The Tiger Explained

[Marek14]

Just found this:

http://www.youtube.com/watch?v=fkaI6meNiI0

Was this done by someone from here?

[ICN5D]

That's a cool video! It has to be someone on here. No one else would know that name. I left a comment, to see who it might be. The transparent graph was really cool, you could almost see it's two pathways.

[ICN5D]

So, I've been playing around with the cut algorithm, starting with the familiar 4D toratopes. I'm seeing now that the sequence for the different radii are, for example the ditorus: (((major)middle)minor) in (((II)I)I) . This means when we cut the minor radius, we get two toruses that differ in their minor, making them cocircular. When we cut the middle radius, we get two toruses that differ in their middle, making them concentric. When we cut the major, we get two toruses that differ in their major, making them displaced. But for a tiger ((II)(II)), we have ((major1)(major2)), and when we cut either one, we get two toruses that differ in a major radius, but in a strange way. They are arranged in a vertical column, separated and parallel. I'm still having a tough time interpreting this arrangement in the cuts. I see the two separate toruses ((II)(I)), but I don't see any sequence in the notation that tells me they are in a vertical column.

(II)I - cylinder
(I)I - square with circle symmetry
(II) - circle with line symmetry

((maj)min)
((II)I) - torus
((I)I) - cutting major radius gives two displaced circles (II)
((II)) - cutting minor gives two concentric circles

((III)I) - torisphere
((II)I) - cut major gives torus that varies in major rad
((III)) - cut minor gives two concentric spheres

((II)II) - spheritorus
((I)II) - cutting major gives two displaced spheres
((II)I) - cutting minor gives torus that varies in minor rad

(((maj)med)min)
(((II)I)I) - ditorus
(((II)I)) - cut minor gives two corcircular toruses
(((II))I) - cut medium gives two concentric toruses
(((I)I)I) - cut maj gives two displaced toruses

((maj)(maj))
((II)(II)) - tiger
((II)(I)) - two vertical stacked toruses

Is coplanar another way to say displaced? What is the cool "co-" word for the vertical column in tiger slices? I like knowing this stuff.

[Marek14]

So, I've been playing around with the cut algorithm, starting with the familiar 4D toratopes. I'm seeing now that the sequence for the different radii are, for example the ditorus: (((major)middle)minor) in (((II)I)I) . This means when we cut the minor radius, we get two toruses that differ in their minor, making them cocircular. When we cut the middle radius, we get two toruses that differ in their middle, making them concentric. When we cut the major, we get two toruses that differ in their major, making them displaced. But for a tiger ((II)(II)), we have ((major1)(major2)), and when we cut either one, we get two toruses that differ in a major radius, but in a strange way. They are arranged in a vertical column, separated and parallel. I'm still having a tough time interpreting this arrangement in the cuts. I see the two separate toruses ((II)(I)), and the fact that they have an additional pair of parentheses.

Well, this is not entirely true. For example, cutting the middle radius can't get you two toruses differing in their middle radius, for the simple reason that toruses don't HAVE one -- only ditoruses do.

For two toruses to differ in a radius, they have to be a different shape. But if you cut the "major" radius of ditorus, you'll get two identical toruses -- just displaced. Also, cutting the tiger also leads two identical toruses, they don't differ in any diameter.

Maybe it will be useful to show the full range of 3D ditoruses. Well, they is really no 3D ditorus, but any ditorus, no matter what dimension, is limited to the following shapes as its 3D cuts:

300-ditorus (((III))): four concentric spheres. First used in 5D as a cut of spheric ditorus (((III)I)I).
210-ditorus (((II)I)): two toruses differing in their minor radius. First used in 4D as a cut of ditorus (((II)I)I).
201-ditorus (((II))I): two toruses differing in their major radius. First used in 4D as a cut of ditorus (((II)I)I).
120-ditorus (((I)II)): two displaced pairs of concentric spheres. First used in 5D as a cut of cylindrical ditorus (((II)II)I).
111-ditorus (((I)I)I): two coplanar toruses. First used in 4D as a cut of ditorus (((II)I)I).
102-ditorus (((I))II): four spheres arranged in a line. First used in 5D as a cut of toracubtorinder (((II)I)II).
030-ditorus ((()III)): empty. First used in 6D as a cut of 231-ditorus (((II)III)I).
021-ditorus ((()II)I): empty. First used in 5D as a cut of cylindrical ditorus (((II)II)I).
012-ditorus ((()I)II): empty. First used in 5D as a cut of toracubtorinder (((II)I)II).
003-ditorus ((())III): empty. First used in 6D as a cut of 213-ditorus (((II)I)III).

