For tiger, the two toruses would somehow continuously turn in two toruses in perpendicular direction, keeping four circles in a vertical plane undisturbed.
ICN5D wrote:I never thought about that, but you're right, it would be cool. I guess it would be a matter of switching the cut marker around, and figuring the transition out with the notation. That's a pretty neat way to trace out the shape, when you put it together in your head.For tiger, the two toruses would somehow continuously turn in two toruses in perpendicular direction, keeping four circles in a vertical plane undisturbed.
Somehow is right! It could be based off the mid-cut of the oblique tiger cross sections. That 4-holed frame would be the mid-shape between transitions. The n-2 cuts seem to be some kind of anchor point for the morphing, in that they can't be moved, interesting.....
Ah, correction: 5 closed toratopes for 4D, 12 closed for 5D, 33 closed for 6D. Have you and wendy figured out the 7D total yet? Is it 90 or 93? There's only one way to truly prove it, and that's a complete enumeration. But, that condensed notation seems to do the trick. I need to learn it to see all of the shapes in them.
ICN5D wrote:Ah, correction: 5 closed toratopes for 4D, 12 closed for 5D, 33 closed for 6D. Have you and wendy figured out the 7D total yet? Is it 90 or 93? There's only one way to truly prove it, and that's a complete enumeration.
I wrote:The number of closed (equivalently, open) toratopes in n dimensions is http://oeis.org/A000669. See Toratope#Counting_toratopes.
The "anchor point" thing has a very simple explanation: Two hyperplane mid-cuts intersect in a plane, and that's why morphing from one hyperplane cut to another will leave one plane invariant.
Keiji wrote:ICN5D wrote:Ah, correction: 5 closed toratopes for 4D, 12 closed for 5D, 33 closed for 6D. Have you and wendy figured out the 7D total yet? Is it 90 or 93? There's only one way to truly prove it, and that's a complete enumeration.
Actually, that's not true. You can now easily find the number of toratopes in n dimensions without enumeration. Quoting myself from the Tiger Explained thread:I wrote:The number of closed (equivalently, open) toratopes in n dimensions is http://oeis.org/A000669. See Toratope#Counting_toratopes.
Marek14 wrote:Well, in 5D, you could already see it with the 2D array: horizontal line on the array is one 4D cut, vertical is another 4D cut, and the oblique lines would correspond to cuts gained through rotation. The central 3D cut would be invariant.
It's similar for 6D.
wendy wrote:Wendy's calculations on the number of toratopes has errors in it. Last night, she dremt the right solution, and it was the same as Marek14's.
ICN5D wrote:Marek14 wrote:Well, in 5D, you could already see it with the 2D array: horizontal line on the array is one 4D cut, vertical is another 4D cut, and the oblique lines would correspond to cuts gained through rotation. The central 3D cut would be invariant.
It's similar for 6D.
Okay, I see what's going on here. The dimensions of the cut array corresponds to the number of reduced dimensions. More reduced dimensions makes the array more complex, one at a time. So, 5D reduced to 3D requires a 2D array to fully represent? I think this is the first time I understood that part about the arrays. I couldn't figure out how you were getting two and three D arrays with 5 and 6D tigroid cuts. Now I understand! That makes it even more attractive to explore 7D tigroids, it would be a 4D array! Represented in a 3D array, of course, it would have to be understood that even this array had an underlying pattern itself. Wow, there would be four mutual ways to move along an axis to morph the cut and do the tiger dance.
Tiger/torus tiger (((II)(II))((II)I))
ICN5D wrote:Tiger/torus tiger (((II)(II))((II)I))
Oh my gosh! That was an unexpected beauty! That's what I mean about seeing the list, the ones that stand out. I guess those types are my favorite, a pure tigroid having two complex tiger symmetry frames. I guess it can be called the 220210-tiger. Don't worry, I won't ask you to spend 4 hours to cut this one down I'm more interested in discovering some sort of immediately identifiable feature about these types, a quick go to for visualizing. Like the case with the 330-tiger: it has two hollow sphere frames, in which the cuts show how both can be interchanged. One hollow sphere becomes the toratope frame, which follows along the other hollow sphere as the tigroid frame. The toratope frame is inflated with a circle, the tigroid frame is inflated with the full toratope.
So, by using this little short cut generalization, a quick map can be made of the toratope along tigoird structures, while maintaining that both can be interchanged with any one of their reduced cuts. The tigroid frame cuts follows the same principle as a circle-inflated toratope cut, but has a whole inflated toratope along its manifold. Cutting far enough will completely separate both hollow frames, in addition to cross-breed hybrid cuts of both superimposed frames. I'm going to meditate on that during my bike ride today. Yep, that's right, I dodge rollerbladers and joggers while doing 6D geometry in my head. Life is great
So in 2D, you should understand circle (II), whose mid-cut is two points.
In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.
anderscolingustafson wrote:So in 2D, you should understand circle (II), whose mid-cut is two points.
