So if you want to do it that way, we would have
S1#(S2xS3) = (S1#S2)xS3 = ((III)I)(III)
S1#(S3xS2) = (S1#S3)xS2 = ((IIII)I)(II)
Besides, there's another restriction on the spheration operator. If there are too many dimensions in the normal space, then spheration is undefined. For example,
(S1xS1xS1) # S1 doesn't exist. You're starting with (II)(II)(II), and you want to put brackets around it in such a way that there are only two things in the brackets. You can have (S1xS1xS1) # S2 or (S1xS1xS1) # S3 or more, but not (S1xS1xS1) # S1.
Wouldn't it be
S2#(S1xS2) = (S2#S1)xS2 = ((III)I)(III)
S3#(S1xS1) = (S3#S1)xS1 = ((IIII)I)(II) ?
But if you mean both of the definitions for min-frames, then how else would you describe min-frames, such as the 3-frame ((II)I)(II) embedded in 5D? It is a valid surface, which can be defined algebraically, apart from open and closed toratopes.
Teragon wrote:I’d like to introduce another systematic mathematical notation for toratopes in any number of dimensions and a systematic nomenclature based on it that I've developed. It's short and by standards of high dimensional geometry it's easy to understand.
Teragon wrote:Are the ditorus and the tiger topolocically one and the same object, i.e. can one be deformed into another without breaking it or turning it inside out?
Teragon wrote:It would be cool to find a criterium to tell if two arbitrary toratopes are topolocially the same
Teragon wrote:... i.e. if there are really 33 different toratopes in 6D or less
ICN5D wrote:I think ((III)(II)) and ((II)(II)I) are duals, but I'm not 100% sure.
ICN5D wrote:Topological Duals:
((III)I) and ((II)II)
Teragon wrote:ICN5D, thanks for the fast reply.
Let's talk only about toratopes constructed using those two operations. It's nice to see that Keiji came to the same number of topologically different objects in 6D as I did with my approach (13). This suggests that from my notation you can immediately tell if two toratopes are topological duals and you could calculate the total number of different topologies in any number of dimensions.ICN5D wrote:I think ((III)(II)) and ((II)(II)I) are duals, but I'm not 100% sure.
It's been a while since I've learned your notation. I don't get the difference between ((II)(II)I) and (((II)(II))I) right now, but as ((III)(II)) contains a 3-loop and ((II)(II)I) doesn't, they can't be duals. (or does it?)
Teragon wrote:I disagree that the torisphere and the spheritorus are topological equivalents. How would you deform a spherical cavity into a circular one?
Teragon wrote:You can't exchange the two loops of the torus in 3D either, except if you add one more dimension, so that there is no closed cavity anymore.
Teragon wrote:Still I think ((III)(II)) and ((II)(II)I) are topologically different, at least in 5D. You can't exchange the innermost loop with any of the outer loops.
Teragon wrote:It's been a while since I've learned your notation. I don't get the difference between ((II)(II)I) and (((II)(II))I) right now
ICN5D wrote:Teragon wrote:I disagree that the torisphere and the spheritorus are topological equivalents. How would you deform a spherical cavity into a circular one?Teragon wrote:You can't exchange the two loops of the torus in 3D either, except if you add one more dimension, so that there is no closed cavity anymore.
Both of these can be answered, or so I believe, by looking at this animation:
See what's going on here? Specifically with the major and minor radius? When the torus turns inside out, the major becomes the minor, while the minor becomes the major. They switch roles.
ICN5D wrote:The real question is if the role switching happens when we have +2 major radii (the entire class of 'tigroids')?
Marek14 wrote:The question is: is cutting the hole in torus absolutely necessary to invert it, or is it just a trick to make it work in 3D? In other words, if you had more dimensions available, could you turn the torus inside out without cutting it?
Teragon wrote:Marek14 wrote:The question is: is cutting the hole in torus absolutely necessary to invert it, or is it just a trick to make it work in 3D? In other words, if you had more dimensions available, could you turn the torus inside out without cutting it?
It is absolutely necessary in 3D, but yes, if you had more dimensions, you could turn it inside out. The same is true for the torisphere and the spheritorus.
I actually don't know if the notion of a topology is defined relative to a given number of dimensions or if you can add a dimension in order to deform it. It would really make sense to define it independ of the number of dimensions. I mean, a spheritorus is still a spheritorus in 5D. In this case, torisphere and the spheritorus would indeed be equivalent.
Marek14 wrote:Well, imagine that you are a 3D being (should be easy) living in the 3D surface of torisphere/spheritorus. Is there any way, without measuring diameters, to determine whether it's a torisphere or spheritorus?
ICN5D wrote:All right, here's my thinking. If we can use 5D space to transform a tiger into a 3-torus, and vice versa:
((II)(II)) <--> (((II)I)I)
then, using a 6D space, should we be able to turn ((II)(II)I) into (((II)I)II) ?
If this is the case, would we also be allowed to turn (((II)I)II) into ((III)(II)) ?
ICN5D wrote:All right, here's my thinking. If we can use 5D space to transform a tiger into a 3-torus, and vice versa:
((II)(II)) <--> (((II)I)I)
ICN5D wrote:then, using a 6D space, should we be able to turn ((II)(II)I) into (((II)I)II) ?
ICN5D wrote:If this is the case, would we also be allowed to turn (((II)I)II) into ((III)(II)) ?
Marek14 wrote:Logically, surface of any toratope should be topologically equivalent to some product of spheres of various dimensions. From the toratope notation, you can read the "signature", as number of terms inside the various parentheses. Both (((II)I)II) and ((III)(II)) would have the same signature, 223 (circle x circle x sphere, 2 fundamental circular loops and 1 fundamental spherical envelope), so they could be transformed.
For 5D toratopes, the signatures are:
5: (IIIII)
42: ((IIII)I), ((II)III)
33: ((III)II)
322: (((III)I)I), (((II)II)I), ((III)(II)), (((II)I)II), ((II)(II)I)
2222: ((((II)I)I)I), (((II)(II))I), (((II)I)(II))
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