#some2 Articles about computer graphics: iquilezles.org/ Ray marching tutorials: / theartofcodeiscool In this video I am trying to construct a visualization of 4D shapes
Nice! I also played with the same 4D models in my new game. Great to see a quality video about the less talked about ones like the duocylinder and tiger. Another interesting fact about the tiger is that it can be used to link a chain in 4D, which can't be done with a regular spheritorus like you might expect.
hey, I just want to let you know, I've been a huge fan of your work (like the fractal marble game) since BEFORE hyperbolica, and I literally checked Steam weekly for MONTHS and showed friends the devlogs since hey started to see if Hyperbolica released because I love (globally) non-Euclidean space and I thought your representation was so accessible that it even helped me explain it to my art-major girlfriend. If that game existed when I was 10 I would've been in heaven and learned so much. Thank you so much for all the work you've put in to interactive education!
The ability to create shapes by multiplying other shapes really is a cool way to do things. I'm curious how you're handling the simulation and rendering. Can't wait to see the 5D version!
Is there another way to understand putting the duocylinder and tiger together? I personally found the cartesian product thing convoluted without visuals for 3D shapes, let alone 4D.
So good to see something that isn't just Flatland repackaged (though of course its impossible not to reference) but actually explains things in a different way. I understand a little more now, and have a million more questions haha
It is, and there's another one credited to Ludwig Schlafli, depicting the regular polychora (pentatope, 16-cell, 24-cell, etc.) You can also check out Marc ten Bosch's work
This is some amazing quality content. I feel like I found a hidden gem. Keep making videos on cool geometry ideas and working with your awesome presentation style! You are definitely going places :D
After all these years, I still consider the tiger to be one of my finest ideas. Though it's just the name for the toratope, not for the product of two cylinders (Wendy Krieger called that "duocylinder margin"). The tiger was found through visualization of mid-cuts of various 4D toruses. (4D ball has to be counted among them -- it's sort of a "zero" element among 4D toruses.) Mid-cut of a 4D ball is a ball. Mid-cut of a spheritorus is either two offset balls or a torus. Mid-cut of a torisphere is either a torus or two concentric spheres. Mid-cut of a ditorus is always two toruses, but they can be concentric (same inner diameter, different outer diameters), co-circular (same outer diameter, different inner diameter, leading to one torus inside of the other), or offset. Tiger fills in the remaining "elementary" mid-cut: two toruses with vertical offset. In fact, all of its coordinate midcuts look like that. Later, I and few others developed the toratopic notation which allows to determine the mid-cuts easily and has some other fun applications. I'd suggest to try duocylinder where the radii of the two generating circles are not the same. The same for tiger. The symmetry is lower, but there would be visibly distinct mid-cuts.
This is B Y F A R the best video on 4d objects I've ever seen! You actually showed how more complex shapes change instead of just showing expanding spheres and cubes and stuff!
Hi, I also use blender for math animations. So far, I only used reflection groups to build intuition for higher dimensions. I learnt a lot from your slicing. Thx
Sweet stuff! You could also 'cheat' and make 4D models that are projections of 5D objects, the way people connect a cube inside a cube in 3D to model a tesseract. I did that with a 5D hypercube mesh and its dual for my 4D app. I haven't built that marvelous 120-cell or 600-cell, though. Lol
I love this. Seeing 4d shapes transform into 3d shapes then 3d to 2d. This is a very well done video visualizing shapes & very good animations & information.
Engineering classes taught me Solidworks, and thus making shapes from extrusions and rotations, but I've never heard of this relationship system even though it should be obvious intuitive if I had only thought about it. cool stuff.
So cool to learn about all these 4D shapes. I can't imaging them at all, but very interesting to know they exist. Very nice graphics and well explained.
The issue with this is that you use 2D images for the 2D cross sections. We live in 3D, but we can only 2D images that move through time. So a 2D being would see a 1D image, i.e. a line. Imagine the jump from 1 line to a painting, and how much more information you get from that additional dimension. We definitely perceive the 3rd dimension, but at any given moment what we see can be reproduced as a 2D image, or 2 rather (1 for each eye). It's like how u can perceive VR as real and 3D but it's a 2D screen.
