Currently the torus product of A and B is defined as follows:
Construct an A.
Let u be the position vector of a point in A.
Construct a new B, with one axis pointing in the direction of u, and all other axes being perpendicular to the axes of A.
Let v be the position vector of a point in B.
The torus product of A and B is the set of all possible vectors u+v.
More succinctly, if A is a kD object, and B is a jD object, then A#B is a (k+j-1)D object, and
A#B = {u + v| u E A(x_1 ... x_k), v E B(u, x_k+1 ... x_k+j-1) }
where E is the membership symbol.
You may not have noticed this, but there is a huge problem with this definition- it creates lots of ugly shapes:
The product of a circle on the origin and another circle is a torus. But what if the first circle is not on the origin? If you work it out properly, you'll see that you don't get a doughnut. In fact you get an ugly looking shape that noone would ever dunk in their coffee.
It's fat at some points, thin at other points, because the position vector doesn't point from the centre.
This also means that the torus product is not associative.
circle#(circle#circle) puts a torus at every point on a circle. This is a nice, ordinary shape.
(circle#circle)#circle on the other hand, is ugly. It starts with a nice torus, and puts a circle at every point. If we take a cross section of the torus, on the xz plane, we see two displaced circles. So the final shape is actually the ugly shape, rotated through the yw plane.
So circle#(circle#circle) does not equal (circle#circle)#circle
Here is the simplest problem with the torus product as it stands.
What do you get if you spherate an arbitrary line (not through the origin) by a circle? A cylinder? No, you get a horrifying little thing who's radius decreases as you follow the line.
r = R/sqrt(1+y^2/x^2)
Where x is the shortest distance between the line and the origin, and y is the distance down the line you go. It looks like gamma from relativity, so I'm going to call it a "relativistic cylinder".
Anyway, the point is it's ugly. That's why we need a new definition for the torus product that can handle objects away from the origin. The key is to construct B so that one axis is parallel to the normal vector, not the position vector. The normal vector is simply the vector perpendicular to the surface.
So our new definition is this:
Construct an A.
Let u be the position vector of a point in A.
Let n be the normal vector of A at a point in A.
Construct a new B, with one axis pointing in the direction of n, and all other axes being perpendicular to the axes of A.
Let v be the position vector of a point in B.
The torus product of A and B is the set of all possible vectors u+v.
A#B = {u + v| u E A(x_1 ... x_k), v E B(n, x_k+1 ... x_k+j-1}
Lets investigate this new product.
Line#circle is always a cylinder.
circle#circle is always a torus.
(circle#circle)#circle = circle#(circle#circle).
Conjecture 1:
The new torus product is associative, that is (A#B)#C = A#(B#C)
I haven't yet found a construction using this product that leads to an ugly shape. I don't yet have a precise definition for "ugly", but you know an ugly shape when you see it.
Conjecture 2:
If A and B are not ugly, then A#B is not ugly.
One nice thing about this is we can now spherate the cylinder.
With the old definition, we'd get a relativistic torus. Now, we just get a torinder.
In my notation, we would write this as (2+1)*2 = 2*2 + 1
In yours, we would say (I think)
(2*1)+1 = (2+1)*1
Finally, there's a very important question we need to ask. What does this mean for the tiger? The tiger is supposed to be the 2D form of the duocylinder, spherated by a circle. But the 2D form of the duocylinder doesn't have a normal vector. A 3D object in 4D has a tangent realm and a normal vector, but a 2D object has a tangent plane and a normal plane. Does this mean the tiger will go extinct?
Actually, it just means the torus product is more complicated than I thought. Rather than spherating, we just place a circle onto the normal plane.
We can redefine the torus product again, to take into account that A may have a normal plane. If we using the torus product along with a subscript, we can can express different objects as the product of the same objects, but in a different dimension. For instance (circle#circle)<sub>3</sub> might be the torus, while (circle#circle)<sub>4</sub> is the duocylinder. More on this when I've worked it out entirely.