Rotopic split proposed

Higher-dimensional geometry (previously "Polyshapes").

Rotopic split proposed

Postby Keiji » Thu Nov 19, 2009 11:59 am

Since people seem to finally be returning to this forum, it's time to ask an important question!

Currently, the set of rotopes includes rotatopes, toratopes and tapertopes. As you should know, the extensions to the rotope construction system caused the introduction of so-called ambiguous, immeasurable and strange rotopes. Strange rotopes have of course been given a proper definition after their discovery but I would like to disclaim the other two sets, as they are just about useless.

Doing this without disturbing the process of rotopic construction is more or less impossible. Therefore I would like to split the set of rotopes up into the following two sets:
  • Toratopes. PWrong's recent research has brought more importance to the set of closed toratopes itself, so it deserves to be standalone and lose the "closed" qualifier (which would then be implicit). This set does of course include the so-called strange rotopes as there is nothing truly strange about them; the "strange" attribute would be dropped as well.
  • Tapertopes. The new definition of tapertopes would be "any combination of Cartesian product and pyramid operations on hyperspheres." Therefore, with this definition, there are some shapes that would (and deserve to) appear in this set which were not in the original set of rotopes, such as the 3-3 duoprism. Cylinders (and variations thereof) would also appear in this set.
This leaves two sets of rotopes out:
  • ambiguous and immeasurable rotopes, which would be disclaimed
  • Cartesian products of non-hyperspheric closed toratopes and rotopes. Unless anyone has any better ideas, it seems a worthy compromise to not include these in either set. In 4D, the only instance of this is the torinder; there are more in 5D and higher.
The only shapes that appear in both sets are the hyperspheres themselves (toratopes would contain hyperspheres from the circle up, whereas tapertopes would additionally contain the point and the digon, considered as hyperspheres). The sets would not be combined and would not have a combined notation. This avoids ambiguous and immeasurable rotopes.

If anyone has any objections to this, please say so. If I don't receive any objections in a week (i.e. by the 26th), I'll update the entire wiki to reflect the changes, archiving the old information. I hope you all see why I am proposing this change and how it would simplify a lot of concepts related to rotopes.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Rotopic split proposed

Postby PWrong » Fri Nov 20, 2009 11:51 am

Making toratopes separate from tapertopes sounds good to me. I'm not bothered about what happens to the tapertopes. I have a couple of points about toratopes though.

As you should know, the extensions to the rotope construction system caused the introduction of so-called ambiguous, immeasurable and strange rotopes.

I'm afraid I've forgotten exactly what these are.

Toratopes. PWrong's recent research has brought more importance to the set of closed toratopes itself, so it deserves to be standalone and lose the "closed" qualifier (which would then be implicit). This set does of course include the so-called strange rotopes as there is nothing truly strange about them; the "strange" attribute would be dropped as well.

I used "closed toratope" to mean one with brackets around it, and open to mean one without. So (21)3 is open and ((21)3) is its closed counterpart. This is useful because it makes it immediately obvious that there is an even number of toratopes in nD. Open toratopes are still useful, since you can calculate homology groups of things like 22. Also each closed toratope is homeomorphic to a (usually higher dimensional) open toratope, e.g. (21) ~ 22 and ((21)1) ~ 222.

Cartesian products of non-hyperspheric closed toratopes and rotopes. Unless anyone has any better ideas, it seems a worthy compromise to not include these in either set. In 4D, the only instance of this is the torinder; there are more in 5D and higher.

The torinder is an open toratope, (21)1. I don't think we should leave these out.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Rotopic split proposed

Postby Keiji » Fri Nov 20, 2009 12:50 pm

PWrong wrote:Making toratopes separate from tapertopes sounds good to me. I'm not bothered about what happens to the tapertopes. I have a couple of points about toratopes though.

As you should know, the extensions to the rotope construction system caused the introduction of so-called ambiguous, immeasurable and strange rotopes.

I'm afraid I've forgotten exactly what these are.


A strange rotope is a rotope which involves a Cartesian product where neither argument is a hypercube. This is a very arbitrary definition and only came about due to enumeration via rotopic notation. It says next to nothing about the shapes themselves, which is why I intend to drop the attribute.

I used "closed toratope" to mean one with brackets around it, and open to mean one without. So (21)3 is open and ((21)3) is its closed counterpart. This is useful because it makes it immediately obvious that there is an even number of toratopes in nD. Open toratopes are still useful, since you can calculate homology groups of things like 22. Also each closed toratope is homeomorphic to a (usually higher dimensional) open toratope, e.g. (21) ~ 22 and ((21)1) ~ 222.

The torinder is an open toratope, (21)1. I don't think we should leave these out.


If we included open toratopes, all non-A/I rotopes would be accounted for. Then the various cylinders would also appear in both sets along with the hyperspheres.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Rotopic split proposed

Postby PWrong » Sat Nov 21, 2009 9:33 am

If we included open toratopes, all non-A/I rotopes would be accounted for

What's non-A/I?

Then the various cylinders would also appear in both sets along with the hyperspheres.

