Not at all, this is not a perspective or orthogonal projection of a tesseract. It's a self-wrapping 3D map of the surface of a tesseract. As the title of the video says too, you are exploring just the surface of the tesseract. The POV is in the surface. As if, by analogy with exploring the surface of a cube, the POV was in the faces, looking straight at the edges with a 1D display.So, this gamer video, http://www.youtube.com/embed/lLg3oLbwRsQ. You can explore the tesseract from inside.
It's so fascinating because you take a wrong assumption for reality. Though reality isn't much less fascinating. The eight cubes face to face constitute a surface with zero 'hyperdepth' enclosing a 4D space. You can't picture it? Well this is 4D! We can't picture it because it's unknown to us. The same way a person who sees only in shades of red and blue can understand the notion of other colors but can't get a faithful mental image of a colorful scene.And so, the other very interesting thing to notice about the tesseract is that this 4d object apparently consists of 8 3d cubes. I.e. the cubes completely fill its 4d space. But they are 3d themselves. I find this fact fascinating.
A cube is flat in 4D, it has zero length on the W axis. Stacking cubes doesn't fill up a 4D space. It's the same as stacking squares in 3D.This is entirely different from planes in 3D. No matter how many planes we stack on top of each other in 3D, they still will amount to 0 thickness. But stacking 3d things in 4D, apparently, can make up a 4d object?
x(3-volume unit) = y(4-volume unit) is a false equation. The unit for 3-volumes and 4-volumes is different.For example. If we cheat a bit and take 8 equal cubes with edge=8 units, then 8 to the 4th = 4096 and this is the 4-volume of a tesseract with edge = 8units. And 8 cubed = 512 == the 3-volume of one of the cubes. Since we have 8 of them, we multiply 512 by 8 and get 4096 as well. So, in this particular (and unique, lol) case where the edge = 8 units, the 8 3-volumes == 1 4-volume.
The area is the 2D analog of the length of a 1D object and of the volume of a 3D object. They are the same thing in a different n-dimensional space, they have analog properties.Area and volume are such different things... they are not compatible.
Really, get this idea out of your head. Don't think that you are somehow seeing in 4D when you look at a projection of a 4D object. There's no way to directly see the 4-volume when looking at a projection of a 4D object, since your POV is in the 3-plane of the projection. It's the same as looking edge-on at the drawing of a cube.The 8 cubes that bound it on 8 sides have their edges aligned in 4D. There is no space in between them.
Ovo wrote:[...] I believe you have been partly misled by the commonly made mistake/missing step in the 3D->2D analogy that I have pointed out here.
[...] Don't think that you are somehow seeing in 4D when you look at a projection of a 4D object. There's no way to directly see the 4-volume when looking at a projection of a 4D object, since your POV is in the 3-plane of the projection. It's the same as looking edge-on at the drawing of a cube.
Ovo wrote:Hey it's me again.Not at all, this is not a perspective or orthogonal projection of a tesseract. It's a self-wrapping 3D map of the surface of a tesseract. As the title of the video says too, you are exploring just the surface of the tesseract. The POV is in the surface. As if, by analogy with exploring the surface of a cube, the POV was in the faces, looking straight at the edges with a 1D display.So, this gamer video, http://www.youtube.com/embed/lLg3oLbwRsQ. You can explore the tesseract from inside.
Ovo wrote:The eight cubes face to face constitute a surface with zero 'hyperdepth' enclosing a 4D space.
Ovo wrote:A cube is flat in 4D, it has zero length on the W axis. Stacking cubes doesn't fill up a 4D space. It's the same as stacking squares in 3D.This is entirely different from planes in 3D. No matter how many planes we stack on top of each other in 3D, they still will amount to 0 thickness. But stacking 3d things in 4D, apparently, can make up a 4d object?
Ovo wrote:The area is the 2D analog of the length of a 1D object and of the volume of a 3D object. They are the same thing in a different n-dimensional space, they have analog properties.Area and volume are such different things... they are not compatible.
Ovo wrote:Really, get this idea out of your head.The 8 cubes that bound it on 8 sides have their edges aligned in 4D. There is no space in between them.
p 59
In a simplex each tetrahedron is adjacent to each of the other four.
