Marek14 wrote:Well, I finally found a way to slice the tetratorus in x or y direction. I uploaded a series of images from Mathematica to http://rapidshare.de/files/27316331/tetratorus.zip.html (No place to host them...)
Rob wrote:Marek14 wrote:Well, I finally found a way to slice the tetratorus in x or y direction. I uploaded a series of images from Mathematica to http://rapidshare.de/files/27316331/tetratorus.zip.html (No place to host them...)
Shouldn't the tetratorus change from two torii to a torus perpendicualr to that?
Interesting pictures anyway. I can make better ones if you give me the cartesian equation
Rob wrote:Well, your equation was mal-formed, but I fixed it, and here are the images:
And now I can see that it is indeed correct, it shouldn't be a perpendicular torus because it would be 4th dimensionally perpendicular and that's what forms the blob you get on the last slice.
What did you use for those pictures, Rob?
Could you render the singular slices, i.e. those where the overall topology changes via a sharp point? There should be three - between your pictures 2 and 3, 4 and 5, and 7 and 8. I don't remember the exact values for y for them, but they should all be integers.
It looks like you used the same diameters as me (4,2 and 1) - the halving ensures that the toratope will never intersect itself.
Take the tiger#circle, ((22)1)
Rob wrote:What did you use for those pictures, Rob?
That's a secretCould you render the singular slices, i.e. those where the overall topology changes via a sharp point? There should be three - between your pictures 2 and 3, 4 and 5, and 7 and 8. I don't remember the exact values for y for them, but they should all be integers.
I think those slices are interesting. With my parameters (see below), the slices are at 0.7, 1.3 and 2.7 respectively.
It looks like you used the same diameters as me (4,2 and 1) - the halving ensures that the toratope will never intersect itself.
No, I used 2, 1, and 0.3.Take the tiger#circle, ((22)1)
((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.
And isn't it a bit too soon to be going for 5D objects?
If you want, I can make image slices for that as well.
Marek14 wrote:Rob wrote:I think those slices are interesting. With my parameters (see below), the slices are at 0.7, 1.3 and 2.7 respectively.
Yes - if we mark the radii A, B, and C, then the transitions are at B-C, B+C and A-C (I think that final shrink to the point should be A+C, 3.3 in your model)
It looks like you used the same diameters as me (4,2 and 1) - the halving ensures that the toratope will never intersect itself.
No, I used 2, 1, and 0.3.Take the tiger#circle, ((22)1)
((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.
Actually, calling this circle#tiger seems very counterintuitive to me. I mean, how does one build it? By taking a tiger, and replacing each point with a circle. You can't START with a circle.
If you want, I can make image slices for that as well.
Well, the 5D possibilities are numerous:
1. Petaglome - x^2+y^2+z^2+w^2+v^2=A^2
All of the 3D midsections are spheres.
2. (41) (Sqrt(x^2+y^2+z^2+w^2)-A)^2+v^2=B^2
xyz midsection are two concentric spheres, but xyv one is a torus.
3. ((31)1) (Sqrt((Sqrt(x^2+y^2+z^2)-A)^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (four concentric spheres), xyw (two nested torii), xyv (two concentric torii) and xwv (two separated torii)
4. (((21)1)1) (Sqrt((Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+w^2)^2-C+v^2=D^2
Possible cuts: xyz (four nested torii), xyw (two concentric pairs of nested torii), xyv (four concentric torii), xzw (two separated pairs of nested torii), xzv (two pairs of concentric torii), xwv (four torii in a line), and zwv (empty in the middle)
5. ((22)1) (Sqrt((Sqrt(x^2+y^2)-A)^2+(Sqrt(z^2+w^2)-B)^2)-C)^2+v^2=D^2
Possible cuts: xyz (two pairs of nested torii atop each other), xyv (empty in the middle) and xzv (four torii in the corners of rectangle)
6. ((211)1) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (two nested torii), xyv (two concentric torii), xzw (two separated pairs of concentric spheres), xzv (two separated torii), zwv (empty in the middle)
7. (32) (Sqrt(x^2+y^2+z^2)-A)^2+(Sqrt(w^2+v^2)-B)^2=C^2
Possible cuts: xyz (empty in the middle), xyw (two torii atop each other), xwv (two torii atop each other - different)
8. ((21)2) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+(Sqrt(w^2+v^2)-C)^2=D^2
Possible cuts: xyz (empty in the middle), xyw (two pairs of concentric torii atop each other), xzw (two stacks of two torii), xwv (stack of four torii), zwv (empty in the middle, but different from xyz)
9. (311) (Sqrt(x^2+y^2+z^2)-A)^2+w^2+v^2=B^2
Possible cuts: xyz (two concentric spheres), xyw (torus), xwv (two separated spheres)
10. ((21)11) (Sqrt((Sqrt(x^2+y^2)-A)^2+z^2)-B)^2+w^2+v^2=C^2
Possible cuts: xyz (two nested torii), xyw (two concentric torii), xzw (two separated torii), xwv (four spheres in a line), zwv (empty in the middle)
11. (221) (Sqrt(x^2+y^2)-A)^2+(Sqrt(z^2+w^2)-B)^2+v^2=C^2
Possible cuts: xyz (stack of two torii), xyv (empty in the middle), xzv (four spheres in corners of rectangle
12. (2111) (Sqrt(x^2+y^2)-A)^2+z^2+w^2+v^2=B^2
Possible cuts: xyz (torus), xzw (two separated spheres), zwv (empty in the middle)
Funny, isn't it?
Well, I finally realized what the general equation is and how to derive it:
((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.
This makes me wonder what the sequence is for the number of possible torii... lol.
PWrong wrote:Well, I finally realized what the general equation is and how to derive it:
I already posted this here.
((22)1) is the circle#tiger. That said, I don't like the definition of spheration anyway.
A#B isn't valid unless B is a sphere in some dimension. The reason is you need unique axes to align the tiger, but not a sphere.
This makes me wonder what the sequence is for the number of possible torii... lol.
We already found this. 1, 1, 2, 5, 12, 33, 90, 261, 766, 2312, ...
http://tetraspace.alkaline.org/forum/viewtopic.php?t=403&start=30
This mathworld page has the final formula for the number of rotopes and the number of torii.
I also wrote a mathematica notebook to list all the rotopes in dimension n.
Rob wrote:Well, that's a bit silly, how is a line curved, let alone a torus?
bo198214 wrote:Hey Marek and Rob,
give you a jerk! Wouldnt it be a great idea to have a program where you can choose out of a list of 4d toratopes (and where you can put in a shape in CSG notation), and it shows you a 3d slice moving through the shape?
That would be novel and a really appreciable distribution to the 4d community.
PS: Programming language java, then you can put it online everywhere.
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