Rotatopes above 9D

Discussion of shapes with curves and holes in various dimensions.

Rotatopes above 9D

Postby Keiji » Thu Dec 03, 2009 11:33 am

How should we represent hyperspheres above 9D in rotatopes?

Listing them as individual numbers is very nice for shorthand. The obvious choice would be to explicitly say 13x6x2 or whatever, but that unnecessarily complicates rotatopes that don't have any hyperspheres above 9D.
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Re: Rotatopes above 9D

Postby PWrong » Fri Dec 04, 2009 2:38 am

We could use x where necessary and not use it elsewhere. Alternatively we could use x to indicate that we're talking about the topologists sphere. So 1x1 = 22. this makes the conjecture easier to state. I'm biased though because I use the S^n notation all the time.
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Re: Rotatopes above 9D

Postby Keiji » Fri Dec 04, 2009 5:47 am

PWrong wrote:Alternatively we could use x to indicate that we're talking about the topologists sphere.


That would get confusing. It's already bad enough having to verbosely say that I mean an n-net-sphere ("topologist's sphere") or an n-bounding-sphere (toratopic hypersphere). And if we "used it where necessary and not elsewhere", any sequence of digits like 31 would be ambiguous because you wouldn't know whether it was 3x1 or a 31-sphere.
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Re: Rotatopes above 9D

Postby PWrong » Fri Dec 04, 2009 7:13 am

We could use a new set of brackets, or quotation marks or something to indicate that what we're talking about has large spheres, even if there's no x.

e.g. "24x5" or "21x14x3x2x1x1" or "31". You could even take things out of the quotation marks if there's way too many small spheres, e.g. "21x14"x3211 or "54x32x12"75532211111
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Re: Rotatopes above 9D

Postby Keiji » Fri Dec 04, 2009 9:27 am

I just had a better idea - anything before an x, or between two x's, is one number, but anything after the last x is individual digits. This allows the natural highest-first ordering.

So your examples would be 24x5, 21x14x3211, 31x, 54x32x12x75532211111.
There there's no ambiguity, and it's nice and short :D
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Re: Rotatopes above 9D

Postby PWrong » Fri Dec 04, 2009 9:35 am

That still leaves "31" and 3x1 with the same notation. Unless you write x31 for a single large hypersphere.
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Re: Rotatopes above 9D

Postby Keiji » Fri Dec 04, 2009 9:40 am

No, the 31-bounding-sphere is 31x and the spherinder is 31. I just edited my post :P
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Re: Rotatopes above 9D

Postby PWrong » Fri Dec 04, 2009 9:51 am

Ok. So you can also have 24x21x as opposed to 24x21
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