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.
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.
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.
(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.
which are equations of two torii in xyz hyperplane, with main circles in xy, and displaced through z.
Rob wrote: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...
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)
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.
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.
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.
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
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?
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?
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.
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.
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.
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.
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...
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...
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?
Hmm, I tried using them, but they seem to be for the duocylinder, as the cross-sections formed were the cylinder expanding and shrinking.
I think that by "margin" she means "two dimensions lower than the original figure".
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