x2 array of smaller 2x2x2 arrays of spheres. That's 8 smaller trigers in a larger cubic array. Which makes the (((II)I)((II)I)((II)I)) easier to understand as S2xC3xC3.

Hope that clarifies some of the things I use and say

-- Philip