The tiger does not exist.

Discussion of shapes with curves and holes in various dimensions.

The tiger does not exist.

Postby Keiji » Mon Jul 17, 2006 8:21 pm

I think I mentioned this before, but nobody noticed it, or something. Whatever. Anyway, in order to make something into a torus, that object needs at least one pair of opposite, flat n-hypercells where the object is (n+1)-dimensional. The duocylinder has two curved cells, and no other cells, so it is impossible to make it into a torus, and since a tiger is a duocylinder made into a torus, the tiger cannot possibly exist.
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Re: The tiger does not exist.

Postby Marek14 » Tue Jul 18, 2006 7:12 am

Rob wrote:I think I mentioned this before, but nobody noticed it, or something. Whatever. Anyway, in order to make something into a torus, that object needs at least one pair of opposite, flat n-hypercells where the object is (n+1)-dimensional. The duocylinder has two curved cells, and no other cells, so it is impossible to make it into a torus, and since a tiger is a duocylinder made into a torus, the tiger cannot possibly exist.


How do you define "make something into a torus"? Tiger is definitely not "a duocylinder made into a torus" - it's defined as a set of points in a set distance from duocylinder margin. In other words, if you consider a circle "base" for torus, then "base" of tiger is not cell of duocylinder, but the margin.

However, its definition doesn't really include any operation - it's just a set of points with given prperty, just like a torus is set of points equidistant from a circle.

Plus, the tiger was originally discovered from its parametric equations - hard thing to do if it doesn't actually exist :)
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Postby Keiji » Tue Jul 18, 2006 9:21 am

Something made into a torus is, in terms of RNS notation, the original object surrounded by brackets.

ie:

cylinder = 21, torus = (21)
cubinder = 211, toracubinder = (211)
spherinder = 31, toraspherinder = (31)
duocylinder = 22, tiger = (22)

In order to make something into a torus you bend it round and attach the ends. But you can only do this if it has at least one pair of opposite flat hypercells. Since the duocylinder has no flat cells, either a) the tiger does not exist or b) the tiger isn't actually a torus.
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Postby pat » Tue Jul 18, 2006 6:26 pm

Why do they have to be flat?
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Postby Keiji » Tue Jul 18, 2006 6:55 pm

Because you can't attach them if they are curved.
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Postby Marek14 » Wed Jul 19, 2006 6:14 am

Rob wrote:Something made into a torus is, in terms of RNS notation, the original object surrounded by brackets.

ie:

cylinder = 21, torus = (21)
cubinder = 211, toracubinder = (211)
spherinder = 31, toraspherinder = (31)
duocylinder = 22, tiger = (22)

In order to make something into a torus you bend it round and attach the ends. But you can only do this if it has at least one pair of opposite flat hypercells. Since the duocylinder has no flat cells, either a) the tiger does not exist or b) the tiger isn't actually a torus.


Hmmm... Actually, I don't think that those parenthesis correspond to any ACTUAL operation. Every torus is assigned to another object, yes, but that's just an artifact of the fact that there is the same number of both. Any operation assigned to it was almost definitely an afterthought - the way I use it, there is NO operation defined, it's just a combinatoric assignment.

I think you are treating a PROPERTY of the assignment, which only works for simple objects, as a part of the definition of torus - this, I think, is incorrect.

Discussion - originally, the various torii are defined via their parametric equations:

(4)
x = A * cos a * cos b * cos c
y = A * sin a * cos b * cos c
z = A * sin b * cos c
w = A * sin c

OR

x = A * cos a * cos c
y = A * sin a * cos c
z = A * cos b * cos c
w = A * sin b * cos c

(both of these sets describe the same object, but precisely because of tiger, it's important to know both)

(31)
x = A * cos a * cos b + B * cos a * cos b * cos c
y = A * sin a * cos b + B * sin a * cos b * cos c
z = A * sin b + B * sin b * cos c
w = B * sin c

You see here that A describes 3-dimensional object in xyz, and B 3+1 dimensional one. Actually, we treat w as containing a 1-dimensional object with value of zero - but we can only introduce 1-dimensional objects this way.

