The missing rotopes

Higher-dimensional geometry (previously "Polyshapes").

The missing rotopes

Postby Keiji » Fri Aug 04, 2006 3:58 pm

In an attempt to understand the quirks in our list of rotopes, I have created this tree:

Image

I'm using | to mean a normal letter, and ' to mean a superscript letter.

Now, if you look at the first three dimensions, the number of rotopes is a power of 3: 1, 3, 9. However, in the 4th dimension, there are two new shapes: the duocylinder and the tiger, highlighted in red. The reason that analogies of these do not occur in lower dimensions is that they only have effect two dimensions later and if you add a (||) to a 1D |, you just get |(||) which is the same as the cylinder.

I'll look further into these, for higher dimensions.
Last edited by Keiji on Sat Jun 16, 2007 5:22 pm, edited 1 time in total.
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Postby Nick » Sat Aug 05, 2006 12:27 am

What's with the parentheses?
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Postby Keiji » Sat Aug 05, 2006 12:58 am

Parenthesis = spheration.

e.g. (||) = (xy) = circle.
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Postby Keiji » Mon Aug 07, 2006 12:14 pm

Well, I went to make a list of all the 5D rotopes. In the process of doing so, I found that we've actually forgotton about two 4D rotopes! These are x<sup>y</sup>(zw), the cartesian product of a triangle and a circle, and its spherated version, (x<sup>y</sup>(zw)).

x<sup>y</sup>(zw) is NOT the same as (xy)z<sup>w</sup>!

(xy)z<sup>w</sup> refers to a cylinder tapered to a point, which is not the same as the cartesian product of a triangle and a circle.

Anyway, the sequence now goes:

1D: 1 rotope
2D: 3 rotopes
3D: 9 rotopes
4D: 31 rotopes
5D: 105 rotopes

The 5D rotopes can be found here: http://fusion-global.org/share/rotopes5d.png
Note that on that picture, I didn't bother connecting the lines up (it would take me all day to do that!) so look at it the same way as the 4D rotopes image I posted at the top of this thread. I also re-ordered the rotopes so they now are in the order of extrude, taper, extrude and spherate.

By the way, can anyone think of a decent name for the x<sup>y</sup>(zw) and (x<sup>y</sup>(zw))? I can't...
Last edited by Keiji on Sat Jun 16, 2007 5:25 pm, edited 1 time in total.
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Postby PWrong » Tue Aug 08, 2006 9:37 am

x<sup>y</sup>(zw) is NOT the same as (xy)z<sup>w</sup>!

This fact disturbs me. We wouldn't have this problem under my notation, but then my notation has other ambiguities. For instance, 1'1' can be either square pyramid or triangular prism pyramid. Maybe we need square brackets to sort things out, so that [11]' is square pyramid, x<sup>y</sup>(zw) would be 1'(11), and (xy)z<sup>w</sup> would be [(11)1]'.

I've just noticed you're still counting the triangular torus as a tapertope. I still think it shouldn't count, and not just because of the torus product problem. If you can find a set of equations that describes it, I'll accept it as a tapertope, and only then.
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Postby moonlord » Tue Aug 08, 2006 9:45 am

PWrong wrote:For instance, 1'1' can be either square pyramid or triangular prism pyramid.


Why so? In Rob's notation it will be a triangular prism pyramid (applies to all the shape before the "comma"), in yours, a square pyramid (applies to anything after the last "comma").

Maybe we need to define operator precendency. Arithmetic and algebra needed it too, though...
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Postby PWrong » Tue Aug 08, 2006 10:36 am

In my notation, 1'1' can be either x<sup>z</sup>y<sup>z</sup> or x<sup>y</sup>z<sup>w</sup>. There's no other way to express triangular prism pyramid in my notation.
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Postby Keiji » Tue Aug 08, 2006 10:47 am

PWrong wrote:This fact disturbs me.


It shouldn't do.

