Algebra of rotopes

[PWrong]

I've started learning a bit of algebra at uni, like groups. I was hoping the set of rotatopes with x might turn out to be a group, but it isn't. Unfortunately a group requires that every object has an inverse.

Anyway, I thought I'd look at the algebra of rotopes with the three products we have. Here's a list of definitions. Note that 'x' is a binary operation, but # and -> aren't, because they don't satisfy closure.

P = the set of rotatopes
R = the set of rotopes without tapering
T = the set of rotopes with tapering
'X' is the cartesian product of sets. (not related to the cartesian product 'x'
{1,2,3} X {a,b} = {(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}

S = the set of hyperspheres
R' = the set of rotopes, except those with open 1's on the end (like 11 and 21, but not (21)).
T' = the set of taper

x: T X T --> T
'->' : T X T --> T
#: R' X S

The reason I define # like this is that otherwise we get ugly shapes. Not just the triangular torus, but things like triangle # circle and square # circle, which intersects itself at the corners.

rules:

AxB = BxA
(AxB)xC = Ax(BxC)

A#B /= B#A
(A#B)#C always exists, but A#(B#C) doesn't.
(AxB)#C /= (A#C) x (B#C)

A->B = B->A
(AxB) -> A and (AxB) -> B always exist.

I'm sure there's a few more rules, which I'll post later if I find them.

[bo198214]

Algebraicly it doesnt look very promising to me. (At least if we omit the computation rule (A₁x...xA_n)# k = (A₁...A_n 1...1) with k-n 1's.)
Also is X#Y not always defined: For example there is no RNS expression that is equal to 222#2.
This is because every RNS expression (X₁...X_n1...1)=(X₁x ... x X_n) # n+d.
So we can only have (X₁x ... x X_n) # m where m >= n (plus crossproducts of them) as RNS expressions. But obviously n=3 and m=2 in 222#2.
This is because 222 is 3dim in 6dim space. So the normal space is 3dim. But 2 is only in 2dim space.

[wendy]

One should note that of the products, there are certain features etc. The cartesian product doesn't really exist (it is divided into the comb and prism products).

The surtope products, such as the pyramid, prism, tegum and comb products, do associate and commute. The pyramid product adds a dimension on each application (simplex = pyramid-power of its vertices), while the comb product is pondering (ie reduces a dimension).

The torus-product is the same as a prism-product of the surfaces, and is thus a comb-product. For example, the product of a pentagon and hexagon gives 30 squares in 4d (which is com/ass). The folding down of this into 3d gives a polyhedron, which is associative product, but not communitive.

The products known as prism, tegum and sphere are all coherent. This means that the content of the product, is the product of content, when these are measured in the corresponding unit.

The pyramid, torus, and comb products do not preserve dimensionality, and therefore coherency does not apply.