Great work! Might want to change the thumbnail to "what do 4d objects in 3d space look like?". A camera in E4 would output multiple 3d views, and there is only 1 3d view projected onto 2d screen in this video. Also, check out the talk "Dual Quaternions Demystified" as a great introduction to Geometric Algebra, you'd definitely enjoy it.
Omg I just had a glance and ur videos look amazing from the topics in the titles and thumbnails. Exactly what I've wanted to srudy and talk about. Let's see what the content is like now lol
@@mihailmilev9909 It is Geometric Algebra. One thing to mention though is that it goes back to rotation around an axis rather than within a plane. This is actually a smart thing, since a plane doesn't actually give a unique rotation unless you only ever want to rotate around the origin. There are more 3D rotations than there are planes, but there are exactly as many rotations as there are axes. In 2D... since when is rotation in 2D around the "z-axis," and what even is this mystical "z-axis?" Again though, there is only one plane for rotations, but who says can't rotate around any given point in the plane? "Point"... axes in 2D aren't lines, they're _points!_ If we accept that axes of rotations are not lines, then 4D certainly has axes like any others; they just happen to be planes. In general, an axis of rotation in N dimensions is an N-2 dimensional manifold. (Yes, _manifold;_ not _subspace._ You can have axes of rotation that aren't subspaces.)
Wow such a good representation! The multiplication especially! I'm very interested in the promised next video. Have you read Greg Egans Diaspora? In there, (digital) people travel to a universe that is 5 dimensional, and the writer describes the planets and stars there to have 2 equators, because there are more degrees of freedom, but it is mindblowing to even thing about it (he also has excerpts and supplemetary math, animations and figures to better understand the concepts in the book - you can find them on his personal website for free;) )
Before you make your way to 5D and 6D it would be interesting to explore a bit more in this 4D realm. For example, what does the Hopf fibration of the the 3-sphere look like? A slightly harder question would be what does the tiling of the 3-sphere by dodecahedron look like. Furthermore, it is a theorem (not hard to show) that every unorientable surface can be embedded inside of 4D space. It would be nice to see how the real projective plane and the Klein bottle look from these descriptions! 😎
Hi Artem! This is such a great video! 🤩 I'm working on a documentary about 4D geometries, and I'd love to include a few seconds (with attribution, of course)!
Actually the tiger is just another type of ditorus. Since the torus isn't radially symmetric, depending on how you rotate it before revolving it with an offset you get different shapes, like the ditorus or the tiger. Thats why tiger cross sections look like two toruses on top of eachother, and ditorus cross sections look like 2 toruses side by side. The tiger can also be created in the way you described, though.
Смотрел видео про 4х-мерный гольф, указали на видео про визуализацию, вспомнил, что видел у Онигири. После этого смотрю, указан этот канал, имя то же, что у автора Онигири, захожу - и тут вы! Надеюсь, что мой комментарий на русском не испортит вам ничего, но было бы интересно посмотреть другие ваши видео на английском.
Hey, is a video about higher dimensions still on the table? I really liked this video, and I understand that these take a ton of work and life gets in the way, but if you're still working on it I'd love to see it
Молодец Артем, теперь представим землю как сфероЦилиндр. Раскручивая цилиндр получаем гравитацию на внутреннем слое. Если наш шарик является цилиндром это обьясняет частично гравитацию. Может мы видим землю круглой изза ограниченного глаза? Поэтому гравитация остается тайной. Вопрос, как вывернуть внутреннюю сторону цилиндра внаружу по типу Мобиус фигуры 😮 получится торус, наружняя поверхность которого есть внутренняя поверхность цилиндра, что в свою очередь есть сфера. Представь как магнитные поля работают на таких фигурах. Магнитное поле в форме торуса как мы его знаем, но с ним можно играть также как с фигурами в итоге загоним его в сферу и поля станут вывернутыми 😅 когда получишь премию не забудь упомянуть меня 😂 раскрась поверхности внутренние и наружние разным цветом. Магнитное поле в форме торуса, это истинная форма обьекта в 5 измерении, цилиндр 4 и шар что мы видим в 3м. Земля тоже в 5м будет торусом, магнитное поле это вещество не нашего измерения с которым мы как дети играем с древности
I noticed that the hypercube’s inner cube orbit looks a bit like an elliptical orbit with the points on the bottom square going slow when it’s in the bckground and then quickly when it’s in the foreground.