As for tiger, it's actually ((major1)(major2)minor). This is why my notation describes tiger with 3 numbers, not 2. However, the 3rd number is allowed to be 0 because while this radius CAN add more dimensions, like in toraduocyldyinder ((II)(II)I), it's not REQUIRED to.

Is coplanar another way to say displaced? What is the cool "co-" word for the vertical column in tiger slices? I like knowing this stuff.

Coplanar is not the same as displaced. Displaced just means they lie in different places, but coplanar means they are in the same plane. Well, more precisely, their main circles are in the same plane.

Removed large unnecessary portion of quote. ~Keiji

[ICN5D]

Well, this is not entirely true. For example, cutting the middle radius can't get you two toruses differing in their middle radius, for the simple reason that toruses don't HAVE one -- only ditoruses do.

Yeah, I wasn't sure how to put that one. I still understand that the effect is concentric pairs, even thought there is no middle radius in a torus. When I think of middle radius cuts while picturing the two concentric toruses, I see them merging. That is, I still see the frame of the ditorus that it traces out.

For two toruses to differ in a radius, they have to be a different shape. But if you cut the "major" radius of ditorus, you'll get two identical toruses -- just displaced. Also, cutting the tiger also leads two identical toruses, they don't differ in any diameter.

This is correct, the toruses themselves don't change at all. I guess what I meant was that they follow along the other major radius, and it is this one that they are tracing out and differing in, when merging or separating.

Maybe it will be useful to show the full range of 3D ditoruses. Well, they is really no 3D ditorus, but any ditorus, no matter what dimension, is limited to the following shapes as its 3D cuts:

Awesome! Thanks for that. It will help me interpret the notation better. It's the subtle arrangements of () and " I " that confuse me right now. I have no idea how you can see a row of four spheres in the sequence (((I))II) . No doubt there is some hidden meaning in there, but I can't see it right now. I will meditate on it, though. Do you have any other cool lists like that, going through all possible combinations?

As for tiger, it's actually ((major1)(major2)minor). This is why my notation describes tiger with 3 numbers, not 2. However, the 3rd number is allowed to be 0 because while this radius CAN add more dimensions, like in toraduocyldyinder ((II)(II)I), it's not REQUIRED to.

That makes sense, now that you pointed out the "0" in the number notation. I knew the minor radius had to be in there somewhere, just couldn't put my finger on it.

[ICN5D]

Keiji, are you crackin' jokes? Because that's kind of funny.

Unfortunately, though, I got to Marek's post before you did!

[Marek14]

Awesome! Thanks for that. It will help me interpret the notation better. It's the subtle arrangements of () and " I " that confuse me right now. I have no idea how you can see a row of four spheres in the sequence (((I))II) . No doubt there is some hidden meaning in there, but I can't see it right now. I will meditate on it, though. Do you have any other cool lists like that, going through all possible combinations?

Well, the row of four spheres is achieved by following the notation from inside out:

We start with an empty set inside the innermost pair of parentheses: This is a point.
(I): Now we "blow up" the point, i.e. replace it by all points with fixed distance from it. Since there's 1 I, which is dimensional marker, we'll do this in 1D. Final result: two points on a line.
((I)): Now we blow up these two points again. But since we still only have 1 I, we're still in 1D. So each point splits in further two and we have four points on a line.
(((I))II): Now we blow up THESE four points, but this time we have three dimensional markers, and so we're in 3D. Each of the points will get blown up in 3D and transforms into a sphere.

And that's how you end with four spheres in a line.

Now let's look at a similar arrangement for tigers:

300-tiger ((III)()): empty. First appearing in 5D in cylspherintigroid ((III)(II)).
210-tiger ((II)(I)): two vertically stacked toruses. First appearing in 4D in tiger ((II)(II)).
201-tiger ((II)()I): empty. First appearing in 5D in toraduocyldyinder ((II)(II)I).
111-tiger ((I)(I)I): four spheres in vertices of rectangle. First appearing in 5D in toraduocyldyinder ((II)(II)I).
102-tiger ((I)()II): empty. First appearing in 6D in 222-tiger ((II)(II)II).
003-tiger (()()III): empty. First appearing in 7D in 223-tiger ((II)(II)III).