Isn't that what we want? I've always assumed that all rotatopes were toratopes. I worked out that formula (it was a big sum over the partitions of n) to count the number of toratopes in nD. The formula includes all the rotatopes, all the closed toratopes and many Cartesian products of non-hyperspheric closed toratopes, like (21)1 and ((21)2)(31)2. I don't see any problem with shapes like these being called toratopes. Is it to do with some objects being both toratopes and tapertopes?
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Rotopic split proposed

Postby Keiji » Sat Nov 21, 2009 2:37 pm

PWrong wrote:
If we included open toratopes, all non-A/I rotopes would be accounted for

What's non-A/I?


non-ambiguous/immeasurable.

Then the various cylinders would also appear in both sets along with the hyperspheres.

Isn't that what we want?


Well, I wasn't saying it was a problem, just pointing it out.

I've always assumed that all rotatopes were toratopes. I worked out that formula (it was a big sum over the partitions of n) to count the number of toratopes in nD. The formula includes all the rotatopes, all the closed toratopes and many Cartesian products of non-hyperspheric closed toratopes, like (21)1 and ((21)2)(31)2. I don't see any problem with shapes like these being called toratopes.


There's no problem with it. I had just imagined open toratopes wouldn't be wanted in the first instance.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Rotopic split proposed

Postby PWrong » Sat Nov 21, 2009 10:44 pm

Ok, that's fine then.

I didn't mention open toratopes in the homology thread because first I thought they would be trivial, and now I'm realising that some frames are much more difficult than closed toratopes.

I'll do the 5D cube and some of the 4D open toratopes today and post my results.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Rotopic split proposed

Postby Keiji » Sun Nov 22, 2009 5:57 pm

I've added the info on tapertopes and toratopes to the wiki:
Tapertope, List of tapertopes, Tapertopic notation.
Toratope, List of toratopes, Toratopic notation.

I'll add the 5D toratopes and a summary table soon.

If this is acceptable, I'll change the info on shape pages and archive the info on rotopes.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Rotopic split proposed

Postby PWrong » Mon Nov 23, 2009 1:47 am

Looks good. I'll put the homology groups on the pages for individual toratopes when I've finished them.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Rotopic split proposed

Postby Keiji » Tue Nov 24, 2009 8:33 pm

I've just finished migrating all the tapertopes and toratopes up to 4D over to the new templates. I also added the two 4D tapertopes that didn't have pages.

I've also rewritten the Rotope page completely and merged a few other pages into it.

I'll migrate all the existing 5D and up tapertopes and toratopes soon.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Rotopic split proposed

Postby PWrong » Wed Nov 25, 2009 2:49 am

I put in the homology groups just for square, circle and cube.
What do you think, would it be better off as a table in the shape template? The circle plus is for the direct sum of groups. 2Z is my shorthand for ℤ⊕ℤ.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Rotopic split proposed

Postby Keiji » Wed Nov 25, 2009 7:17 am

It'd probably be better to use the shorthand in the articles and explain what it means in a page about homology groups. Wouldn't it be a waste of space to keep writing ℤ⊕ℤ⊕ℤ⊕ℤ⊕ℤ⊕...?

I don't really want to crowd up the shape template any further with a table of homology groups; it looks fine as it is now.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Rotopic split proposed

Postby PWrong » Wed Nov 25, 2009 8:05 am

Wouldn't it be a waste of space to keep writing ℤ⊕ℤ⊕ℤ⊕ℤ⊕ℤ⊕...?

Yes, but I don't know the correct way to abbreviate it. We never did anything more difficult than the torus, klein bottle and projective plane in the algebraic topology unit, so we never needed a shorthand.
I've been saying 3Z, but that would literally mean the set {..., - 3, 0, 3, 6, 9, ...}.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Rotopic split proposed

Postby Keiji » Wed Nov 25, 2009 5:04 pm

PWrong wrote:I've been saying 3Z, but that would literally mean the set {..., - 3, 0, 3, 6, 9, ...}.


Yet Z2 = Q and 2Z is a power set. Why should multiplication go to the elements?
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Rotopic split proposed

Postby PWrong » Thu Nov 26, 2009 2:44 am

That's more of a set theory notation. nℤ is ok, as long as we're clear about what it means. Problem is that people sometimes write the cyclic group with two elements as ℤ/2ℤ (although a better notation is ℤ2). This is the quotient group of the integers by the even integers, and it comes up in the homology groups of the klein bottle.

Also Z2 = Q isn't strictly true. There is a bijection between them so they have the same cardinality, but that doesn't mean they're equal. It's like saying {1,2,3} = {a,b,c}.
User avatar
PWrong
Pentonian
 
Posts: 1599
Joined: Fri Jan 30, 2004 8:21 am
Location: Perth, Australia

Re: Rotopic split proposed

Postby Keiji » Thu Nov 26, 2009 5:04 am

Well, for all I care, Z2 = Z as well. But I couldn't think of any other examples to prove my point. :P
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England


Return to Other Geometry

Who is online

Users browsing this forum: No registered users and 9 guests