Very great video! You are wrong in saying that the duo cylinder and tiger can’t be constructed by revolution, they can. A duo cylinder is constructed by revolving a cylinder through the fourth dimension and a tiger can be constructed by revolving a torus in the fourth dimension at a different orientation than the ditorus.
If you stare at 1 blue dot in the visualization it goes in an ellipse orbit, kinda like you are a Black Hole but instead of going behind u it goes right infront of u.
What is the relationship between this set of 4D tori and the Clifford torus? You mentioned that the "tiger" is the Cartesian product of two circles, but so is the Clifford torus (with the specification that these circles must lie in two separate two-dimensional subspaces of R4). So where does the Clifford torus fit here? Great video, very explanatory and the shapes interest me.
To my understanding, the Clifford torus is a 2-torus (a 2D surface) and is topologically equivalent to the surface of a normal 3D donut. The difference is that the donut has some points with negative curvature in one part, and positive curvature in another, but the Clifford torus, which can only be embedded in 4D or higher, would be "flat" at all points, just like how the surface of a cylinder is "flat". The Clifford torus also appeared as a face on the duocylinder. By the way, it wasn't mentioned in the video, but there's one more torus/cylinder shape in 4D which is the torinder. It's the extrusion of the 3D donut into 4D, giving it some 4D "height". You can construct the ditorus if you take the torinder, bend it into a loop, and join the opposing donut cells.
A rotation is a translation in some space that has been restricted about a fixed reference. It has nothing at all to do with axis or planes unless we choose those things, specifically, as our fixed reference. There is no real reason at all to believe the universe is orthogonal.... It probably isn't and there are many different kinds of spaces to choose from. Orthogonal just happens to be the easiest for us to comprehend and relate to with our limited experience.
Mathematically 3d rotations can be described with 2 angles - phi, and theta. This is because all you get from a rotation is a direction - i.e. a unit vector pointing in the direction of a particular point of the 2d surface of a sphere. The one, three, and six used here seem to be a result of wanting to describe them with cartesian coordinates rather than their own units. The number needed should just be n-1 where n is the number of dimensions, since the angles describe a point on the n-1 dimensional surface of an n dimensional sphere.
An in 4D, the "axis" from the center to the surface isn't an axis at all, since that's a line segment, while 4D axes of rotation are 2D planes, so you'd need another point on the unit sphere just to describe which plane the rotation is in. Oh hey: (4-1) * 2 = 6.
Wait, then wouldn't 4D space begin to have something foreign to us "above" planes? A thing that would be described by 4 parameters. With pm meaning parameter, here's how rotational options each dimension should have. 4D has ways to rotate "around a surface" which in 3D just amounts to sitting still, like rotating "around the point" of a 1D line. (1pm = point, 2pm = plane, 3pm = surface, 4pm = ???) 1D - 1pm(1) [X] 2D - 1pm(2) [X,Y], 2pm(1) [XY] 3D - 1pm(3) [X,Y,Z], 2pm(3) [XY, XZ, YZ], 3pm(1) [XYZ] 4D - 1pm(4) [W,X,Y,Z], 2pm(6) [WX, WY, WZ, XY, XZ, YZ] 3pm(4) [WXY, WXZ, WYZ, XYZ] 4pm(1) [WXYZ]
Like... there are ways to have beings composed entirely of higher dimensions and still not sharing any with ours. 3 dimensional objects and realities composed entirely of dimensions we cannot interact with, but acting the same. Like paper on a floor vs a wall. Maybe different laws of physics there?
Nice! I also played with the same 4D models in my new game. Great to see a quality video about the less talked about ones like the duocylinder and tiger. Another interesting fact about the tiger is that it can be used to link a chain in 4D, which can't be done with a regular spheritorus like you might expect.
hey, I just want to let you know, I've been a huge fan of your work (like the fractal marble game) since BEFORE hyperbolica, and I literally checked Steam weekly for MONTHS and showed friends the devlogs since hey started to see if Hyperbolica released because I love (globally) non-Euclidean space and I thought your representation was so accessible that it even helped me explain it to my art-major girlfriend. If that game existed when I was 10 I would've been in heaven and learned so much. Thank you so much for all the work you've put in to interactive education!