Now let's look at the 111-tiger more closely to see how it's built:

(I), as we've seen, is a pair of points.
(I)(I) is a cartesian product of two pairs of points, i.e. four points in vertices of rectangle.
((I)(I)I) is blowing up this cartesian product in 3 dimensions. Thus, four spheres.

[ICN5D]

I do in fact see some logic in the (I) notation. If (I) means two points at vertices of a line, (I)(I) is four points at vertices of square, then (I)(I)(I) is eight points at vertices of cube

(I) is related to a " I " , where () means at the corners of a I, line
(I)(I) is related to a " II ", where () means at the corners of a II, square
(I)(I)(I) is related to a " III ", where () means at the corners of a III, cube
(I)(I)(I)(I) would then be related to a IIII, where sixteen points are at the corners of a tesseract IIII

I am noticing that the sequence ((I)(I)I) has a sphere (III) in it, as well as the (I)(I), corners of a square. The (I)(I) is the arrangement, (III) is what's left over when omitting the two ()().

Would it be impossible to have sixteen spheres at the vertices of a tesseract? If the sequence ((I)(I)(I)(I)I) means sixteen pentaspheres at the vertices of a tesseract, how could only three " I " markers fit in there? They probably don't, since there would be empty sets. At some point, though, the pentaspheres could be cut down to spheres, but I'm not sure how that would look.

If (I)(I) is the cartesian product of two point pairs, (I)(I)(I) would be the cart prod of three pairs, six in total. One would be incorrect to conclude that there are only six points. There are actually eight, in vertices of a cube, derived from (I)(I)(I) being related to a III. It seems like (I) means a hollow line, and the product of three hollow lines would be identical to the product of three solid lines, making a cube. This interpretation makes more sense, and is probably the way you are seeing it.

According to (((I))II), I believe that (((I))I) would be four circles along a line. It's the effect of the extra parentheses that I'm trying to learn. If only having so many dimension markers, I guess this would sort of lock it in to a particular arrangement, forcing the pairs to be within a certain n-plane. If so, then what would ((((I))I)I) be? I see a linear arrangement , and the ditorus, but the extra () around the first marker stumps me. It looks like a 4D cut of a tetratorus. A cut of the major, followed by a cut of the 1st middle according to ((((maj)mid1)mid2)min). This seems like a row of two concentric ditorus pairs, four in total.

Would I just simply need to memorize ((II)(I)) is two vertical stacked toruses? Omitting the () around the "I" will make ((II)I), a torus, but the placement of the (I) has to be the key ingredient here. It's within the torus, taken out from a circular radius. This will give the torus pair a circular symmetry with their merging, the tiger dance, as you like to call it Or, is there some relationship with the hidden minor radius that can conclude this arrangement? ((II)(I)) gives us three dimensions, ((II)I) gives us a torus, (I) gives us two of these toruses. Where is the key part in the vertical stacking? The endpoints of the (I) are along Z, where the torus pair sits. It must be simple familiarizing and memorizing.

[Marek14]

I do in fact see some logic in the (I) notation. If (I) means two points at vertices of a line, (I)(I) is four points at vertices of square, then (I)(I)(I) is eight points at vertices of cube

(I) is related to a " I " , where () means at the corners of a I, line
(I)(I) is related to a " II ", where () means at the corners of a II, square
(I)(I)(I) is related to a " III ", where () means at the corners of a III, cube
(I)(I)(I)(I) would then be related to a IIII, where sixteen points are at the corners of a tesseract IIII

I am noticing that the sequence ((I)(I)I) has a sphere (III) in it, as well as the (I)(I), corners of a square. The (I)(I) is the arrangement, (III) is what's left over when omitting the two ()().

Would it be impossible to have sixteen spheres at the vertices of a tesseract? If the sequence ((I)(I)(I)(I)I) means sixteen pentaspheres at the vertices of a tesseract, how could only three " I " markers fit in there? They probably don't, since there would be empty sets. At some point, though, the pentaspheres could be cut down to spheres, but I'm not sure how that would look.

