Animations of 4D torii cross-sections

Discussion of shapes with curves and holes in various dimensions.

Animations of 4D torii cross-sections

Postby Plasmath » Thu Feb 11, 2021 12:29 am

I made some animations of cross-sections of the 4D torii (and the torinder) that were based on some of the renders by Polyhedron Dude.
Last edited by Plasmath on Sun Feb 14, 2021 12:36 pm, edited 1 time in total.
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Re: Animations of 4D torii cross-sections

Postby Plasmath » Thu Feb 11, 2021 1:20 pm

Okay, so I've been able to get some higher-quality images from using external links.

Tiger
Image

Ditorus
Image

Image

Image

Torisphere
Image

Image

Spheritorus
Image

Image

Torinder
Note: the first cross-section is just a boring torus for the entire time because it is a prism, however the other two look pretty neat:
Image

Image

I hope this makes it easier to visualize the 4D torii, although I haven't figured out how to make oblique cross-sections yet.
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Re: Animations of 4D torii cross-sections

Postby Plasmath » Sat Feb 13, 2021 1:51 am

Okay, so I've made some oblique cross-sections of all the 4D torii (all at a 45° angle), and although the tiger looks the best, I still think that the others look pretty cool.

Tiger
Image


Here's a slower version of the animation, there's a lot going on.
Image

Ditorus
Image

Image

Spheritorus
Image

Image

Torisphere
I didn't find much of anything interesting when exploring the torisphere. I might look at it in further detail later, but it takes a really long time to poke through the cross-sections.

Torinder
Image

Image
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Re: Animations of 4D torii cross-sections

Postby Marek14 » Sun Feb 14, 2021 12:14 am

Look nice -- but I think the speed is set too high. They should change, say, 2-3 times slower.
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Re: Animations of 4D torii cross-sections

Postby Marek14 » Sun Feb 14, 2021 12:43 pm

I know that there are also some interesting rotation animations (between two mid-sections), in 4D and higher.

The interesting ones should be:

Torisphere ((xyz)w): xyz/xyw, two concentric spheres vs. torus, with two concentric circles in xy fixed through the rotation.
Spheritorus ((xy)zw): xyz/xzw, torus vs. two spheres, with two circles in xz fixed through the rotation.
Tiger ((xy)(zw)): xyz/xzw, stack of two torii vs. stack of two torii in different orientation, with 2x2 circles in xz fixed through the rotation.

Ditorus (((xy)z)w):
1) xyz/xyw: two cocircular torii vs. two concentric torii, with four concentric circles in xy fixed through the rotation.
2) xyz/xzw: two cocircular torii vs. two torii, with two pairs of concentric circles in xz fixed through the rotation.
3) xyw/xzw: two concentric torii vs. two torii, with four circles in xw fixed through the rotation.
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Re: Animations of 4D torii cross-sections

Postby Plasmath » Mon Feb 15, 2021 6:29 pm

Here are some rotations of the torii:

Tiger ((xy)(zw)): xyz/xzw, stack of two torii vs. stack of two torii in different orientation, with 2x2 circles in xz fixed through the rotation.

Image

Spheritorus ((xy)zw): xyz/xzw, torus vs. two spheres, with two circles in xz fixed through the rotation.

Image

Torisphere ((xyz)w): xyz/xyw, two concentric spheres vs. torus, with two concentric circles in xy fixed through the rotation.

Image

Ditorus (((xy)z)w):

Image

Image

I couldn't find the final rotation.

Here's also some rotations of the torinder:
Image

Image
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Re: Animations of 4D torii cross-sections

Postby Marek14 » Mon Feb 15, 2021 11:09 pm

The final one is between two concentric torii and two cocircular torii. Each of them occurs in one of your animations.
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Re: Animations of 4D torii cross-sections

Postby icebreaker » Fri Feb 19, 2021 7:49 pm

The animations look very good, visually. But I wouldn't show them to a novice in 4D. I agree with Maker14, time period should be increased. As for the tiger animation, its 'flashing' texture prevents from seeing its very interesting structure, especially for the untrained viewer. If you want to leave stripped texture, it might be well to make colors not so contrast. Also, it's easily to mix up ditorus with spheritorus because of speed and especially lack of transparency. The same problem is visible in torisphere (looks like a glome).

Thank you, didn't see Torinder cross-secitons before. It's a pleasure. How did you get so clear edges? I'm interested what software are you using.
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Re: Animations of 4D torii cross-sections

Postby Plasmath » Sat Feb 20, 2021 2:09 am

I used POV-Ray for the animations, it has the max function implemented so that's how I made the edges.

I could definitely make the tiger less flashy, although making animations longer or transparent would be more difficult because both would make rendering take a much longer time, but I could try to do that.
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Re: Animations of 4D torii cross-sections

Postby ICN5D » Thu Mar 18, 2021 9:09 pm

These are great! You might be able to slow down the animations by opening them in a gif maker, like gifmoviegear. Then, select all the frames and set a longer duration to them (in seconds, like 0.1 s). Hopefully the program doesn't mess up the colors, because it might (it could downgrade the color spectrum). I don't use povray, so I don't know how to make the transparent surfaces (other than using calcplot). I'm sure some of the adobe programs could modify the timescale, but those will cost $$$ monthly to license one.
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Re: Animations of 4D torii cross-sections

Postby Plasmath » Sat Apr 03, 2021 2:18 am

Okay, this took much longer than I expected, but I've finally re-rendered everything, now with 91 frames in each because just slowing the frames down looked really bad. I think I made some mistakes before and also forgot some sections before, but I've added them here.

I also made some of these transparent, and the tiger a bit less flashy.

Tiger
Orthogonal cross-sections
Image

Oblique cross-sections
Image

Rotations
Image

Ditorus
Orthogonal cross-sections
Image
Image
Image

Oblique cross-section
Image
Image
Image

Rotations
Image
Image
Image

Spheritorus
Orthogonal cross-sections
Image
Image

Oblique cross-sections
Image

Rotations
Image

Torisphere
Orthogonal cross-sections
Image
Image

Oblique cross-sections
Image

Rotations
Image

Torinder
Orthogonal cross-sections
Image
Image

Oblique cross-sections
Image
Image
Image

Rotations
Image
Image
Image
Last edited by Plasmath on Sat Apr 03, 2021 9:26 pm, edited 1 time in total.
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Re: Animations of 4D torii cross-sections

Postby ICN5D » Sat Apr 03, 2021 2:06 pm

Torisphere definitely has some neat looking oblique slices! Especially when getting close to the concentric spheres slice, between the 67.5 deg and 90 deg angles.

Image
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Re: Animations of 4D torii cross-sections

Postby Plasmath » Sat Apr 03, 2021 9:26 pm

Okay, I think I just forgot to render the oblique sections of the torisphere. I'll edit the post so that it includes that.
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Re: Animations of 4D torii cross-sections

Postby icebreaker » Sun Apr 04, 2021 5:55 am

Plasmath wrote:Okay, this took much longer than I expected, but I've finally re-rendered everything, now with 91 frames in each because just slowing the frames down looked really bad. I think I made some mistakes before and also forgot some sections before, but I've added them here.

I also made some of these transparent, and the tiger a bit less flashy.

...


I think it was worth it! Look so smooth that I would like to touch these, if could. And they have become more informative, too.
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