((21)1)
x = A * cos a + B * cos a * cos b + C * cos a * cos b * cos c
y = A * sin a + B * sin a * cos b + C * sin a * cos b * cos c
z = B * sin b + C * sin b * cos c
w = C * sin c

Here, we start with 2D object in xy, then enlarge it to (2+1)D in xyz, THEN enlarge it once again to ((2+1)+1)D object

(211)
x = A * cos a + B * cos a * cos b * cos c
y = A * sin a + B * sin a * cos b * cos c
z = B * sin b * cos c
w = B * sin c

Here, we start with 2D object in xy, and 1D in z and w, then combine it all in just one step (as opposed to previous equation) - that's why we have just two radii (A and B) instead of three

Now we're coming to tiger. Tiger (22) has equations:

x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = B * cos b + C * cos b * sin c
w = B * sin b + C * sin b * sin c

The new thing is that we start with no assumed 1D objects - we simply start with two independent 2D objects (xy and zw), THEN combine them.

From this point of view, the tiger certainly IS a torus, since it is obtained through the same basic construction. There is, however, one more clue - we obtain this if we try to slice every polytope with coordinate hyperplanes:

Slicing glome, (4), we always get a single sphere.
Slicing (31), we get EITHER a single torus, OR two concentric spheres.
Slicing ((21)1), we get either:
1: two concentric torii differing only in their inner radii
2: two identical torii displaced in xy plane (xy plane is their main plane - i.e. the plane where their outer radius lies)
3: two concentric torii differing only in their OUTER radii
Slicing (211), we get EITHER a single torus, OR two identical, displaced spheres.

Taking all possible combinations, we find that one is missing: none of there three toruses can produce two dorii displaced in z direction when sliced. Happily, this is exactly what a tiger produces if you slice it:

x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = B * cos b + C * cos b * sin c
w = B * sin b + C * sin b * sin c

Solve for w = 0:

sin b * (B + C * sin c) = 0

We assume that B + C * sin c cannot be zero (this is basically the assumption of having "decent" radii to make the figure non-intersecting, and it's the same we make for 3D torii). This leaves sin b = 0, making cos b either 1 or -1, leading to:


x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = B + C * cos b * sin c
w = 0

and

x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = -B + C * cos b * sin c
w = 0

which are equations of two torii in xyz hyperplane, with main circles in xy, and displaced through z.
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Postby Keiji » Wed Jul 19, 2006 9:01 am

Bear in mind, I don't understand parametric equations; what do all the letters (apart from xyzw) stand for?

Marek14 wrote:Hmmm... Actually, I don't think that those parenthesis correspond to any ACTUAL operation. Every torus is assigned to another object, yes, but that's just an artifact of the fact that there is the same number of both. Any operation assigned to it was almost definitely an afterthought - the way I use it, there is NO operation defined, it's just a combinatoric assignment.


What about the spheration operation? Doesn't circle # x mean the same as (x)?

(4)


The glome is just 4, not (4).

Now we're coming to tiger. Tiger (22) has equations:

x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = B * cos b + C * cos b * sin c
w = B * sin b + C * sin b * sin c

The new thing is that we start with no assumed 1D objects - we simply start with two independent 2D objects (xy and zw), THEN combine them.


What are these objects, then?

which are equations of two torii in xyz hyperplane, with main circles in xy, and displaced through z.


So these torii are oriented perpendicular to the plane of their major circumference? Certainly that would produce an infinite genus figure?

If you could give me a surcell equation (like x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + w<sup>2</sup> = 1, for the glome), I would be able to have a look at this object...
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Postby Marek14 » Wed Jul 19, 2006 2:01 pm

Rob wrote:Bear in mind, I don't understand parametric equations; what do all the letters (apart from xyzw) stand for?


The capital letters (A,B etc.) represent various radii of the figure. They are considered to be fixed. The lowercase letters (a, b, c) are parameters. To get a point on the figure, you select a value for a, b, and c (from <0, 2pi) each, let's say), then compute x,y,z,w coordinates corresponding to these values. We have always three parameters, that's why the result is object with three internal dimension (surface of torus)


Marek14 wrote:Hmmm... Actually, I don't think that those parenthesis correspond to any ACTUAL operation. Every torus is assigned to another object, yes, but that's just an artifact of the fact that there is the same number of both. Any operation assigned to it was almost definitely an afterthought - the way I use it, there is NO operation defined, it's just a combinatoric assignment.