You can't have sixteen spheres at vertices of tesseract since sphere is a 3D object and tesseract is 4D; once you make cartesian product of four pairs of point, you're in 4D, so any blowing up will naturally end up as 4D at least.

If (I)(I) is the cartesian product of two point pairs, (I)(I)(I) would be the cart prod of three pairs, six in total. One would be incorrect to conclude that there are only six points. There are actually eight, in vertices of a cube, derived from (I)(I)(I) being related to a III. It seems like (I) means a hollow line, and the product of three hollow lines would be identical to the product of three solid lines, making a cube. This interpretation makes more sense, and is probably the way you are seeing it.

According to (((I))II), I believe that (((I))I) would be four circles along a line. It's the effect of the extra parentheses that I'm trying to learn. If only having so many dimension markers, I guess this would sort of lock it in to a particular arrangement, forcing the pairs to be within a certain n-plane. If so, then what would ((((I))I)I) be? I see a linear arrangement , and the ditorus, but the extra () around the first marker stumps me. It looks like a 4D cut of a tetratorus. A cut of the major, followed by a cut of the 1st middle according to ((((maj)mid1)mid2)min). This seems like a row of two concentric ditorus pairs, four in total.

The main operation for construction is the spheration or "blowing up": This means replacing a given set of points by set of points with certain distance from the original set, i.e. the lowest distance from new point to any of the old is given.

((((I))I)I) can be constructed as following:

(I) is 1D circle, hollow, i.e. pair of points. Its radius will be A. Every pair of parentheses is connected to one radius. You can have these points at A and -A.

((I)) is a spheration of these two points, while we still stay in 1D. This means that we'll get four points, one pair lies in distance B from the first point, the other one in distance B from second point. These points are at:
A+B, A-B, -A+B, -A-B

(((I))I) introduces radius C. We're spherating previous result, four points, but this time we're adding one additional dimension. Therefore, each of the four points is replaced by a circle of radius C and we have four circles in a line.

Finally, ((((I))I)I) introduces radius D. We're spherating the previous result, four circles, and we're adding one extra dimension. The result is that each of the four circles will be replaced by a torus.

Final result: four toruses in a line.

Would I just simply need to memorize ((II)(I)) is two vertical stacked toruses? Omitting the () around the "I" will make ((II)I), a torus, but the placement of the (I) has to be the key ingredient here. It's within the torus, taken out from a circular radius. This will give the torus pair a circular symmetry with their merging, the tiger dance, as you like to call it Or, is there some relationship with the hidden minor radius that can conclude this arrangement? ((II)(I)) gives us three dimensions, ((II)I) gives us a torus, (I) gives us two of these toruses. Where is the key part in the vertical stacking? The endpoints of the (I) are along Z, where the torus pair sits. It must be simple familiarizing and memorizing.

As for ((II)(I)), it's derived like this:

(II) is a circle.
(I) is a pair of points.
(II)(I) is a cartesian product of the two, i.e. two circles in parallel planes.
((II)(I)) is spheration of previous result without adding any additional dimensions, i.e. two toruses whose main circles lie in parallel planes.

[ICN5D]

Aha! I see it now! Your notation uses a combination of cartesian products and spheration. Both of which I understand at this point. I see your method of building up the sequence. That's the best way for me see it. Now I get the vertical stacking.


120-ditorus (((I)II)): This is supposed to make two pairs of concentric spheres

(I) - hollow circle, two points

(I)II - cartesian product of two points and a square, two squares along a line

((I)II) - adding 0 dim, spherating two squares, making two spheres along a line, same process as spherating a cubinder into a torisphere (II)II --> ((II)II)

(((I)II)) - adding 0 dim = staying in 3D, spherating the two spheres, making similar pattern with hollow circle, but acting on spheres. This will make 2 spheres in same position as one, manifesting the concentric arrangement.


I think I see how they become concentric. It looks like the cartesian product of a hollow circle and two spheres. If I understand it correctly, ((III))(I) should also make the same thing. Looks like a cut from a cylspheritorinder, or ((III)I)(II) 312-cyltorinder. Not sure what the most current naming would be.

