Stereoscopic 4D Klein Bottle
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- เผยแพร่เมื่อ 23 ส.ค. 2013
- This animation is a stereogram, so if you view the left image with your left eye and the right image with your right eye, it should produce the illusion of being 3D. Focusing your eyes properly for this effect can take some practice. If you're unsure how to do this, I would recommend viewing on a small screen (like a smart phone) and start with it very close to your eyes and gradually move it away until it comes into focus.
The 3D image you see is the "shadow" of a four dimensional Klein bottle that has no self-intersection, which needs a minimum of four dimensions to achieve. A Klein bottle is a surface which is similar to a Mobius strip in that it only has one side, except the Klein bottle is completely closed; it has no edges. If you think of the Mobius strip as an annulus (flat cylinder) that twists in the third dimension, you can think of the Klein bottle as a torus (donut shape) that twists in the fourth dimension.
The first half of the animation shows a straight orthogonal projection into 3D space, and the second half shows a stereographic projection by normalizing all points to the unit 3-sphere and then projecting into R^3 with the usual stereographic projection map.
Although it appears that the Klein bottle is changing shape, it is actually just rotating in four dimensional space. This is similar to when you rotate a 3D object and look at its 2D shadow. Although the object stays the same shape, its shadow looks like it is deforming.
Angular velocity for rotations was determined by a random walk in R^6 (there are 6 planes of rotation in 4D space), and primary color was determined by a random walk in the unit cube (for RGB values).
This was made in Mathematica 9.0.
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The music is "Let Yourself Be Huge" by Cloudkicker, available under the Creative Commons Attribution license.
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/ bmsharp - วิทยาศาสตร์และเทคโนโลยี
I made a Klein bottle hat recently (see pic)
Tibees I think you're wearing it inside out (best hat ever btw).
@@RichardHennigan hahaha
Teacher: So did you study?
Me: Yes! I remember everything!
My brain during exam:
Looks amazing through Google cardboard! You can easily see the two mobius strips with the added depth.
+Lawrence Simmons works with just crossing your eyes as well xD
Deeply moving. Thanks.
Fantastic work, thanks !
Very good. It's very easy come in 3d with my eyes and it's beautiful
the Klein bottle have a 2d boundary, so that even it existing in 4d a 3d slice will output a line-curve. There is any 4d generalization that works the same way as the mobius strip, i.e has one side, but with a 3d boundary? such that a 3d slice give a surface? and if it exists how could I parametrize that?
by the way thanks for the parametric equations in the beginning.
Simple 3d object 🌝
Passei por uma deficiência na minha vista direita em que a visão estará nos 90 por cento no centro.Para observar espectogramas está muito complicado e depois fiquei surpreso quando observei este Steroscopic mas através do estrabismo,magnífico! Basta observar as imagens em movimento a olhar para a ponta do nariz com ele direccionado para o centro das imagens,toda aberta!
Faszinierend 👍👍👍
hi beautiful work....I would love to use your videos for an art project/theater piece I am involved in. I do not know how to contact you! Would this be something you would be ok with if we credited you?
when I was on dmt it was nothing like this. it was inside and outside at the same time
it was beautifull
Feeling nauseous,hallucinated, and hypnotized.Anyone call the paramedics?
How does one project a Klein bottle onto a 3 sphere? I don't understand how you can bring its surface into a sphere. I mean like, how can you even do that for a Möbius strip using a sphere in the first place?!
3-sphere? Isn't it projected in R^3?
@@sender1496 My bad. I was curious to know about how the stereographic projection was made.
I'm not even sure about a stereographic projection of a Möbius strip as well.
I assumed that we use the method where we map the edges of the strip to a 2 sphere
So we'll just scale up a dimension in the case of a Klein bottle.
Please correct me if I was wrong.
I so wish someone with the time and resources would make a 4D rendering engine. I would be happy to help... and have offered to many times. It's sad that people can't yet get to see what 4D really looks like.
someone made a game that renders the 4d in 3d.... its still close? meigakure is the name btw
Do you know the game "4d Toys"?
It's insanely hard. The closest people have done is Miegakure, 4D Toys, and 4D Miner
This is cool, but it definitely looks like surfaces are colliding if not intersecting.
song
IS THIS THE REAL LIFE
I tried and I think I got it, but now my eyes hurt..
Tell me about it
Is this vr---
CROSS YOUR EYES SO THE TWO OF THEM ARE ALIGNED IN THE MIDDLE AHH
Koala Bear you would need to have myopia problem in other to do this (which I do) or else it will be really blurry for normal eyes
@@ertyiopas1514 it creates '3' images, its just crossing ur eyes, its normal right?
You can both cross your eyes outward and inward. This is like when you wear a vr headset to create the 3D effect
Slow down! Sheesh!
Apparently only horses can see this stereogram...
i fell sick
Always the same mistake in 3D videos: showing the images distal instead of proximal. PLEASE: re-upload with the RIGHT-eye image on the LEFT, and LEFT-eye image on the RIGHT. The reason is that it's MUCH easier to do a cross-eye than to stare "beyond infinity" (the divergent cross-eye). All books on stereochemistry use the proximal images for that reason.
JanPBtest I disagree. I have been doing the infinity thing for years now even to the point where I eliminate the need for google cardboard because I use my fingers to block the other image. I can’t stand crossing my eyes
Interesting. I'm exactly the opposite (and all the stereochemistry textbooks :-) ).
Shapes in Ohio are so weird💀
It's totally distracting that the two images are not complete. They need to be completely separate for the 3D effect to really work. Try again.
+photondance So how does one represent that on screen, please ? Is it sufficient just to put a small space between the two images ?
+Richard Swan That's a good idea. I wish I had the software to try that.
Ideally, for this demonstration, I would have two complete images on the 2d screen, that never intersected, or extended beyond their borders.
+photondance It seems (?) the 2nd animation has a "singular" point where is loops out to infinity. That makes it impossible to limit the animation except by truncation. I don't have time right now to check if that is exactly true; it is merely an initial impression for the time being.
+photondance It seems (?) the 2nd animation has a "singular" point where is loops out to infinity. That makes it impossible to limit the animation except by truncation. I don't have time right now to check if that is exactly true; it is merely an initial impression for the time being.
+Richard Swan That's exactly right. When the Klein bottle passes over the north pole of the 3-sphere, it becomes infinitely large in the projected space.