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A teacher of mine told us, "Don't try to imagine 4D space. You can't. If you think you can imagine a 4D hypersphere, you're wrong. It's just not possible." And

A teacher of mine told us, "Don't try to imagine 4D space. You can't. If you think you can imagine a 4D hypersphere, you're wrong. It's just not possible." And everytime someone talks about how to imagine higher-dimensional space, I have to think back, and I'm convinced that my teacher was right. Our intuitive understanding of 2D and 3D just doesn't generalise to higher dimensions. A hypersphere is nothing like a sphere, just like a sphere is nothing like a circle, which is again nothing like a line segment. If you talk about multidimensional geometries, stick to the math, and don't try imagining it. Analogies to 2D or 3D objects (it's smooth, but like a spike) are pointless and don't lead to new insights. We really need to stick to precise, mathematical language if we want to work with higher dimensional geometry.

Retric 10y ago
I find some value from picturing 4D objects as 3D objects moving and changing size and or shape through space. In the ball example. At T:1/4 and T:3/4 I see the same 3D ball shape with different balls. At time 0, 0.5, and 1 they are gone. As such it's natural for a 4D where to fit in the middle as it can be centered at T:1/2 and you just need to find the intersection point in 4D space where the balls touch. Now, this does not work for rotations, but it's better than nothing.
jacobolus 10y ago
However there is a 4-dimensional sphere (3-sphere) that you can easily access: the possible rotations of any object in 3-dimensional space is topologically and metrically isomorphic to a (half of a) 3-sphere, in the same way as the rotations of an object in 2-dimensional space are isomorphic to a 1-sphere (circle). Think of an object’s current orientation as the “north pole” of our abstract 4-dimensional sphere. Rotating the object around any axis pushes you toward the equator, which you reach once you have rotated the object halfway around (the equator is made from all 180° turns, and is topologically a 2-sphere). Continuing to rotate the object takes you towards the “south pole”, which is where you get if you rotate your object 360°. To get back to the north pole, rotate by 360° again. Cf. https://en.wikipedia.org/wiki/Versor https://en.wikipedia.org/wiki/Versor https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati... and https://en.wikipedia.org/wiki/Rotation_group_SO(3) https://en.wikipedia.org/wiki/Rotation_group_SO(3) * * * One other way you can get at the 3-sphere is by taking a stereographic projection into all of 3-dimensional space (plus a point at infinity). If you have some 3-dimensional shapes drawn on the surface of your 3-sphere, these will get distorted under the stereographic projection, but locally angles will be preserved, just like a stereographic projection of the 2-sphere onto a piece of paper.
- JadeNB 10y ago
  I think that your terminology "3-sphere" (which is standard) as synonymous with "4-dimensional sphere" (which is not) may be confusing. Of course the 3-sphere is 3-dimensional, hence the name; it is just that it is canonically embedded in 4-dimensional Euclidean space (and not in 3-dimensional, flat Euclidean space).
  - jacobolus 10y ago
    Fair enough. Hopefully nobody was confused. I was trying to meet the grandparent poster halfway. His comment was “[...] Don't try to imagine 4D space. You can't. If you think you can imagine a 4D hypersphere [...]”
Eerie 10y ago
>A teacher of mine told us, "Don't try to imagine 4D space. You can't. If you think you can imagine a 4D hypersphere, you're wrong. It's just not possible." Your brain is a computer. If a computer can manipulate 4D hypersphere, so can you. You can't visualize it, but that's a different thing. You can't visualize a 3D sphere either, only 2D projection.
- rebuilder 10y ago
  I'm not sure I agree you can't visualize a 3d object. You can't draw one, I'll give you that, but we spend all our time in 3d space. Saying the brain's representation of that is 2D seems dubious.
  - Eerie 10y ago
    Not the brain, the eye's representation is 2D. The brain's can be 3D, despite the brain not receiving any 3D input, so I don't see why the brain can't represent 4D as well.
jobigoud 10y ago
I think he was wrong. It's pretty hard but not impossible. You don't visualize them directly as 3D objects though, you visualize 3D objects with metadata attached everywhere. While you navigate the space in your mind's eye you have an extra layer of information that flows with you. Consider Alicia Boole Stott. She created cardboard models of the various cross-sections of the 6 regular polytopes. Including the 24-cell which has no equivalent in 3D, the 120-cell and the 600-cell which are super complex figures. I think if you read her story you'll also be convinced that she indeed visualized the figures. [1] I spent quite a lot of time earlier this year on this visualization exercise on the 4 simplest regular polytopes while working on an toy application to manipulate 4D objects and tesseractic honeycomb in VR. You can have a good grasp of where every edge go and the angle between every faces. I could count the number of vertices and edges of a 16-cell or 24-cell just by navigating around the figure in my head. The tesseract and 16-cell are good starting points. 1: http://www.sciencedirect.com/science/article/pii/S0315086007000973 http://www.sciencedirect.com/science/article/pii/S0315086007...
rotten 10y ago
I've been thinking about the 4D visualization problem and I am becoming increasingly convinced we can develop some cool tools to help wrap our mind around it using VR goggles. There are two ways to draw a 3D object in 2D. One is to draw slices, or projections. The other is to use tricks of "perspective" (points of infinity and shading) It is very hard to visualize a 3D object simply from the slices model. When we try to draw 4D objects in 3D, we often simply construct models that are slices of the 4D objects and then try to extrapolate back in our minds what the 4D object "looks like". This is even more difficult to wrap our brains around. I postulate that we can develop similar tricks of perspective and shading that will help use see a 4D object if we draw on a 3D canvas. I don't think we can do so on a 2D canvas and make any sense of it. However, I'm hopeful that using VR goggles and drawing in 3D might be the trick. I've only really just started thinking about this idea. I have a ton of books on constructing perspective for 3D objects on 2D surfaces. Much has been written about it since the Renaissance. It is only in the last couple of years that we've had an effective and accessible 3D canvas to work with. (constructing hundreds of sculptures is not really practical) I don't think a lot of work has been done with VR goggles yet for 4D visualizations.