You've already noted how ((III)) is two concentric spheres, the reverse way can also be done here, by adding a (I) into the preexisting ((III)), to make (((I)II)). But, the build method from the ground up is shedding more logical light on it. So, (((III))) is four concentric spheres and ((((III)))) is eight concentric spheres. (((I)(I)I)) is four concentric sphere pairs in vertices of a square, (((I)(I)(I))) is eight concentric sphere pairs in vertices of a cube. Starting to make sense, which is quite awesome.

[Keiji]

This has been incredibly helpful for me too!

I've created a page at Extended toratopic notation, is this all correct?

[Marek14]

Aha! I see it now! Your notation uses a combination of cartesian products and spheration. Both of which I understand at this point. I see your method of building up the sequence. That's the best way for me see it. Now I get the vertical stacking.


120-ditorus (((I)II)): This is supposed to make two pairs of concentric spheres

(I) - hollow circle, two points

(I)II - cartesian product of two points and a square, two squares along a line

((I)II) - adding 0 dim, spherating two squares, making two spheres along a line, same process as spherating a cubinder into a torisphere (II)II --> ((II)II)

No, doesn't work quite like that. With this method, you only use steps which are in parenthesis. So from (I), you go right into ((I)II) and you spherate the two points in 3 dimensions. But yes, you'll get two spheres along a line.

(((I)II)) - adding 0 dim = staying in 3D, spherating the two spheres, making similar pattern with hollow circle, but acting on spheres. This will make 2 spheres in same position as one, manifesting the concentric arrangement.

Exactly.

I think I see how they become concentric. It looks like the cartesian product of a hollow circle and two spheres. If I understand it correctly, ((III))(I) should also make the same thing. Looks like a cut from a cylspheritorinder, or ((III)I)(II) 312-cyltorinder. Not sure what the most current naming would be.

No, ((III))(I) can't be the same thing -- for one thing, this has four dimensions while (((I)II)) has three. ((III))(I) would be two pairs of concentric spheres, yes, but displaced in a fourth dimension -- not in a dimension the spheres lie in.

You've already noted how ((III)) is two concentric spheres, the reverse way can also be done here, by adding a (I) into the preexisting ((III)), to make (((I)II)). But, the build method from the ground up is shedding more logical light on it. So, (((III))) is four concentric spheres and ((((III)))) is eight concentric spheres. (((I)(I)I)) is four concentric sphere pairs in vertices of a square, (((I)(I)(I))) is eight concentric sphere pairs in vertices of a cube. Starting to make sense, which is quite awesome.

Yes, it works like that.

Keiji: Yes, seems correct, though I'd emphasize more that there's a difference between (II)II and ((II)II). The notation was developed primarily for closed toratopes and there are some problems with open ones: the cuts of open toratopes depend on if you take only surfaces of toratopes or insides as well for the product.

Also, the spheration operation, as you define it, might need a better definition. As you use it, then spherating a circle into 3D will get you solid torus (sum of spheres centered on each point of circle). But any spheration operation works only on surface! So you will need to specify what is the surface of resulting set and discard the rest.

You also say that the notation allows better representation of cuts. Of course, cuts are achieved by leaving out various I's, but that is not actually said on the page -- it probably should be

[Keiji]

There, I've amended the wiki page to address those issues now

[ICN5D]

Actually, I spent some time on using the cut algorithm on open toratopes. It's not that bad at representing what they are. I think a symbol should be used for the I that was taken out. This will help distinguish the cuts from another preexisting shape notation. It will also place a marker on it's symmetry when moving along the axis. Marek, I know you use the letter i in the current notation. Here, I'm going to use it to identify the part that was cut. I hope that somehow parallels your method I'm only going to cut one axis in each distinct coordinate plane. I noticed that the notation can't express a glomolatrix. If possible, it would be helpful in determining the end shape before disappearance, for certain closed toratopes.


- If cutting inside an n-sphere, the cut has n-spheric symmetry, N-1 = number of axes with identical symmetry of the cut, i.e. (Ii)II

- If cutting a line, or inside an n-cube, cut has n-cubic symmetry, like a prism, N-1 = number of axes with identical symmetry of the cut, i.e. (II)Ii

- For closed toratopes, when cutting a particular radius, cut has n-spheric symmetry of that radius***, i.e. ((Ii)I) has major radius circle symmetry, ((IIi)I) has major radius spherical symmetry, etc.

