You might know me from the CRF-polytopes section. I'm quite new to this section (and not plannig to contribute to this as much as the CRFs), and thus don't know much about toratopes. As far as I get it, toratopes are uniquely defined by the toratope notation, which works as follows:

-every I means another variable

-a set of parentheses mean you assign a radius to the distance of whatever is in these parentheses to the origin.

example: ((II)II)

start at the inside: (II)- two variables (x and y) are assigned a radius (thus x

^{2}+y

^{2}=R

_{1}

^{2}<=> sqrt(x

^{2}+y

^{2})-R

_{1}=0)

next set of parentheses: the old equation (sqrt(x

^{2}+y

^{2})-R

_{1}) and two new variables (z and w) are assigned a radius: (sqrt(x

^{2}+y

^{2})-R

_{1})

^{2}+z

^{2}+w

^{2}=R

_{2}

^{2}

If we rewrite the "a

^{2}+b

^{2}+c

^{2}+d...=R

_{n}

^{2}"-equation to |a,b,c,d,...|=R

_{n}, we get the following:

||x,y|-R

_{1},z,w|=R

_{2}

This is almost the same as ((II)II), so I guess this is how you derived the notation (no I am not going to read all 23 pages 'bout the tiger )

Furthermore you should know that we have, not too long ago, found an interestion function. This function works on dynkin-style notation of polytopes, and makes this notation isiomorph to the coxeter group that is used. (a Coxeter-group is a kind of symmetry-group in this context). This has shown to have interesting properties.

I think I have discovered a way to connect these two branches in some manner, yielding toratopes with other symmetries.

The symmetry of toratopes basically is .2.2.2.-symmetry. This symmetry uses the normal coordinate system with (x,y,z,w)-coordinates. This symmetry also means that if (x,y,z,w) is on the toratope, then (±x,±y,±z,±w) is also on the toratope. There are thus exactly 2

^{4}=16 points for every point that is on this (4-dimensional) toratope. These 16 points correspond to the elements of the coxeter-group .2.2.2. In general a coordinate in this group can be written as (x)2(y)2(z)2(w). The function that changes this coordinate into one of 16 others takes one number (say x), and change it into (-x). Furthermore the other values should be incremented with 2sin(90-180/n), where n is the number between these two values. (in this case n=2, so 2sin(90-180/n)=2sin(0)=0. and the values y,z and w shouldn't be changed at all)

Now instead of the .2.2.2.-group, one could try to use another group (say .4.3.3., the tesseractic group). Now coordinates are generally given by (a)4(b)3(c)3(d). Any coordinate (a)4(b)3(c)3(d) also implies the coordinates (-a)4(b+a*sqrt(2))3(c)3(d), (a+b*sqrt(2))4(-b)3(c+b)3(d), (a)4(b+c)3(-c)3(d+c) and (a)4(b)3(c+d)3(-d). This can then be re-used recursively to find a total of 48 implied coordinates. To define toratopes here we can take the derived definition: ||x,y|-R

_{1},z,w|=R

_{2}. The ||-function should now be defined differently (or rather more generally). The distance between the origin according to a number of variables can be calculated in a way, but I don't know how. (Klitzing has a spreadsheed that does this, but that's pure magic to me (it uses matrices etc, way above my vector calculus )). The matrix that is used for the magic has the same numbers 2sin(90-180/n) as in the group-function, so if we are lucky, the new shape has the given symmetry instantly. If not, we can define the shape locally (a>0 and b>0 and c>0 and d>0), and then make the whole shape by expanding this part using the group-function

This is the general idea I had to make toratopes with other symmetries. I'm not sure if it will work directly this way, but we will see.