Retric 10y ago

I find some value from picturing 4D objects as 3D objects moving and changing size and or shape through space. In the ball example. At T:1/4 and T:3/4 I see the same 3D ball shape with different balls. At time 0, 0.5, and 1 they are gone. As such it's natural for a 4D where to fit in the middle as it can be centered at T:1/2 and you just need to find the intersection point in 4D space where the balls touch. Now, this does not work for rotations, but it's better than nothing.

jacobolus 10y ago

However there is a 4-dimensional sphere (3-sphere) that you can easily access: the possible rotations of any object in 3-dimensional space is topologically and metrically isomorphic to a (half of a) 3-sphere, in the same way as the rotations of an object in 2-dimensional space are isomorphic to a 1-sphere (circle). Think of an object’s current orientation as the “north pole” of our abstract 4-dimensional sphere. Rotating the object around any axis pushes you toward the equator, which you reach once you have rotated the object halfway around (the equator is made from all 180° turns, and is topologically a 2-sphere). Continuing to rotate the object takes you towards the “south pole”, which is where you get if you rotate your object 360°. To get back to the north pole, rotate by 360° again. Cf. https://en.wikipedia.org/wiki/Versor https://en.wikipedia.org/wiki/Versor https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati... and https://en.wikipedia.org/wiki/Rotation_group_SO(3) https://en.wikipedia.org/wiki/Rotation_group_SO(3) * * * One other way you can get at the 3-sphere is by taking a stereographic projection into all of 3-dimensional space (plus a point at infinity). If you have some 3-dimensional shapes drawn on the surface of your 3-sphere, these will get distorted under the stereographic projection, but locally angles will be preserved, just like a stereographic projection of the 2-sphere onto a piece of paper.

JadeNB 10y ago
I think that your terminology "3-sphere" (which is standard) as synonymous with "4-dimensional sphere" (which is not) may be confusing. Of course the 3-sphere is 3-dimensional, hence the name; it is just that it is canonically embedded in 4-dimensional Euclidean space (and not in 3-dimensional, flat Euclidean space).
- jacobolus 10y ago
  Fair enough. Hopefully nobody was confused. I was trying to meet the grandparent poster halfway. His comment was “[...] Don't try to imagine 4D space. You can't. If you think you can imagine a 4D hypersphere [...]”

Eerie 10y ago

>A teacher of mine told us, "Don't try to imagine 4D space. You can't. If you think you can imagine a 4D hypersphere, you're wrong. It's just not possible." Your brain is a computer. If a computer can manipulate 4D hypersphere, so can you. You can't visualize it, but that's a different thing. You can't visualize a 3D sphere either, only 2D projection.

rebuilder 10y ago
I'm not sure I agree you can't visualize a 3d object. You can't draw one, I'll give you that, but we spend all our time in 3d space. Saying the brain's representation of that is 2D seems dubious.
- Eerie 10y ago
  Not the brain, the eye's representation is 2D. The brain's can be 3D, despite the brain not receiving any 3D input, so I don't see why the brain can't represent 4D as well.

jobigoud 10y ago

I think he was wrong. It's pretty hard but not impossible. You don't visualize them directly as 3D objects though, you visualize 3D objects with metadata attached everywhere. While you navigate the space in your mind's eye you have an extra layer of information that flows with you. Consider Alicia Boole Stott. She created cardboard models of the various cross-sections of the 6 regular polytopes. Including the 24-cell which has no equivalent in 3D, the 120-cell and the 600-cell which are super complex figures. I think if you read her story you'll also be convinced that she indeed visualized the figures. [1] I spent quite a lot of time earlier this year on this visualization exercise on the 4 simplest regular polytopes while working on an toy application to manipulate 4D objects and tesseractic honeycomb in VR. You can have a good grasp of where every edge go and the angle between every faces. I could count the number of vertices and edges of a 16-cell or 24-cell just by navigating around the figure in my head. The tesseract and 16-cell are good starting points. 1: http://www.sciencedirect.com/science/article/pii/S0315086007000973 http://www.sciencedirect.com/science/article/pii/S0315086007...

rotten 10y ago

I've been thinking about the 4D visualization problem and I am becoming increasingly convinced we can develop some cool tools to help wrap our mind around it using VR goggles. There are two ways to draw a 3D object in 2D. One is to draw slices, or projections. The other is to use tricks of "perspective" (points of infinity and shading) It is very hard to visualize a 3D object simply from the slices model. When we try to draw 4D objects in 3D, we often simply construct models that are slices of the 4D objects and then try to extrapolate back in our minds what the 4D object "looks like". This is even more difficult to wrap our brains around. I postulate that we can develop similar tricks of perspective and shading that will help use see a 4D object if we draw on a 3D canvas. I don't think we can do so on a 2D canvas and make any sense of it. However, I'm hopeful that using VR goggles and drawing in 3D might be the trick. I've only really just started thinking about this idea. I have a ton of books on constructing perspective for 3D objects on 2D surfaces. Much has been written about it since the Renaissance. It is only in the last couple of years that we've had an effective and accessible 3D canvas to work with. (constructing hundreds of sculptures is not really practical) I don't think a lot of work has been done with VR goggles yet for 4D visualizations.