Do you have a video about the chain? If so I'd love to see it.
@@lexinwonderland5741 I'm not such a dedicated fan but I'm with this guy (you who I'm responding to)
The ability to create shapes by multiplying other shapes really is a cool way to do things. I'm curious how you're handling the simulation and rendering. Can't wait to see the 5D version!
Is there another way to understand putting the duocylinder and tiger together? I personally found the cartesian product thing convoluted without visuals for 3D shapes, let alone 4D.
I also made an online constructor for 4D shapes: artemonigiri.github.io/4D-Shapes/
5:59 Should be “extrusion” instead of “revolution”
6:59?
Pin your comment, it's a great addition to that great video!
oh wow, awesome video! It's fascinating how many types of 4D cylinders and toruses there are.
CARY!
So good to see something that isn't just Flatland repackaged (though of course its impossible not to reference) but actually explains things in a different way. I understand a little more now, and have a million more questions haha
I think this might be the best video about 4D shapes that I've ever seen. Bravo! And thank you.
This was absolutely fantastic and answered questions I didn't even know I had -- I can't wait for the next one!!
This video finally made 4D click! For the first time i feel like i truly understand the concept. Thank you very much!
This video has more quality than some channels with over a million subscribers, you’ve definitely earned a sub from me
Last year, I was looking for a lot of videos on 4D shapes, and I couldn’t find any good ones. This is probably one of the best ones I’ve seen..
It is, and there's another one credited to Ludwig Schlafli, depicting the regular polychora (pentatope, 16-cell, 24-cell, etc.)
You can also check out Marc ten Bosch's work
@@coopergates9680 thank you!
The duocylinder can be made by spinning a cylinder in the plane containing its axis.
It works! Thank you!
higher dimensional gay barrels
Cool! We are waiting for the next video!
This is such a well made video/explanation/visualization, amazing job!
This is an incredible video! This is the most fascinating and helpful visualization of 4D geometry that I have ever seen, good job.
This is some amazing quality content. I feel like I found a hidden gem. Keep making videos on cool geometry ideas and working with your awesome presentation style! You are definitely going places :D
Incredible animation and excellent explanation..
The only problem with this video is that I got a little sad when I noticed there are no other videos on this channel.
Awesome video, there is so much energy coming from you via the commentary or the editing, I am very interested in seeing more from you
That was a spectacular 4D safari and the closest I've ever been to groking these exotic shapes. Thanks for the trip.
After all these years, I still consider the tiger to be one of my finest ideas. Though it's just the name for the toratope, not for the product of two cylinders (Wendy Krieger called that "duocylinder margin").
The tiger was found through visualization of mid-cuts of various 4D toruses. (4D ball has to be counted among them -- it's sort of a "zero" element among 4D toruses.) Mid-cut of a 4D ball is a ball. Mid-cut of a spheritorus is either two offset balls or a torus. Mid-cut of a torisphere is either a torus or two concentric spheres. Mid-cut of a ditorus is always two toruses, but they can be concentric (same inner diameter, different outer diameters), co-circular (same outer diameter, different inner diameter, leading to one torus inside of the other), or offset. Tiger fills in the remaining "elementary" mid-cut: two toruses with vertical offset. In fact, all of its coordinate midcuts look like that.
Later, I and few others developed the toratopic notation which allows to determine the mid-cuts easily and has some other fun applications.
I'd suggest to try duocylinder where the radii of the two generating circles are not the same. The same for tiger. The symmetry is lower, but there would be visibly distinct mid-cuts.
12:45 btw id love to watch that second part video with higher dimensional objects sir 🙏🏻
I want to live a day in 4D now
Hi
I believe you are competent at this topic.
Please do not be discouraged from visualizing 5D shapes
Er... it is a bit agonizing. For some shapes it is ok, but others...
This is B Y F A R the best video on 4d objects I've ever seen! You actually showed how more complex shapes change instead of just showing expanding spheres and cubes and stuff!
