I want to make sure I understand spheration properly, and then I have a few questions about it.
First, I'll enumerate the 4D toratopes in product notation (at least, the product notation that's on the list of ro
topes page, which could be nonsense for all I know
Tesseract - 1x1x1x1
Glome - 4
Cubinder - 2x1x1
Toracubinder - 2#3
Duocylinder - 2x2
Tiger - (2x2)#2
Spherinder - 3x1
Toraspherinder - 3#2
Torinder - (2#2)x1
Ditorus - (2#2)#2
Now, the first thing I notice is that those numbers don't all add up to 4. If # counted as -1, then everything apart from the tiger would - with the tiger being 5.
So, I'm imagining that the Cartesian product of two circles used in the tiger is really referring to the 2-frame duocylinder, and that 2D surface is being spherated by a circle to produce the 3-net, 4-bounding-space tiger.
Apart from hyperspheres, all closed toratopes must have a spheration operator at the lowest level of precedence (i.e. the "outermost" spheration operator). This is simple to show as all closed toratopes have a group around the entire shape and only hyperspheres have only 1s inside that group. So the dual of a toratope (i.e. turning it inside out) is given by reversing the operands of the outermost spheration operator:
dual toracubinder = 3#2 = toraspherinder
dual toraspherinder = 2#3 = toracubinder
dual tiger = 2#(2x2) = ?
dual ditorus = 2#(2#2) = ?
Now, what is the dual tiger and the dual ditorus? They must exist, since they are given by taking the tiger or ditorus and turning them inside-out (a la this
, except in 4D). For the ditorus, if spheration is associative (it certainly isn't commutative), it could be self-dual, since associativity would mean (2#2)#2 = 2#(2#2).
However, I have no idea how to represent the dual tiger, 2#(2x2), in toratopic notation - this shape would be "putting a duocylinder at every point in a circle, oriented perpendicular to the circle". To me, this sounds like the shape would be 5D, but of course we are referring to the 2-frame duocylinder, so the resulting shape would be 3-frame. Then the question is, can the resulting shape be embedded in four dimensions without self intersections?
So, my questions:
1. Is everything I've said above correct?
2. Is the spheration operator associative (and thus the ditorus self-dual)?
3. How do you find A#B where B is not a hypersphere?
4. What is the dual of the tiger? If the tiger is self-dual, then why is 2#(2x2) equivalent to (2x2)#2?
5. If the dual of the tiger isn't a toratope, does this mean the set of toratopes is too restrictive?