Some contradiction in wiki re: expanded rotatopes

Discussion of shapes with curves and holes in various dimensions.

Some contradiction in wiki re: expanded rotatopes

Postby Kwiitope » Wed Mar 30, 2022 4:28 pm

Hello, newbie here (well, new member anyway; I first looked at this site years ago). Some confusion/contradiction from the wiki (which I like very much overall):

From the Rotatope article ( http://hi.gher.space/wiki/Rotatope ): "Rotatopes of two dimensions or higher also function as expanded rotatopes: a rotatope which is (at its minimal frame) homeomorphic to a set of toratopes of lower or equal dimension." (Note that the only qualification of rotatopes there is that they be of two dimensions or higher.)

From the "Expanded rotatope" article ( http://hi.gher.space/wiki/Expanded_rotatope ): "The expanded rotatope of a closed toratope is the unique rotatope that is homeomorphic to the toratope." (emphasis mine)

The "List of toratopes" article ( http://hi.gher.space/wiki/List_of_toratopes ) lists an expanded rotatope for each toratope (both open and closed) listed (of course, this list only lists the toratopes from dimensions 2 through 5)

So my question is, can/should the adjective "closed" before the word "toratope" in the "Expanded rotatope" article be removed or replaced with "of two dimensions or higher" after the word "toratope" (if the expanded rotatopes have to be at least 2D than so do the rotatopes which have expanded toratopes).

Thanks in advance to anyone who can be of assistance with this.
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Re: Some contradiction in wiki re: expanded rotatopes

Postby wendy » Fri Apr 01, 2022 9:17 am

This is not something I had to deal a lot with. I am more influencial in the 'brackettopes' (it was my suggestion for the <> and rss() function, as well as the tegum product used here).
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Re: Some contradiction in wiki re: expanded rotatopes

Postby Keiji » Fri Apr 08, 2022 9:56 pm

Hmm... I'm a bit rusty in this area.

The following statements are definitely correct:

A. "The expanded rotatope of a closed toratope is the unique rotatope that is homeomorphic to the toratope." (from expanded rotatope)
B. "To find the expanded rotatope of a [closed] toratope written in toratopic notation, look at each group, including nested groups, and write down the number of elements in the group." (from expanded rotatope)
C. "[A rotatope] of two dimensions or higher [...] is (at its minimal frame) homeomorphic to a set of toratopes of lower or equal dimension." (from rotatope)

I suspect the word closed in statement A is unnecessary and that it applies just as well to open toratopes, since the page later states that (when dealing with an open toratope) you can simply "append any outer digons to the result", i.e. if ((II)I) gives you 22, then ((II)I)I gives you 221, so you would expect the rotatope 221, at its minimal frame, to be homeomorphic to the open toratope ((II)I)I. However, I'd have to be more knowledgeable in homeomorphisms to verify this. I'd be happy to drop the word closed from that statement if someone with relevant expertise agrees.

I've added closed to statement B to indicate I'm only stating it's definitely true for closed toratopes. If statement A is true for open toratopes too, then so is statement B, with the "Then append any outer digons to the result." instruction added.

The phrase "of two dimensions or higher" is included in statement C because in 2D and above, toratopes fall neatly into a symmetry of one open toratope for every closed toratope and vice versa. In 1D, technically, there is exactly one toratope - the digon "I", which is open, so there is one open toratope and no closed toratopes. (A closed toratope must have a single outer group.) In 0D, technically, there is exactly one toratope - the point "", which I suppose also counts as open, so again there is one open toratope and no closed toratopes. I've always simply defined toratopes as only existing in 2D and higher to avoid having to deal with that messiness. But, yes, figures are homeomorphic to themselves last I checked, so if you count the point and the digon as toratopes, then in 0D and 1D the statement does hold for the single possible figure in each.
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