Hi, I also use blender for math animations. So far, I only used reflection groups to build intuition for higher dimensions. I learnt a lot from your slicing. Thx
Generate shadow projections too as well as the cross-section slices
Finnaly someone showed other rotating 4d shapes and not just a hyper cube
Great video! One of the best I've seen!
Fantastic video - clear explanation - beautiful animations - I absolutely love it! 😍
Great vid bro! Keep it up 💪🏼
Very cool, great professional video
this was amazing! I can't wait for the next video!
Thank you Artem, very cool!
Sweet stuff! You could also 'cheat' and make 4D models that are projections of 5D objects, the way people connect a cube inside a cube in 3D to model a tesseract. I did that with a 5D hypercube mesh and its dual for my 4D app. I haven't built that marvelous 120-cell or 600-cell, though. Lol
Amazing video! thank you!
This is amazing ! :)
I love this. Seeing 4d shapes transform into 3d shapes then 3d to 2d. This is a very well done video visualizing shapes & very good animations & information.
Engineering classes taught me Solidworks, and thus making shapes from extrusions and rotations, but I've never heard of this relationship system even though it should be obvious intuitive if I had only thought about it. cool stuff.
So cool to learn about all these 4D shapes. I can't imaging them at all, but very interesting to know they exist. Very nice graphics and well explained.
Absolute cinema
Amazing video !
The issue with this is that you use 2D images for the 2D cross sections. We live in 3D, but we can only 2D images that move through time. So a 2D being would see a 1D image, i.e. a line. Imagine the jump from 1 line to a painting, and how much more information you get from that additional dimension. We definitely perceive the 3rd dimension, but at any given moment what we see can be reproduced as a 2D image, or 2 rather (1 for each eye). It's like how u can perceive VR as real and 3D but it's a 2D screen.
Excellent!
Great work! Might want to change the thumbnail to "what do 4d objects in 3d space look like?". A camera in E4 would output multiple 3d views, and there is only 1 3d view projected onto 2d screen in this video.
Also, check out the talk "Dual Quaternions Demystified" as a great introduction to Geometric Algebra, you'd definitely enjoy it.
Wait, I've seen that video thumbnail or heard of it. It's about Geometric Algebra? I didn't know
Come to think of it I think I've seen ur channel before
Omg I just had a glance and ur videos look amazing from the topics in the titles and thumbnails. Exactly what I've wanted to srudy and talk about. Let's see what the content is like now lol
@@mihailmilev9909 It is Geometric Algebra. One thing to mention though is that it goes back to rotation around an axis rather than within a plane. This is actually a smart thing, since a plane doesn't actually give a unique rotation unless you only ever want to rotate around the origin. There are more 3D rotations than there are planes, but there are exactly as many rotations as there are axes. In 2D... since when is rotation in 2D around the "z-axis," and what even is this mystical "z-axis?" Again though, there is only one plane for rotations, but who says can't rotate around any given point in the plane? "Point"... axes in 2D aren't lines, they're _points!_ If we accept that axes of rotations are not lines, then 4D certainly has axes like any others; they just happen to be planes. In general, an axis of rotation in N dimensions is an N-2 dimensional manifold. (Yes, _manifold;_ not _subspace._ You can have axes of rotation that aren't subspaces.)
Actually, woah. Having multiple views of the same shape seems like such a great idea for visualisation
This is really cool 🎉
wow this is very nice, very nice animations
Amazingly systematic video
Finally, after playing 4d toys a while ago, I know what the tiger is
Nice job!
Super Cool stuff
Great video!!!! We want more videos talking about the 4d
Awesome!
So cool
Wow such a good representation! The multiplication especially! I'm very interested in the promised next video. Have you read Greg Egans Diaspora? In there, (digital) people travel to a universe that is 5 dimensional, and the writer describes the planets and stars there to have 2 equators, because there are more degrees of freedom, but it is mindblowing to even thing about it (he also has excerpts and supplemetary math, animations and figures to better understand the concepts in the book - you can find them on his personal website for free;) )
Before you make your way to 5D and 6D it would be interesting to explore a bit more in this 4D realm. For example, what does the Hopf fibration of the the 3-sphere look like? A slightly harder question would be what does the tiling of the 3-sphere by dodecahedron look like. Furthermore, it is a theorem (not hard to show) that every unorientable surface can be embedded inside of 4D space. It would be nice to see how the real projective plane and the Klein bottle look from these descriptions! 😎
Amazing vid!