What about the spheration operation? Doesn't circle # x mean the same as (x)?


Actually, I'm not sure what this means. This notation was introduced by someone else.

(4)


The glome is just 4, not (4).


ah

Now we're coming to tiger. Tiger (22) has equations:

x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = B * cos b + C * cos b * sin c
w = B * sin b + C * sin b * sin c

The new thing is that we start with no assumed 1D objects - we simply start with two independent 2D objects (xy and zw), THEN combine them.


What are these objects, then?


First has the equations:
x = A * cos a
y = A * sin a

These are equations of a circle. The equations containing B, likewise, are equations of a circle.

which are equations of two torii in xyz hyperplane, with main circles in xy, and displaced through z.


So these torii are oriented perpendicular to the plane of their major circumference? Certainly that would produce an infinite genus figure?

Huh? No, imagine it as two tires lying on top of each other - or one hovering above the other. This is as opposed to two tires lying next to each other.
If you could give me a surcell equation (like x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + w<sup>2</sup> = 1, for the glome), I would be able to have a look at this object...


The thing is that this is exactly the hard part with torii. Their parametric equations are easy to derive, but the implicit equation is much harder.

Take ordinary 3D torus, for example. Its parametric equations are:

x = A * cos a + B * cos a * cos b
y = A * sin a + B * sin a * cos b
z = B * sin b

What does this REALLY represent, though? We can split it in A terms and B terms, to get
x = A * cos a
y = A * sin a
which is a circle, and
x = B * cos a * cos b
y = B * sin a * cos b
z = B * sin b
which is a sphere. A naive way to look at this would be to take the circle, and then "add" the sphere to every of its points. This is naive because it assumes a and b for the sphere indepentent, when in fact a is fixed by choice of the point at the main circle. So for every point, only a slice of the sphere, i.e. a circle is added, which is definition of torus.

The thing is that this cannot be readily transformed in implicit equation, at least I don't see a way :(
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Postby Keiji » Wed Jul 19, 2006 2:15 pm

Marek14 wrote:
Rob wrote:Bear in mind, I don't understand parametric equations; what do all the letters (apart from xyzw) stand for?


The capital letters (A,B etc.) represent various radii of the figure. They are considered to be fixed. The lowercase letters (a, b, c) are parameters. To get a point on the figure, you select a value for a, b, and c (from <0, 2pi) each, let's say), then compute x,y,z,w coordinates corresponding to these values. We have always three parameters, that's why the result is object with three internal dimension (surface of torus)


And I imagine the values of the lowercase letters represent local coordinates?

Marek14 wrote:
Rob wrote:
Marek14 wrote:Hmmm... Actually, I don't think that those parenthesis correspond to any ACTUAL operation. Every torus is assigned to another object, yes, but that's just an artifact of the fact that there is the same number of both. Any operation assigned to it was almost definitely an afterthought - the way I use it, there is NO operation defined, it's just a combinatoric assignment.


What about the spheration operation? Doesn't circle # x mean the same as (x)?


Actually, I'm not sure what this means. This notation was introduced by someone else.


well, neither am I, lol. Spheration (the correct name is "torus product", which I forgot to write in my last post) doesn't make much sense to me either.

Marek14 wrote:
Rob wrote:
Marek14 wrote:Now we're coming to tiger. Tiger (22) has equations:

x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = B * cos b + C * cos b * sin c
w = B * sin b + C * sin b * sin c

The new thing is that we start with no assumed 1D objects - we simply start with two independent 2D objects (xy and zw), THEN combine them.


What are these objects, then?


First has the equations:
x = A * cos a
y = A * sin a

These are equations of a circle. The equations containing B, likewise, are equations of a circle.


Yes, but where did the "+ C * cos a * cos c" bit come from, then?

Marek14 wrote:
Rob wrote:
Marek14 wrote:which are equations of two torii in xyz hyperplane, with main circles in xy, and displaced through z.