*** In the instance of the cut not having the same kind of radius as it's labeled, this symmetry marker is only used for establishing how the cut behaves when moving along the axis, how it collapses or merges. For example: the ditorus (((II)I)I), cut the middle radius (((II)i)I) gives two concentric toruses ((II))I) that merge into one, being labelled as having "middle radius symmetry". The concentric toruses do not have a middle radius, but the ditorus cut through it's middle radius causes the concentric pair to merge into one torus.


Determining the Cuts and Symmetry:

Open Toratopes:

1) If cutting in a 2-sphere, remove the i and only the () it's inside will give the cut

2) If cutting in a +3-sphere, remove only the i will give the cut

3) If cutting inside the (), removing the i and the entire term of (...) it's inside will give the end shape before disappearing. This also helps identify which part of the cut collapses

4) If cutting a line ( inside an n-cube ), removing the i and all terms will be analogous to abrupt disappearance as the end shape, because there is none



Closed Toratopes:

1) Remove the i will give the cut in the current notation used

2) Removing the i and the innermost () will give the end shape before deflating and disappearing ( for some because the hollow circle cannot be represented )



xyzwv

II - square
Ii - line with line symmetry, moving along Y will stay the same length and abruptly disappear

(II) - circle
(Ii) - hollow circle (I) = two points, with circle sym, will merge

(II)I - cylinder
(Ii)I - square II with circle sym, will collapse to line I
(II)i - circle (II) with line sym, will maintain and disappear

(II)II - cubinder
(Ii)II - cube III with circ sym, will collapse to square II
(II)Ii - cylinder with square symm, will maintain then disappear

(III)I - spherinder
(IIi)I - cylinder (II)I with spherical sym, will collapse raduis to a line I of equal length
(III)i - sphere with line symm, will maintain then disappear

((II)I) - torus
((Ii)I) - displaced circles ((I)I) with circle symmetry, will merge into one circle (II)
((II)i) - concentric circles ((II)) with circle symmetry, will merge into one circle (II)

(II)III - tesserinder
(Ii)III - tesseract IIII with circle symm, will collapse height to cube III
(II)IIi - cubinder (II)II with cube symm, will maintain then disappear

(III)II - cubspherinder
(IIi)II - cubinder (II)II with sphere symm, will collapse to square II along a 2-plane
(III)Ii - spherinder with square symm, will maintain then disappear

(II)(II) - duocylinder
(II)(Ii) - cylinder (II)I with circle symm, will collapse to circle (II)

(III)(II) - cylspherinder
(IIi)(II) - duocylinder (II)(II) with spheric symm, will collapse to circle along a 2-plane
(III)(Ii) - spherinder (III)I with circ symm, will collapse to sphere (III)

((II)II) - spheritorus
((Ii)II) - two displaced spheres ((I)II) with circ symm, will merge to one sphere (III), then deflate to point and disappear
((II)Ii) - torus ((II)I) with spherical symm on minor rad, will collapse to thin ring and disappear

((III)I) - torisphere
((IIi)I) - torus ((II)I) with spherical symm, collapses maj radius to sphere (III), then deflate and disap
((III)i) - two concentric spheres ((III)) with minor circ symm, merge into one sphere (III), then deflate and disap

(((II)I)I) - ditorus
(((Ii)I)I) - two displaced toruses (((I)I)I) with circ symm on maj, will merge into one torus ((II)I)
(((II)i)I) - two concentric toruses (((II))I) with cir symm on mid, will merge into one torus ((II)I)
(((II)I)i) - two cocircular toruses (((II)I)) with circ symm on minor, will merge into one torus ((II)I)

((II)(II)) - tiger
((II)(Ii)) - two vertical stacked toruses ((II)(I)) with circ symm on maj2 radius, will merge into one torus, then deflate to thin ring and disappear

[ICN5D]

I've been thinking about the hollow circle. Couldn't it be represented by ((I)(I))? What is this thing if not? Spherating a quartet or points, while still maintaining 2D, will seem like the II to (II) analogy where a solid square spherates into a solid circle. If (I)(I) means four points in vertices of a square, what would spherating do to those points? Maybe turn them into a single line around a circle, the glomolatrix?