Hi Artem! This is such a great video! 🤩
I'm working on a documentary about 4D geometries, and I'd love to include a few seconds (with attribution, of course)!
Great Describing Brother
Amazing
Actually the tiger is just another type of ditorus. Since the torus isn't radially symmetric, depending on how you rotate it before revolving it with an offset you get different shapes, like the ditorus or the tiger. Thats why tiger cross sections look like two toruses on top of eachother, and ditorus cross sections look like 2 toruses side by side. The tiger can also be created in the way you described, though.
Very interesting, thanks!
Very good video, good work, very informative, proud of you. (:
Смотрел видео про 4х-мерный гольф, указали на видео про визуализацию, вспомнил, что видел у Онигири. После этого смотрю, указан этот канал, имя то же, что у автора Онигири, захожу - и тут вы! Надеюсь, что мой комментарий на русском не испортит вам ничего, но было бы интересно посмотреть другие ваши видео на английском.
an absolute brilliant gem about multi-dimensional geometry.
Do you already have any plans for the announced n-D geometry-Video?
Oh, hi Onigiri!
I would love to see you visualize 3D and more of fractals. Some people did it but I would love your explanation on it better.
A tiger can be generated by revolving an offset torus. In fact, it is topologicaly homeomorphic to a ditorus
Hey, is a video about higher dimensions still on the table? I really liked this video, and I understand that these take a ton of work and life gets in the way, but if you're still working on it I'd love to see it
Молодец Артем, теперь представим землю как сфероЦилиндр. Раскручивая цилиндр получаем гравитацию на внутреннем слое. Если наш шарик является цилиндром это обьясняет частично гравитацию. Может мы видим землю круглой изза ограниченного глаза? Поэтому гравитация остается тайной. Вопрос, как вывернуть внутреннюю сторону цилиндра внаружу по типу Мобиус фигуры 😮 получится торус, наружняя поверхность которого есть внутренняя поверхность цилиндра, что в свою очередь есть сфера. Представь как магнитные поля работают на таких фигурах. Магнитное поле в форме торуса как мы его знаем, но с ним можно играть также как с фигурами в итоге загоним его в сферу и поля станут вывернутыми 😅 когда получишь премию не забудь упомянуть меня 😂 раскрась поверхности внутренние и наружние разным цветом. Магнитное поле в форме торуса, это истинная форма обьекта в 5 измерении, цилиндр 4 и шар что мы видим в 3м. Земля тоже в 5м будет торусом, магнитное поле это вещество не нашего измерения с которым мы как дети играем с древности
For exactly 2 seconds i could comprehend the 4th dimension.
I noticed that the hypercube’s inner cube orbit looks a bit like an elliptical orbit with the points on the bottom square going slow when it’s in the bckground and then quickly when it’s in the foreground.
Very great video! You are wrong in saying that the duo cylinder and tiger can’t be constructed by revolution, they can. A duo cylinder is constructed by revolving a cylinder through the fourth dimension and a tiger can be constructed by revolving a torus in the fourth dimension at a different orientation than the ditorus.
Could you try making the slices as 2--D slices in a 2-D array? I think this might be helpful for gaining intuition about 4-D rotations.
Lovely
Wait it's yours first video?!
If start so good then future even better. I subscribe!
Ну, не первое видео, у него на русском канале «Onigiri» 250000 подписчиков
No. I guess he created this channel just for SoME, cuz you need English language for SoME
New 4D Video plzz. We want it !
Great video thanks !. would it make any sense to consider 4 dimensional versions of conic sections (ie a 4D Hyperbola or revolution )
Yes! I should probably add it to the next video :D
So, with Cartesian product operation you only need a line and N-dimensional spheres for all N > 2, to get all basic figures of some kind?
Yes! And then other shapes can be constructed by merging these basic objects
And the line is just the 1-ball.
If you stare at 1 blue dot in the visualization it goes in an ellipse orbit, kinda like you are a Black Hole but instead of going behind u it goes right infront of u.
