## Toratopes as equivalence classes.

Discussion of shapes with curves and holes in various dimensions.

### Toratopes as equivalence classes.

I've thought of a way to make our ideas more rigorous using equivalence classes.

If you're not familiar with these, read up on equivalence relations and equivalence classes on Wikipedia. It's an extremely useful concept in mathematics that isn't taught early enough in my opinion. A simple example is the equivalence relation "has the same birthday" on the set of people. This says that two people are "equivalent" if they have the same birthday. It partitions the set of all people into 365 equivalence classes, each of which contains millions of people who have the same birthday. Each person is in exactly one equivalence class.

My basic idea is that our toratope notation can be used to describe an equivalence class. For example, "(II)I" is the equivalence class of cylinders. Any cylinder, whether a tall thin solid cylinder, or a short wide hollow cylinder, or a pair of parallel circles, is an element of the set (II)I. Similarly, any tiger, solid or hollow, and for any values of the three radii, is a member of the set ((II)(II)). Two sets will be "equivalent" if they are in the same toratope class. We could come up with a more rigorous definition of this equivalence.

I think this will make it much easier to talk about toratopes without getting bogged down too early in a discussion of frames and parameters, so it will make our paper more interesting. We can add something on frames and parameters later. PWrong
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### Re: Toratopes as equivalence classes.

Torotopes are essentially comb products, with the actual product implemented twice.

Something like ((ii)i) is actually a product of two circles (ii)(oi), and these can be replaced by any polytopee, eg (12)(o,3).

The main difference with the tiger is that both of the shared axies end in the same polytope, eg (12)(12)(oo 5)=((ii)(ii)). Doing it like that shows the close relation this has with the tritorus (((ii)i)i) = (ii)(oi)(oi), into which it might be distorted topologically.
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### Re: Toratopes as equivalence classes.

Interesting. Does this relate to the toratope groups of equal surface and volume?
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### Re: Toratopes as equivalence classes.

Does this relate to the toratope groups of equal surface and volume?

You could define another equivalence relation by "A ~ B if they have the same surface area and volume", but I'm not sure if that would be useful. I haven't kept up with the thread on volumes and surface areas.

Doing it like that shows the close relation this has with the tritorus (((ii)i)i) = (ii)(oi)(oi), into which it might be distorted topologically.

The tiger ((II)(II)) and the 3-torus ((21)1) are homeomorphic to each other (i.e. topologically equivalent), and to the 6D shape (II)(II)(II). Similarly, the torus ((II)I) is homeomorphic to the duocylinder (II)(II). A homeomorphism is also an equivalence relation, but it's not the one I'm using here to distinguish between toratopes.

So how do we define the toratope equivalence relation properly? If we want to say that rectangles are members of the class II, but parallelograms are not, it will be difficult. It might be easier to allow any linear transformation. This would also mean that ellipses are members of the class (II).

Frames might be more difficult. A while ago we came up with an operation ∂ that transforms, for example, a solid cube into a hollow cube, a hollow cube into a wireframe cube (12 lines), and a wireframe cube into a pointframe cube (8 points). So we could say that A and B are equivalent if A = ∂B or B = ∂A. But this won't work, there are many strange sets A such that ∂A is a wireframe cube. The operation ∂ is not a bijection, it has no natural inverse. However the convex hull brings shapes back up to their solid form, from which you can go back down with ∂. So I've come up with this conjectured definition:

A ~ B if and only if their convex hulls are equal up to linear transformation, AND there exist nonnegative integers i, j such that ∂i A and ∂j B are nonempty and equal up to linear transformation.

With this relation, there are many equivalence classes that are not toratopes. However I think I can prove that it neatly divides up the toratopes into the categories we've defined. PWrong
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### Re: Toratopes as equivalence classes.

PWrong wrote:The tiger ((II)(II)) and the 3-torus ((21)1) are homeomorphic to each other (i.e. topologically equivalent), and to the 6D shape (II)(II)(II)

Yeah, I noticed that, too. It's the 3-frame of (II)(II)(II) that can be made by folding a cube into 4,5,6D. Same thing with a torus and Clifford torus, both made by hose-linking or sock-rolling a hollow tube from a square. The square to 2-frame of (II)(II) is the same as cube to 3-frame of (II)(II)(II). But, I'm also thinking that the 3-frame of ((II)I)(II) is homeomorphic as well. And, then we get to the 4-cube. Folding a 4-cube can produce (((II)(II))I) , (((II)I)(II)) , ((((II)I)I)I) , the 4-frame of (II)(II)(II)(II) , the 4-frame of ((II)I)(II)(II) , the 4-frame of (((II)I)I)(II) , the 4-frame of ((II)I)((II)I) , the 4-frame of (((II)(II))I)I , the 4-frame of ((((II)I)I)I)I , and the 4-frame of ((((II)I)(II))I , as far as I can tell.
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### Re: Toratopes as equivalence classes.

This is easy to see when you write them with the x and # products.

(((II)(II))I) = ((S1 x S1 ) # S1) # S1
(((II)I)(II)) = ((S1 # S1) x S1) # S1
((((II)I)I)I) = S1 # S1 # S1 # S1
(II)(II)(II)(II) = S1 x S1 x S1 x S1
((II)I)(II)(II) = (S1 # S1) x S1 x S1
(((II)I)I)(II) = ((S1 # S1) # S1) x S1
((II)I)((II)I) = (S1 # S1) x (S1 # S1 )

I'm not sure about your last three examples. Let A be the 5-frame of (((II)(II))I), then the 4-frame of (((II)(II))I)I is ∂A x ∂I, which has two disjoint set. None of the shapes I listed above are disconnected. PWrong
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### Re: Toratopes as equivalence classes.

You might have to deal with something like (xx)(oxx) = ((ii)ii) vs (xxx)(ox) = ((iii)i).

Topologically these surfaces are identical, in as far as they're sphere-circle combs, and the surface is the repeated product of the surfaces of a sphere and circle.

But their interior is different. The first is a spherated circle, a wheel with a spherical cross-section. The second is a spherated sphere, that is, a ring that would fit around a linear pole. The surface may be topologically equal, but the interiors are not. However if you spherate both of these in 5D, you get something that can be turned into each other, because 'turning inside out' of a spherated margin is straightforward.
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### Re: Toratopes as equivalence classes.

PWrong wrote:I'm not sure about your last three examples. Let A be the 5-frame of (((II)(II))I), then the 4-frame of (((II)(II))I)I is ∂A x ∂I, which has two disjoint set. None of the shapes I listed above are disconnected.

Yeah, after thinking about it, there's one dimension left over, uncurled. Like comparing a duocylinder prism to a cube.
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