Torus product discussion

Discussion of shapes with curves and holes in various dimensions.

Torus product discussion

Postby PWrong » Sun Jul 30, 2006 4:30 am

Like I said before: since (x1) is every point of a circle replaced with x

Where'd you get that idea? (31) is sphere#circle, ((21)1) is torus#circle, (211) is circle#sphere.

In general, (A1) is A#circle, i.e. every point in A is replaced by a circle.
(A11) is A#sphere, and I'm not sure about (AB) or (AB1).

((22)1) is every point of a circle replaced with a tiger.

It can't be, because we can easily find the parametric equations of ((22)1), but circle#tiger doesn't have any equations, just like circle#triangle.

I know that. But 1 is a line, and (1) is the same as 1. There is sadly no RNS notation for the two points, unless someone wants to reform it...

It doesn't need one. The object 1 comes in two forms, a line and a pair of points.

Split from "Slicing toratopes with hyperplanes" by Rob.
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Postby Marek14 » Sun Jul 30, 2006 5:26 am

PWrong wrote:
Like I said before: since (x1) is every point of a circle replaced with x

Where'd you get that idea? (31) is sphere#circle, ((21)1) is torus#circle, (211) is circle#sphere.

In general, (A1) is A#circle, i.e. every point in A is replaced by a circle.
(A11) is A#sphere, and I'm not sure about (AB) or (AB1).

Thanks. I couldn't remember who created this notation in the first place, and if I remember it right.

BTW, how do you pronounce "#"? I found that I call it "sur" lately, as in "circle sur sphere". It's a shortcut for "surrounded" (each point of a circle is surrounded by sphere), but I think it has some other meanings too.

(AB) means "replace each point of AxB with a circle". AxB is a cartesian product of SURFACES of A and B. One different thing is that now the circle has no extra dimension to go to (in torus, on the other hand, the minor circle has always one dimension perpendicular to the plane where major circle lies).

(AB1) means "replace each point of AxB with a sphere". A single 1 means that the replacing shape reaches one dimension higher than the base cartesian product.


((22)1) is every point of a circle replaced with a tiger.

It can't be, because we can easily find the parametric equations of ((22)1), but circle#tiger doesn't have any equations, just like circle#triangle.

Mainly because we don't know how it should look :)
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Postby PWrong » Sun Jul 30, 2006 5:47 am

(AB) means "replace each point of AxB with a circle".

For that to be true, there must be conditions on A and B. Otherwise (21) would be cylinder#circle, and (11) would be square#circle. Maybe we should require that A /= 1 and B /=1.

Mainly because we don't know how it should look

More importantly, there are several possible objects we could call circle#triangle or circle#tiger, and they all have different equations.
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Postby Marek14 » Sun Jul 30, 2006 6:13 am

PWrong wrote:
(AB) means "replace each point of AxB with a circle".

For that to be true, there must be conditions on A and B. Otherwise (21) would be cylinder#circle, and (11) would be square#circle. Maybe we should require that A /= 1 and B /=1.

Well, I meant (from the context), that A and B are greater than one. But in fact, you are incorrect, because 1 doesn't mean "line" but "two points". (21) parsed like this would result in cartesian product of circle and two points, resulting in two circles in parallel planes. Both would then get "inflated" to get two torii. The same way, (11) would lead to four circles in this way. So you would still get the same shape, just in larger numbers. If we say that "1" here denounces a single point and every larger number denounces surface of sphere, then everything is ok.

Mainly because we don't know how it should look

More importantly, there are several possible objects we could call circle#triangle or circle#tiger, and they all have different equations.
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Postby Keiji » Sun Jul 30, 2006 8:30 am

PWrong wrote:
Like I said before: since (x1) is every point of a circle replaced with x

Where'd you get that idea? (31) is sphere#circle, ((21)1) is torus#circle, (211) is circle#sphere.


(31) is every point in a circle replaced by a sphere.

