PWrong wrote:This is fantastic, good luck with the tiger and ditorus. Do you think the (xy)^2 terms will be enough though? Given that these are octic surfaces I thought you'd need to add something more complicated.

Thanks. I'm making an educated guess on the oblique terms. Those will be the first to test out. Since it has to define the intercepts of two degree-4 tori, maybe it does need more. I have a few ideas brewing on what else it could be. If it is different, it may work for both octics, which would be an interesting find. One thing to point out, is the similarity between the 3D and 4D terms used so far. One simply adds a few more 2-D planes, 2((xw)^2 + (yw)^2 + (zw)^2) for a 4D compliment. But, these may just be a degree-4 thing, that works for ((II)I), ((II)II) , and ((III)I).

Ah, yeah, I keep forgetting about that. I checked it out again, and it made a little more sense this time. The concept of symmetric subgroups has everything to do with toratopes. These shapes and their equations have amazingly high symmetry, which leads to complex patterns in the subgroups. I wonder if that's ever been explored before? I'm imagining some kind of root finding algorithm that lets you derive the intercepts algebraically. You'd end up with the factored out clusters of lower toratopes or something like that.

Hey I'm confused. I was under the impression that the x±b±a terms were derived from solving the equation with all variables except one set to zero. But for the torus, the solutions for z with x=0 and y=0 are

z = ±\sqrt{a^2 - b^2}

Have I misunderstood the process?

Well, the way I see it, is this:

((xy)z) : full torus

(sqrt(x^2 + y^2) - a)^2 + z^2 - b^2

(()z) : z-cut of torus, cancel out x and y

(sqrt() - a)^2 + z^2 - b^2

Which, from what I can see will cancel out the sqrt, too, into

(- a)^2 + z^2 - b^2

This cut is the Z axis going through the hole, which is the interior complex plane of the torus. The ring is real and the hole is imaginary, in terms of solutions. Think of a parabola that stands above the x-axis in +y space. Solving for x will make 2 complex solutions, with imaginary numbers as the x-intercept. The imaginary value is locating the parabola in a higher dimension, of y-space. Kind of like a "projective intercept value"

Same thing is going on here, with z. Radially outwards from the z-axis are the real solutions of the ring. We could move the torus by the value of " a " to get z = ± b, as we cut through a circle and get 2 points in a row. But, there is another circle as well, spaced by " 2a " in the complex plane. We have to account for the second circle intercept, even if we don't see it in the real plane. The value of " a " defines the size of the complex plane, and becomes " ai ".

And, since we can move the torus back and forth to both circles, making two points in a row in two locations, we assign the range as ± ai . Now we get 2 points ± b at the position ± ai , which simply becomes

z = ± b ± ai

or

z = -b-ai , -b+ai , +b-ai , +b+ai

as the 4 complex solutions when solving for z. Shifting the torus by the value of +ai will make two real solutions and two complex, becoming

z = - b , + b , -b+2ai , +b+2ai

In relating this concept to toratopic notation and the implicit definition, we get a clear layout of the solutions, in real and complex planes. For z-cut again,

(- a)^2 + z^2 - b^2

will equate and simplify to z = ± b ± ai

The (-a)^2 term will become ± ai, through a similar process to your equation.

z^2 = b^2 - (-a)^2

z = sqrt(b^2 - (-a)^2)

z = ± b ± ai

I don't think I've seen this relation anywhere before. As in, how sqrt(b^2 - (-a)^2) simplifies into ± b ± ai . It's very interesting, and toratopes tell it all.

The first test with complex numbers was the torus, and it worked out very well. It confirmed my suspicion with imaginary numbers and the empty cut through a hole. And from this basic principle, tested and verified, we can easily extrapolate to higher toratopes.