рад смотреть тебя и на английском ютубе. Странно почему твой видос не залетел(
Dude what's your Shadertoy?
What is the relationship between this set of 4D tori and the Clifford torus? You mentioned that the "tiger" is the Cartesian product of two circles, but so is the Clifford torus (with the specification that these circles must lie in two separate two-dimensional subspaces of R4). So where does the Clifford torus fit here? Great video, very explanatory and the shapes interest me.
To my understanding, the Clifford torus is a 2-torus (a 2D surface) and is topologically equivalent to the surface of a normal 3D donut. The difference is that the donut has some points with negative curvature in one part, and positive curvature in another, but the Clifford torus, which can only be embedded in 4D or higher, would be "flat" at all points, just like how the surface of a cylinder is "flat". The Clifford torus also appeared as a face on the duocylinder.
By the way, it wasn't mentioned in the video, but there's one more torus/cylinder shape in 4D which is the torinder. It's the extrusion of the 3D donut into 4D, giving it some 4D "height". You can construct the ditorus if you take the torinder, bend it into a loop, and join the opposing donut cells.
The Cartesian product of 2 circles is a Clifford Torus. You get the tiger by fattening it
I need more
Does anybody know another way to understand how to build up understanding the duocylinder and tiger? The Cartesian way's too confusing personally.
A rotation is a translation in some space that has been restricted about a fixed reference. It has nothing at all to do with axis or planes unless we choose those things, specifically, as our fixed reference. There is no real reason at all to believe the universe is orthogonal.... It probably isn't and there are many different kinds of spaces to choose from. Orthogonal just happens to be the easiest for us to comprehend and relate to with our limited experience.
Mathematically 3d rotations can be described with 2 angles - phi, and theta. This is because all you get from a rotation is a direction - i.e. a unit vector pointing in the direction of a particular point of the 2d surface of a sphere.
The one, three, and six used here seem to be a result of wanting to describe them with cartesian coordinates rather than their own units. The number needed should just be n-1 where n is the number of dimensions, since the angles describe a point on the n-1 dimensional surface of an n dimensional sphere.
A direction in 3d space can be described with 2 angles, but rotation requires one more angle to rotate an object around its own axis.
@@ArtemYashin Ah ok, yeah I see what I did wrong now. Thanks.
An in 4D, the "axis" from the center to the surface isn't an axis at all, since that's a line segment, while 4D axes of rotation are 2D planes, so you'd need another point on the unit sphere just to describe which plane the rotation is in. Oh hey: (4-1) * 2 = 6.
Isn't there an specific name for circles with thickness?
Annuli, singular annulus.
3:50 quaternions: am i a joke to you
Wait, then wouldn't 4D space begin to have something foreign to us "above" planes? A thing that would be described by 4 parameters.
With pm meaning parameter, here's how rotational options each dimension should have. 4D has ways to rotate "around a surface" which in 3D just amounts to sitting still, like rotating "around the point" of a 1D line.
(1pm = point, 2pm = plane, 3pm = surface, 4pm = ???)
1D - 1pm(1) [X]
2D - 1pm(2) [X,Y], 2pm(1) [XY]
3D - 1pm(3) [X,Y,Z], 2pm(3) [XY, XZ, YZ], 3pm(1) [XYZ]
4D - 1pm(4) [W,X,Y,Z], 2pm(6) [WX, WY, WZ, XY, XZ, YZ] 3pm(4) [WXY, WXZ, WYZ, XYZ] 4pm(1) [WXYZ]
Like... there are ways to have beings composed entirely of higher dimensions and still not sharing any with ours. 3 dimensional objects and realities composed entirely of dimensions we cannot interact with, but acting the same. Like paper on a floor vs a wall. Maybe different laws of physics there?
I found this using the #SoME2 tag
That's onigri's english channel?
very good English! :)
A spinning top has precession , what does precession lool like in 4D
очень качественно сделано, объяснено и показано.
My brain isn't braining
неожиданно
8:40
Cuboids.
Онигири! Не думал что увижу тебя тут.
Hello 🍙)
12:45 when's soon?
On @onigiriscience channel