[Marek14]

2) Removing the i and the innermost () will give the end shape before deflating and disappearing ( for some because the hollow circle cannot be represented )

Hm, I'm afraid this is not true... For tiger, for example, we talk about two toruses merging into one and then disappearing, but that's not actually true. The shapes formed in this case are Cassini ovals rotated along a nonbisecting axis, and though they have TOPOLOGY of torus, there is no point where you'd get an exact single torus. It will be always deformed in some way. Only the middle cut will be exact since the extreme Cassini oval is exactly two circles.

I've been thinking about the hollow circle. Couldn't it be represented by ((I)(I))? What is this thing if not? Spherating a quartet or points, while still maintaining 2D, will seem like the II to (II) analogy where a solid square spherates into a solid circle. If (I)(I) means four points in vertices of a square, what would spherating do to those point? Maybe turn them into a single line around a circle, the glomolatrix?

((I)(I)) is four circles in vertices of square. When you finish your homework (remember that?), you'll find it when working on tiger

[ICN5D]

Aha, I see what you mean. I suppose the two toruses will have their main circles also subjected to Cassini oval deformation in addition to merging. Polyhedron Dude ought to update that on the tiger slices. It shows up well in the oblique slices, though.

((I)(I)) is four circles in vertices of square. When you finish your homework (remember that?), you'll find it when working on tiger

Touché, Marek! I need to get the algorithm down pat, first

[wendy]

Does this notation extend to the tiger-like thing with one, three or four holes. In essence, the symmetry of the tiger includes the swirl prism symmetry, and therefore you ought be able to construct the swirl-prism of a hollow sphere with 2, 3, 4, ... holes poked into it. The actual tiger is the swirl of a houla hoop.

[Marek14]

Not sure, wendy. How exactly IS a two-holed torus defined? Is there a parametric or implicit equation, or is it understood solely as topological object?

[wendy]

The tiger has two holes. You can see this, by considering the spherated single face of a duo-cylinder, for example. This becomes a single hole. You can make one with four holes, by having a hollow {3,4,3}, and the holes following four cycles of six octahedra.

[Marek14]

The tiger has two holes. You can see this, by considering the spherated single face of a duo-cylinder, for example. This becomes a single hole. You can make one with four holes, by having a hollow {3,4,3}, and the holes following four cycles of six octahedra.

Ah, I see. I don't think the notation supports this at this point...

[Keiji]

The tiger has two holes. You can see this, by considering the spherated single face of a duo-cylinder, for example. This becomes a single hole. You can make one with four holes, by having a hollow {3,4,3}, and the holes following four cycles of six octahedra.

Wait, I thought we already had established the tiger has one (torus-shaped) hole?

And if the spherated single face of a duocylinder becomes the single hole of the tiger then didn't you just say the tiger has one hole anyway?

[Marek14]

Well, probably depends on how a "hole" is defined. It's actually pretty vague term...

I've posted a "block tiger" a while back (a tiger-like object built from hypercubes). It turns out to have a fairly simple description:

Take a 3x3x3x3 hypercube where each cube has coordinates from (1,1,1,1) to (3,3,3,3). Then remove every cube for which either first two coordinates or last two coordinates (or all four) are equal to 2.

Is there a definition of "hole" that describes the interior? You get a topological torus if you take a cylinder and remove a smaller cylinder from within it. However, for tiger topology, you have to remove two 1x1x3x3 boards from the full hypercube, OR you can remove just one object since the two boards cross in the middle.

So it might be useful talking about two holes since the simplest description involves removing two objects, but it can be equally well described by removing one object only.

[ICN5D]

The tiger, as I understand it, has one location where two sectioned off pathways go through. It's a very strange kind of hole. I see it as two "tunnels" that cross around each other in a ways, pointing in different directions at 90 degrees. It's almost like a ditorus, with the middle radius hole running around the circumference of the major. But, the tiger was made by holding the torus flat and doing the same rotation, into 4D. I don't see it having a torus-shaped hole as in a ditorus, though. It seems like it has two separate circular-holes walled off from each other, but in the same general location in the middle.