((21)1) is every point in a circle replaced by a torus.

(211) is every point in a sphere replaced by a circle.

That's how I see it anyway. I would understand if you switched (31) and (211), but there is no possible way you can replace every point in a torus with a circle and expect to get the tetratorus.
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Postby Marek14 » Sun Jul 30, 2006 9:13 am

Rob wrote:
PWrong wrote:
Like I said before: since (x1) is every point of a circle replaced with x

Where'd you get that idea? (31) is sphere#circle, ((21)1) is torus#circle, (211) is circle#sphere.


(31) is every point in a circle replaced by a sphere.

((21)1) is every point in a circle replaced by a torus.

(211) is every point in a sphere replaced by a circle.

That's how I see it anyway. I would understand if you switched (31) and (211), but there is no possible way you can replace every point in a torus with a circle and expect to get the tetratorus.


I am sorry, but that's exactly how it is. Maybe you should stop being so categorical :)

Work with me here:

Tetratorus (A,B,C) is a set of all points in 4-space with a set distance C from the surface of torus (A,B). (said torus is, similarly, set of all points in a 3-space with a set distance B from a circle with radius A)

Now, what would be the same set defined only in 3-space? In 3-space, set of points with a given distance from torus are two more torii - one outside, one inside of the original one. In other words, we are replacing each point with a 1-D circle, i.e. pair of points, and this 1D circle is lying on a normal to surface at each given point.

Now, in 4D, the normal to 2D surface is also a 2D surface, i.e. a plane. Points of tetratorus lie in these normal planes - and in each of these planes, they are points of the same distance from origin (where the plane intersects the mother torus), i.e. circles.

Note: maybe more intuitive would be to read (x1) as "surface of x, BLOWN UP in D(x)+1-dimensional space". Thus torus is what you get when you blow up a circle in 3D, (31) is when you blow up a surface of sphere in 4D, and tetratorus is what you get when you blow up a surface of torus in 4D. Blowing up in n additional dimension is simply replacing each point with (n+1)-dimensional sphere, although I agree it might look a bit counterintuitive.
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Postby Marek14 » Mon Aug 07, 2006 4:58 pm

BTW, Rob, do you understand now how to replace each point of torus with a circle? :) You never replied to my explanation.
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Postby Keiji » Mon Aug 07, 2006 5:32 pm

Yes, but what about the cubinder? The cubinder actually has edges, so surely it wouldn't form a decent torus...?
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Postby Marek14 » Mon Aug 07, 2006 5:45 pm

Rob wrote:Yes, but what about the cubinder? The cubinder actually has edges, so surely it wouldn't form a decent torus...?


Huh? What's your question?
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Postby Keiji » Mon Aug 07, 2006 6:20 pm

I can see how a circle "blown up" can form a torus, that is obvious, since a circle is just a torus with zero minor radius.

similarly, I can see how a sphere blown up and a torus blown up form the toraspherinder and tetratorus.

But how would a toracubinder work like this? Unlike the sphere and torus, a cylinder has edges. I cannot see how you would "blow up" a cylinder, and get a toracubinder.
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Postby Marek14 » Mon Aug 07, 2006 6:58 pm

Rob wrote:I can see how a circle "blown up" can form a torus, that is obvious, since a circle is just a torus with zero minor radius.

similarly, I can see how a sphere blown up and a torus blown up form the toraspherinder and tetratorus.

But how would a toracubinder work like this? Unlike the sphere and torus, a cylinder has edges. I cannot see how you would "blow up" a cylinder, and get a toracubinder.


Well, the thing is that the word "toracubinder" doesn't occur anywhere in this thread before your post. So I don't know which shape you mean by that term.
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Postby bo198214 » Mon Aug 07, 2006 7:05 pm

*lol*
I suppose its the cubinder blown up.
I would further guess, that there is a problem with the normal space at the edges and nondifferentiable points. So the # definition is not exactly applicable.
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Postby Keiji » Mon Aug 07, 2006 7:43 pm

Marek14 wrote:
Rob wrote:I can see how a circle "blown up" can form a torus, that is obvious, since a circle is just a torus with zero minor radius.

similarly, I can see how a sphere blown up and a torus blown up form the toraspherinder and tetratorus.