So these torii are oriented perpendicular to the plane of their major circumference? Certainly that would produce an infinite genus figure?

Huh? No, imagine it as two tires lying on top of each other - or one hovering above the other. This is as opposed to two tires lying next to each other.


I knew that. But I meant, wouldn't the tiger itself have an infinite genus?

Marek14 wrote:
Rob wrote:If you could give me a surcell equation (like x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + w<sup>2</sup> = 1, for the glome), I would be able to have a look at this object...


The thing is that this cannot be readily transformed in implicit equation, at least I don't see a way :(


Damn. Is it possible to work them out?
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Postby Marek14 » Wed Jul 19, 2006 6:40 pm

Rob wrote:
Marek14 wrote:
Rob wrote:Bear in mind, I don't understand parametric equations; what do all the letters (apart from xyzw) stand for?


The capital letters (A,B etc.) represent various radii of the figure. They are considered to be fixed. The lowercase letters (a, b, c) are parameters. To get a point on the figure, you select a value for a, b, and c (from <0, 2pi) each, let's say), then compute x,y,z,w coordinates corresponding to these values. We have always three parameters, that's why the result is object with three internal dimension (surface of torus)


And I imagine the values of the lowercase letters represent local coordinates?


Well, kind of local coordinates. You have to compute the real coordinates from them. They are kinda like latitude, longitude, and some other -itude. I'm too lazy to think up a name. Hmm... lazitude?

Marek14 wrote:
Rob wrote:
Marek14 wrote:Hmmm... Actually, I don't think that those parenthesis correspond to any ACTUAL operation. Every torus is assigned to another object, yes, but that's just an artifact of the fact that there is the same number of both. Any operation assigned to it was almost definitely an afterthought - the way I use it, there is NO operation defined, it's just a combinatoric assignment.


What about the spheration operation? Doesn't circle # x mean the same as (x)?


Actually, I'm not sure what this means. This notation was introduced by someone else.


well, neither am I, lol. Spheration (the correct name is "torus product", which I forgot to write in my last post) doesn't make much sense to me either.


Well, this kind of product simply means (as I understand it), "take every point of shape #1 and replace it with shape #2". For example, circle # circle would replace every point in a circle with another circle, so you would get a torus.

Marek14 wrote:
Rob wrote:
Marek14 wrote:Now we're coming to tiger. Tiger (22) has equations:

x = A * cos a + C * cos a * cos c
y = A * sin a + C * sin a * cos c
z = B * cos b + C * cos b * sin c
w = B * sin b + C * sin b * sin c

The new thing is that we start with no assumed 1D objects - we simply start with two independent 2D objects (xy and zw), THEN combine them.


What are these objects, then?


First has the equations:
x = A * cos a
y = A * sin a

These are equations of a circle. The equations containing B, likewise, are equations of a circle.


Yes, but where did the "+ C * cos a * cos c" bit come from, then?

That's the parenthesis :) No, seriously, try this:

1. Select a fixed a and b.
2. Compute coordinates
x = A * cos a
y = A * sin a
z = B * cos b
w = B * sin b

3. NOW, leave a and b fixed, vary c, and compute with complete parametric formula. You find that you draw a circle in 4D space around the point you found in step 2. The values of a and b (still fixed) determine not only location of that circle, but also its orientation.

In other words, A and B terms are the "skelet" of the figure (much as a circle is the skelet of torus), and C term is what you must add to actually get to the figure's surface.


Marek14 wrote:
Rob wrote:
Marek14 wrote:which are equations of two torii in xyz hyperplane, with main circles in xy, and displaced through z.


So these torii are oriented perpendicular to the plane of their major circumference? Certainly that would produce an infinite genus figure?

Huh? No, imagine it as two tires lying on top of each other - or one hovering above the other. This is as opposed to two tires lying next to each other.


I knew that. But I meant, wouldn't the tiger itself have an infinite genus?


I think tiger is topologically equivalent to ((21)1). ((21)1) has torus as a skelet, while tiger has duocylinder margin as its skelet, and these two figures are topologically equivalent. But I'm not even completely sure how is "genus" defined for 4D figures.