In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.
I think I understand what a Toratope is now. So basically a Toratope is a circular shape cut in half so that if you cut a donut, circle, or sphere in half you get a Toratope? Is that the correct definition of a Toratope Marek14?
quickfur wrote:<br abp="659">No, a toratope is just a torus-like shape (torus = doughnut shape). The cutting in half part is only for the purposes of visualization, it has nothing to do with the shape itself. In 3D, the main torus-like shape is the torus itself (i.e., the doughnut), but 4D and above, it is possible to have many different kinds of torus-like shapes (hence the name toratope), including shapes with different kinds of "doughnut holes".anderscolingustafson wrote:<br abp="657"><br abp="658">I think I understand what a Toratope is now. So basically a Toratope is a circular shape cut in half so that if you cut a donut, circle, or sphere in half you get a Toratope? Is that the correct definition of a Toratope Marek14?So in 2D, you should understand circle (II), whose mid-cut is two points.<br abp="656">In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.
anderscolingustafson wrote:So do spheres count as Toratopes or does a shape have to have holes in the middle to be a Toratope? Are there Toratopes in 2d and if so what would be some 2d Toratopes?
anderscolingustafson wrote:quickfur wrote:<br abp="659">No, a toratope is just a torus-like shape (torus = doughnut shape). The cutting in half part is only for the purposes of visualization, it has nothing to do with the shape itself. In 3D, the main torus-like shape is the torus itself (i.e., the doughnut), but 4D and above, it is possible to have many different kinds of torus-like shapes (hence the name toratope), including shapes with different kinds of "doughnut holes".anderscolingustafson wrote:<br abp="657"><br abp="658">I think I understand what a Toratope is now. So basically a Toratope is a circular shape cut in half so that if you cut a donut, circle, or sphere in half you get a Toratope? Is that the correct definition of a Toratope Marek14?So in 2D, you should understand circle (II), whose mid-cut is two points.<br abp="656">In 3D, torus ((II)I), whose mid-cuts are two separate circles or two concentric circles.
So do spheres count as Toratopes or does a shape have to have holes in the middle to be a Toratope? Are there Toratopes in 2d and if so what would be some 2d Toratopes?
ICN5D wrote:I'm sorry, I can't help it. The (((II)(II))((II)I)) is just too darn interesting. I played with some 3D cuts of the 220210:
(((I)(I))((I))) - 110100 - tigric-torus tiger
Now, this one is interesting. I've seen this kind of arrangement before, with () over every ' I " in a torus layout. It's related to a (((I)(I))(I)) 11010-double tiger, 8 torii at vert of cube. By removing the pairs of brackets that make it an octet, a little method I just found, we can derive the base cut as being ((II)(I)) 210-tiger, 2 vert stacked torii. So, these modifying brackets that make it an octet act on this initial vertical column arrangement.
Making (((I)(I))((I))) an octet of 2 vertical stacked torii in vertices of a cube.
(((II)())((I))) - 200100
Okay, this one was tough. There's a tricky sequence of () that can be removed to reveal the initial state, and I eventually found ((II)(I)) to be it, again.
The reduction series went like this:
(((II)())((I))) - 200100
((II)()((I))) - 20100 : removed brackets that make a concentric arrangement of 2 torii
((II)()(I)) - 2010 : removed brackets that make vertical column of 4 torii
((II)(I)) - 210 : vertical column of 2 torii, since the () is empty
That makes the 200100 a vertical column of four groups of 2 concentric torii each.
((()(I))((I)I)) - 010110
And once again, we end up with a ((II)(I)) . But, this one is modified differently:
Reduction tree:
(((I))((I)I)) - 10110 : removed empty ()
((I)((I)I)) - 1110 : removed () that make 4 vert stack torii
((I)(II)) - 120 : removed () that make 2 displaced torii
Which makes 010110 two displaced columns of four torii each.
Now, I'm not sure how to apply the arrays yet. There's more to learn about it, but I'll get it soon. I know it's related to 'what part of what was cut', and the array traces out the missing structure. I'll be back soon with the arrays of these cuts, it's becoming very clear to me how it's made.
quickfur wrote:anderscolingustafson wrote:So do spheres count as Toratopes or does a shape have to have holes in the middle to be a Toratope? Are there Toratopes in 2d and if so what would be some 2d Toratopes?
I suppose spheres would be trivial toratopes, so in 2D you at least have the circle as a toratope. Most of the interesting things happen in higher dimensions, though, especially 4D and beyond.
Marek14 wrote:This means that we'll encounter one stack of 4 ditoruses from the direction of medium dimension. THIS is the stack of 4 pairs of concentric toruses. Your mistake is that this is not the cut in the middle, but something that gets revelaled a bit further.
On each side, we'll encounter four tigers and mid-cut of each of them will look as you say - two displaced columns of four torii each. But once again, this will be encountered on each side of the mid-cut, not exactly there.
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