[Marek14]

The tiger, as I understand it, has one location where two sectioned off pathways go through. It's a very strange kind of hole. I see it as two "tunnels" that cross around each other in a ways, pointing in different directions at 90 degrees. It's almost like a ditorus, with the middle radius hole running around the circumference of the major. But, the tiger was made by holding the torus flat and doing the same rotation, into 4D. I don't see it having a torus-shaped hole as in a ditorus, though. It seems like it has two separate circular-holes walled off from each other, but in the same general location in the middle.

The thing with tiger is that its holes are NOT circular, though.

You can take an infinite pole (with other two dimensions limited) and stick it through a circular hole, like in torus.

For tiger, you can take an infinite 2D BOARD (again, with other two dimensions limited) and stick it through. And you can stick two such boards simultaneously, one in xy plane and one in zw plane, and they will intersect in a small region in the middle.

[ICN5D]

That's true as well. I suppose I'm using the term "circular" loosely here. Such is the case of slicing a spheritorus ((II)II), one of them is two spheres that can have this 2D board slid through. This cut is through the major radius, in a similar fashion to a 3D torus. Since the tiger has two major radii, it can have two of these cut directions, in addition to the infinite axles. No matter how the tiger is cut, other than the oblique, it will be through one of its major radii. This is what I meant by circular, through a 2-spheric major radius. Now that I think about it, can a tiger be cut through its minor radius? As in, this kind will give the two concentric circles from a 3D torus cut? It might be two cocircular and hollow spherated margins of a duocylinder, but would require a full 4 dimensions to represent.

[Marek14]

That's true as well. I suppose I'm using the term "circular" loosely here. Such is the case of slicing a spheritorus ((II)II), one of them is two spheres that can have this 2D board slid through. This cut is through the major radius, in a similar fashion to a 3D torus. Since the tiger has two major radii, it can have two of these cut directions, in addition to the infinite axles. No matter how the tiger is cut, other than the oblique, it will be through one of its major radii. This is what I meant by circular, through a 2-spheric major radius. Now that I think about it, can a tiger be cut through its minor radius? As in, this kind will give the two concentric circles from a 3D torus cut? It might be two cocircular and hollow spherated margins of a duocylinder, but would require a full 4 dimensions to represent.

Well, you can have a cut (((II)(II))) of (((II)(II))I) which leads to two tigers differing in minor radius.

[ICN5D]

That would make sense with a tiger torus, for sure. So, the tiger can't be cut that way? From a top to bottom slicing, that is through neither of the major radii?

Been cutting the cyltorinder today:

((II)I)(II)

((II)I)(I) - torinder, height has circular symm, will collapse to torus then disappear
((II))(II) - 2 concentric duocylinders, will merge into one then disappear
((I)I)(II) - 2 displaced duocylinders, will merge into one then disappear. But, I'm not sure if these two are in a row at the endpoints of a line or displaced. ((I)I) means displaced circles, then the cartesian product with a circle turns both into duocylinders.

[Marek14]

That would make sense with a tiger torus, for sure. So, the tiger can't be cut that way? From a top to bottom slicing, that is through neither of the major radii?

Been cutting the cyltorinder today:

((II)I)(II)

((II)I)(I) - torinder, height has circular symm, will collapse to torus then disappear
((II))(II) - 2 concentric duocylinders, will merge into one then disappear
((I)I)(II) - 2 displaced duocylinders, will merge into one then disappear. But, I'm not sure if these two are in a row at the endpoints of a line or displaced. ((I)I) means displaced circles, then the cartesian product with a circle turns both into duocylinders.

Well, tiger has one major radius in xy plane and other in zw plane, and you'll always cut one of these... maybe I don't quite see what you mean by cutting a radius.

((I)I)(II) has, let's say, two circles in xy plane, which then become duocylinders with a second circle in zw plane. So you have two displaced duocylinders and if you displace duocylinders in coordinate direction, there is actually only one way to do it.

[ICN5D]

By cutting a radius, one or more is left intact, while the sliced radius is divided. This leads to displaced vs concentric circles with 3D torus cuts. Cutting through the major radius will make two displaced circles, leaving the minor intact. Cutting through the minor radius will make two concentric circles, leaving the major radius intact.

((II)I) - ((major)minor) 21-torus

((II)) - 20-torus, cutting minor, major is untouched
((I)I) - 11-torus, cutting major, minor is untouched

But, the tiger has a hidden minor in your notation, ((II)(II)), 220-tiger. Cutting any part will always be through a major radius, like you said.