Marek14 wrote:
Rob wrote:If you could give me a surcell equation (like x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + w<sup>2</sup> = 1, for the glome), I would be able to have a look at this object...


The thing is that this cannot be readily transformed in implicit equation, at least I don't see a way :(


Damn. Is it possible to work them out?

Not sure.
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Postby Keiji » Wed Jul 19, 2006 7:27 pm

Marek14 wrote:
Rob wrote:
Marek14 wrote:
Rob wrote:
Marek14 wrote:Hmmm... Actually, I don't think that those parenthesis correspond to any ACTUAL operation. Every torus is assigned to another object, yes, but that's just an artifact of the fact that there is the same number of both. Any operation assigned to it was almost definitely an afterthought - the way I use it, there is NO operation defined, it's just a combinatoric assignment.


What about the spheration operation? Doesn't circle # x mean the same as (x)?


Actually, I'm not sure what this means. This notation was introduced by someone else.


well, neither am I, lol. Spheration (the correct name is "torus product", which I forgot to write in my last post) doesn't make much sense to me either.


Well, this kind of product simply means (as I understand it), "take every point of shape #1 and replace it with shape #2". For example, circle # circle would replace every point in a circle with another circle, so you would get a torus.


Oh yes, forgot that... so circle # x would be the same as (x1). The tiger is supposedly defined as duocylinder # circle, but that never made any sense to me.

Marek14 wrote:
Rob wrote:
Marek14 wrote:These are equations of a circle. The equations containing B, likewise, are equations of a circle.


Yes, but where did the "+ C * cos a * cos c" bit come from, then?

That's the parenthesis :) No, seriously, try this:

1. Select a fixed a and b.
2. Compute coordinates
x = A * cos a
y = A * sin a
z = B * cos b
w = B * sin b

3. NOW, leave a and b fixed, vary c, and compute with complete parametric formula. You find that you draw a circle in 4D space around the point you found in step 2. The values of a and b (still fixed) determine not only location of that circle, but also its orientation.

In other words, A and B terms are the "skelet" of the figure (much as a circle is the skelet of torus), and C term is what you must add to actually get to the figure's surface.


I see now, that makes a lot more sense, but what happens if you set C close to zero? :?

Marek14 wrote:
Rob wrote:
Marek14 wrote:
Rob wrote:
Marek14 wrote:which are equations of two torii in xyz hyperplane, with main circles in xy, and displaced through z.


So these torii are oriented perpendicular to the plane of their major circumference? Certainly that would produce an infinite genus figure?

Huh? No, imagine it as two tires lying on top of each other - or one hovering above the other. This is as opposed to two tires lying next to each other.


I knew that. But I meant, wouldn't the tiger itself have an infinite genus?


I think tiger is topologically equivalent to ((21)1). ((21)1) has torus as a skelet, while tiger has duocylinder margin as its skelet, and these two figures are topologically equivalent. But I'm not even completely sure how is "genus" defined for 4D figures.


The tetratorus, ((21)1) has a pocket and a hole. The tiger, as far as I can see, has no pockets and infinite holes. But you raise the point that there may be an extra kind of hole in 4D, as in 1D there is no type of hole, in 2D there is one, and in 3D there are two. Hmm...
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Postby Marek14 » Wed Jul 19, 2006 8:21 pm

Rob wrote:
Marek14 wrote:
Rob wrote:well, neither am I, lol. Spheration (the correct name is "torus product", which I forgot to write in my last post) doesn't make much sense to me either.


Well, this kind of product simply means (as I understand it), "take every point of shape #1 and replace it with shape #2". For example, circle # circle would replace every point in a circle with another circle, so you would get a torus.


Oh yes, forgot that... so circle # x would be the same as (x1). The tiger is supposedly defined as duocylinder # circle, but that never made any sense to me.



Take duocylinder MARGIN, not duocylinder as such. If you replace each point with a circle, you get tiger. However, it is not as intuitive, because the orientation of circle at each point depends on two variables, and not just one.


Marek14 wrote:
Rob wrote:
Marek14 wrote:These are equations of a circle. The equations containing B, likewise, are equations of a circle.


