I've been learning quite a bit in the last few days about parametric equations. It all finally clicked: how to derive them for toratopes, convert to implicit, and how rotate. In another thread, you (Marek) outlined a linear construction technique for parametric. That's when it all made sense.

I also found this, a doubly-periodic curve :

x = cos(t)cos(nt)

y = sin(t)cos(nt)

z = sin(nt)

set n=3:

https://www.wolframalpha.com/input/?i=solve+x+%3D+cos(t)cos(3t)+,y+%3D+sin(t)cos(3t),+z+%3D+sin(3t)The rotation curve is exactly what I've been imagining that might create 3-prong multitorus and the mantis. It does have the criss-crossing cage bars, but a large enough embedded circle/torus should smooth it out. An important part are the intercepts on the xy-plane. They are at the exact locations of an equilateral triangle, and the 3rd roots of unity (projected from the Re, Im plane to x,y). That mathematical property is exactly what we're looking for, in regards to solutions. (doesn't seem to work for 4th roots, at n=4, though)

Another nice thing is the criss crossing at the poles, where we don't get problems with half-circles. But, try to implicitize the 3-prong curve, and you'll run into astronomically complex terms. I don't know, maybe with the help of mathematica, there really is some very elegant equation that defines a 3-prong frame implicitly. What would be better, is to find a way to sharpen the curve at the poles, and form more well-defined semi-circles. To get a single equation that describes an n-prong curve ... well, one can only dream (if that is even what we're after).

In the little I've learned so far, I see much greater potential. The implicit has some advantages, but the parametric can manipulate surfaces on a much deeper level.

What I've been trying to find out, is if it's possible to convert an equation with 3 time variables to just two. Calcplot only accepts 2, which seems really limiting. If there's no workaround, I will have to find another program.

Another thing that I found was accidental, when trying to make a cross section eq of a rotating torus. I tried the same technique as the implicit, by setting the whole equation to zero, for one axis. In other words:

rotating torus equation:

x = ((2 + cos(v))*cos(u))*cos(a) + (sin(v))*sin(a)

y = (2 + cos(v))*sin(u)

z = (sin(v))*cos(a) - ((2+ cos(v))*cos(u))*sin(a)

set to:

x = 0

y = (2 + cos(v))*sin(u)

z = (sin(v))*cos(a) - ((2+ cos(v))*cos(u))*sin(a)

What I unexpectedly got was a 2D image, a

projection image of the whole torus! Whoa! I had no idea that that's the way it worked. I should have seen it from the wikipedia article, but it didn't click.

So, it really got me thinking...... if this method creates an n-1D projection of the whole body (not a cross section), then I can feasibly create 3D projections, of the whole body of a 4D toratope. But, the problem still remains regarding the 3 time variables. All four of the 4D toratopes have 3.