I've found a simple method for finding the cartesian equation for a toratope. I'm using Marek14's notation without the plusses.

First, note that (11) = (x,y) means that x^2 + y^2 = r^2.

That is, sqrt(x^2 + y^2) - r = 0

Now let A and B be toratopes with equations A=0 and B=0 respectively. Then the equation of (A, B) is sqrt(A^2 + B^2) - r = 0

Let's look at the 3D torus ((11)1) = ((x,y),z)

We have

(x,y): sqrt(x^2 + y^2) - r_1 = 0

so,

((x,y),z): sqrt((sqrt(x^2 + y^2) - r_1)^2 + z^2) - r_2 = 0

Similarly, (A,B,C) means sqrt(A^2 + B^2 + C^2) - r = 0

and AB means (A=0 and B=0)

Here's a more complicated example. (321)211 = ((xyz)(wv)u)(ts)pq

3: sqrt(x^2 + y^2 + z^2) - r_1 = 0

(321): sqrt( (sqrt(x^2 + y^2 + z^2) - r_1)^2 + (sqrt(w^2+v^2) - r_2)^2 + u^2) - r_2 = 0

(311)11: sqrt( (sqrt(x^2 + y^2 + z^2) - r_1)^2 + (sqrt(w^2+v^2) - r_2)^2 + u^2) - r_3 = 0 and sqrt(t^2 + s^2) - r_4 = 0 and |p| - r_5 = 0 and |q| - r_5 = 0

Does this all make sense to everyone? I'll try to implement this into my mathematica program somehow.