Yes, but where did the "+ C * cos a * cos c" bit come from, then?

That's the parenthesis :) No, seriously, try this:

1. Select a fixed a and b.
2. Compute coordinates
x = A * cos a
y = A * sin a
z = B * cos b
w = B * sin b

3. NOW, leave a and b fixed, vary c, and compute with complete parametric formula. You find that you draw a circle in 4D space around the point you found in step 2. The values of a and b (still fixed) determine not only location of that circle, but also its orientation.

In other words, A and B terms are the "skelet" of the figure (much as a circle is the skelet of torus), and C term is what you must add to actually get to the figure's surface.


I see now, that makes a lot more sense, but what happens if you set C close to zero? :?


Then you get closer and closer to C=0, which is exactly the duocylinder margin, i.e. cartesian product of two circles (NOT discs!)

Marek14 wrote:
Rob wrote:
Marek14 wrote:
Rob wrote:
Marek14 wrote:which are equations of two torii in xyz hyperplane, with main circles in xy, and displaced through z.


So these torii are oriented perpendicular to the plane of their major circumference? Certainly that would produce an infinite genus figure?

Huh? No, imagine it as two tires lying on top of each other - or one hovering above the other. This is as opposed to two tires lying next to each other.


I knew that. But I meant, wouldn't the tiger itself have an infinite genus?


I think tiger is topologically equivalent to ((21)1). ((21)1) has torus as a skelet, while tiger has duocylinder margin as its skelet, and these two figures are topologically equivalent. But I'm not even completely sure how is "genus" defined for 4D figures.


The tetratorus, ((21)1) has a pocket and a hole. The tiger, as far as I can see, has no pockets and infinite holes. But you raise the point that there may be an extra kind of hole in 4D, as in 1D there is no type of hole, in 2D there is one, and in 3D there are two. Hmm...


The thing is that tetratorus, sliced by coordinate hyperplane, always results in two torii, in various mutual orientations. Tiger has only one orientation, but it slices into two torii, likewise. I think similar case could be built as (31) and (211) having the same topology.
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Postby Keiji » Wed Jul 19, 2006 9:05 pm

By the way, can you please stop posting "duocylinder margin"? The something-frame adjectives were coined for clarification purposes - do you mean a diframe or a triframe duocylinder?
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Postby PWrong » Thu Jul 20, 2006 4:33 am

If you could give me a surcell equation (like x2 + y2 + z2 + w2 = 1, for the glome), I would be able to have a look at this object...

this thread should help you find the cartesian equations.
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Postby Keiji » Thu Jul 20, 2006 9:31 am

Hmm, I tried using them, but they seem to be for the duocylinder, as the cross-sections formed were the cylinder expanding and shrinking.
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Postby Marek14 » Thu Jul 20, 2006 12:33 pm

Rob wrote:By the way, can you please stop posting "duocylinder margin"? The something-frame adjectives were coined for clarification purposes - do you mean a diframe or a triframe duocylinder?


I took that term from Wendy - I think that by "margin" she means "two dimensions lower than the original figure". So duocylinder margin is, I guess diframe duocylinder, the 2D object curved into 4D that separates both "cells".
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Postby PWrong » Fri Jul 21, 2006 1:15 am

Hmm, I tried using them, but they seem to be for the duocylinder, as the cross-sections formed were the cylinder expanding and shrinking.

The equations for the duocylinder are
x^2+y^2 < r<sub>1</sub>^2
z^2+w^2 < r<sub>2</sub>^2

The equation for the tiger is:
(sqrt(x^2+y^2) - r<sub>1</sub>)^2 + (sqrt(z^2+w^2) - r<sub>2</sub>)^2 < r<sub>3</sub> ^2

I think that by "margin" she means "two dimensions lower than the original figure".

I always took "margin" to mean the lowest dimensional object possible. So the margin of a cube is a set of points, and the margin of a cylinder is a pair of circles.
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Postby Keiji » Fri Jul 21, 2006 10:22 am

Ah, now I get the tiger.

It seems that that extra hole isn't a hole at all. The tiger still has genus 1.
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