RNS is done in order, with the exception of parentheses which work in the same way as math. You seemed perfectly happy with the fact that x<sup>y</sup>zw is not the same as xyz<sup>w</sup>, so why the discrepancy when we spherate two of the dimensions? :roll:

wendy: How was that relevant? Interesting, yes; relevant, no.
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Postby PWrong » Tue Aug 08, 2006 11:04 am

Rob wrote:RNS is done in order, with the exception of parentheses which work in the same way as math. You seemed perfectly happy with the fact that x<sup>y</sup>zw is not the same as xyz<sup>w</sup>, so why the discrepancy when we spherate two of the dimensions? :roll:

Hmm, good point. I guess there's no problem then.
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Postby Keiji » Tue Aug 08, 2006 11:08 am

In rotopes that aren't tapertopes, parts can be in any order, because of the fact that AxB = BxA, and that (AxB)xC = (BxC)xA = (CxA)xB (where x = cartesian product). (AxB)->C is not the same as (B->C)xA.
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Postby Keiji » Sat Jun 16, 2007 7:54 pm

Okay, I'm reviving this topic. Wendy's stuff has been split out to a topic called "Wendy's product notation", because it was totally off topic.

Anyway, yesterday I set up the FGwiki, and imported several pages to do with geometry from the Tetraspace wiki (though most of these pages was stuff that I wrote in the first place, so please don't get all hating me for ripping you off or anything :lol: ). Today, I was looking through the rotopes and remembered about my 5D rotopes chart. Now, seeing this again for the first time in 10 months, I suddenly remembered about the two obscure four-dimensional rotopes that I discovered that aren't in the wiki.

These are 1<sup>1</sup>2, the cartesian product of a triangle and a circle, and (1<sup>1</sup>2), whatever the above is spherated. Firstly, can someone help me with finding out exactly what a "spherated cartesian product of a triangle and a circle" is, and can anyone think up decent names for both of these?

I'm also aware that there are two 5D rotopes missing from the chart: 1<sup>1</sup>3 and (1<sup>1</sup>3).
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Postby pat » Mon Jun 18, 2007 2:23 am

Keiji wrote:x<sup>y</sup>(zw) is NOT the same as (xy)z<sup>w</sup>!

(xy)z<sup>w</sup> refers to a cylinder tapered to a point, which is not the same as the cartesian product of a triangle and a circle.


Alright, let me start by again confessing that I haven't followed this whole notation thing very closely. That said, you're telling me that the exponent doesn't bind to the closest term? That (xy)z<sup>w</sup> is effectively {(xy)z}<sup>w</sup>. If that's the case, then I think superscripting was a bad choice.

Ahhh.. but it seems that I answered the rest of the my question with your wiki page when I finally noticed the difference between Rototope and Rotatope. I was mislead by the word "Rototope" into thinking that they would roll and I was confused how x<sup>y</sup> had gotten there at all.
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Postby Keiji » Mon Jun 18, 2007 6:39 am

Well, as it says on the strange rotope page, I've found a systematic name for the two 4D rotopes I was stuck on (the cyltrianglinder and cyltrianglintigroid), but I'm now at a loss for the name of two 5D rotopes (Rotopes 115 and 116).

pat wrote:Alright, let me start by again confessing that I haven't followed this whole notation thing very closely. That said, you're telling me that the exponent doesn't bind to the closest term? That (xy)z<sup>w</sup> is effectively {(xy)z}<sup>w</sup>. If that's the case, then I think superscripting was a bad choice.


Well, in this case, a superscript isn't interpreted as an exponent. It's just a way of building the whole thing up. CSG notation is able to cope with a lot more than group notation, except it can't be used for tigroids.

Ahhh.. but it seems that I answered the rest of the my question with your wiki page when I finally noticed the difference between Rototope and Rotatope. I was mislead by the word "Rototope" into thinking that they would roll and I was confused how x<sup>y</sup> had gotten there at all.


It's rotopes, not rototopes ;)
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