But how would a toracubinder work like this? Unlike the sphere and torus, a cylinder has edges. I cannot see how you would "blow up" a cylinder, and get a toracubinder.


Well, the thing is that the word "toracubinder" doesn't occur anywhere in this thread before your post. So I don't know which shape you mean by that term.


Toracubinder = (211), the spherated version of a cubinder.
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Postby Marek14 » Mon Aug 07, 2006 8:16 pm

Rob wrote:
Marek14 wrote:
Rob wrote:I can see how a circle "blown up" can form a torus, that is obvious, since a circle is just a torus with zero minor radius.

similarly, I can see how a sphere blown up and a torus blown up form the toraspherinder and tetratorus.

But how would a toracubinder work like this? Unlike the sphere and torus, a cylinder has edges. I cannot see how you would "blow up" a cylinder, and get a toracubinder.


Well, the thing is that the word "toracubinder" doesn't occur anywhere in this thread before your post. So I don't know which shape you mean by that term.


Toracubinder = (211), the spherated version of a cubinder.


Well, you still seem to have some kind of idea that the version in parentheses and the version without them must have something in common. They don't. (211) is circle#sphere, a circle blown up in four dimension instead of three. Actually, this is a spherinder bent with its ends glued together.
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Postby Keiji » Mon Aug 07, 2006 9:18 pm

I see!

So, when the last symbol in x is not a 1, and y is a sequence of 1s, and z is the number of 1s in y plus one, (xy) means x#z.

That makes much more sense.

But now the question is: how do we convert (xy) to a#b where y is not a sequence of 1s?
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Postby Marek14 » Mon Aug 07, 2006 9:50 pm

Rob wrote:I see!

So, when the last symbol in x is not a 1, and y is a sequence of 1s, and z is the number of 1s in y plus one, (xy) means x#z.

That makes much more sense.

But now the question is: how do we convert (xy) to a#b where y is not a sequence of 1s?


Well, you didn't really state what can x and y be...

But the general rule is like this:

1. Parentheses evaluate from inside out.
2. Parentheses of form a1111... with b 1's evaluate to a#(b+1)
3. Other parentheses containing any 1's evaluate to a#(b+k) where a#b is evaluation of the same parentheses without 1's, and k is the number of 1's.
4. Parentheses containing terms a,b,c etc. with at least two terms and none of them equal to 1 evaluate to (a x b x c x ...)#n where n is the number of terms.

So, for example, a sphere spheritiger (321) evaluates like this:
(32) -> (3x2)#2 (Rule 4), so by Rule 3, (321) -> (3x2)#3

Easy!
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Postby Keiji » Tue Aug 08, 2006 11:00 am

We should tell PWrong about this, he seems to be confused about the conversions too... :P
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Postby bo198214 » Tue Aug 08, 2006 11:09 am

It seems that Marek and I found quite similar rules for RNS -> product notation. Though I think this should be better discussed in the RNS and product notations thread.

Rule 2, 3 and 4 can be subsumed in my general formula (yes the notation is a bit different but the meaning is the same):

(A<sub>1</sub>....A<sub>n</sub>1...1) with d 1's is always equal to (A<sub>1</sub>x...xA<sub>n</sub>) # S<sub>n+d-1</sub>

This formula is valid for any n and d including 0. (Though of course we should not allow () as a valid RNS expression, shouldnt we?) We dont need the restriction of rule 4 that there needs to be at least 2 sub terms.
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Postby Keiji » Tue Aug 08, 2006 11:28 am

I'm going to lock this topic, since as Bo said, we already have another topic for this.
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