It may not be very difficult to think of two adjacent tetrahedrons, even though they lie in different hyperplanes...
p 66
In a simplex, for example, there are 5 tetrahedra whose 20 faces fit together in pairs, each tetrahedron having a face in common with each of the other 4. We can take any one of these tetrahedra and place the other 4 upon it, all in one hyperplane, and then we can turn the 4 outside tertrahedra away from this hyperplane without separating them from the 5th or distorting them in any way, until we have brought together every pair of corresponding faces. The 5 tetrahedra together with their interiors now enclose a portion of hyperspace.
p 14
There are also many properties in which space of an even number of dimensions differ from spaces of an odd number of dimensions, and these differences would hardly be recognized if we had only the ordinary geometries. Thus in space of an even number of dimensions roatation takes place around a point, a plane, or some other axis-space of an even number of dimensions, while in space of an odd number of dimensions the axis of a rotation is always of an odd number of dimensions (chap.IV)
Ovo wrote:Hey it's me again.It's so fascinating because you take a wrong assumption for reality. Though reality isn't much less fascinating. The eight cubes face to face constitute a surface with zero 'hyperdepth' enclosing a 4D space. You can't picture it? Well this is 4D! We can't picture it because it's unknown to us. The same way a person who sees only in shades of red and blue can understand the notion of other colors but can't get a faithful mental image of a colorful scene.And so, the other very interesting thing to notice about the tesseract is that this 4d object apparently consists of 8 3d cubes. I.e. the cubes completely fill its 4d space. But they are 3d themselves. I find this fact fascinating.A cube is flat in 4D, it has zero length on the W axis. Stacking cubes doesn't fill up a 4D space. It's the same as stacking squares in 3D.This is entirely different from planes in 3D. No matter how many planes we stack on top of each other in 3D, they still will amount to 0 thickness. But stacking 3d things in 4D, apparently, can make up a 4d object?The area is the 2D analog of the length of a 1D object and of the volume of a 3D object. They are the same thing in a different n-dimensional space, they have analog properties.Area and volume are such different things... they are not compatible.Really, get this idea out of your head. Don't think that you are somehow seeing in 4D when you look at a projection of a 4D object. There's no way to directly see the 4-volume when looking at a projection of a 4D object, since your POV is in the 3-plane of the projection. It's the same as looking edge-on at the drawing of a cube.The 8 cubes that bound it on 8 sides have their edges aligned in 4D. There is no space in between them.
http://books.google.com/books?id=hZULAA ... &q&f=false
p 210 - 211
A spherical tetrahedron has 6 edges, each lying in the edge of a spherical dihedral angle whose interior contains the interior of the tetrahedton. The interior of one of these spherical dihedral angles contains also the interiors of 3 of the 15 tetrahedrons associated with the given tetrahedron as explained above, and its volume is = to the sum of the volumes of the 4 tetrahedrons whose interors are within it.
Writing A' for the opposite point to A, the other extremity of the diameter to A, and so for other points, we let T denote the volume of the tetrahedron ABCD, T1 the volume of A'BC,T12 the volume of A'B'CD, and so on. ABC'D' is congruent to A'B'CD, and we have T34 = T12 etc.
The interior of the dihedral angle C-AB-D contains the interiours of the 4 tetrahedrons whose volumes are T, T1, T2 and T12. If Ф12 is the measure of the dihedral angle AB in terms of a right dihedral angle, and if we take fo unit of volume 1/16 of the volume of the hypersphere, we shall have the relation
T + T1 + T2 + T12 = 4 Ф12
There are 6 if these equations, and in addition one equation expressing the fact that the sum of the 8 different volumes is equal to the volume of a half-hypersphere, namely,
T + Sum(T1) + Sum(T12) = 8
4Dspace wrote:For those who missed it, it appears, one can stack 3d objects in 4D in such a way (stacking them into the W direction) so that in the end they entirely fill the 4D space.
4Dspace wrote: And why is it not obvious that 3D is a subspace of 4D, i.e. it's a part of it, and simply adding 3d cubes or 3d trapezoids in all 4 directions will fill the 4D volume.
4Dspace wrote: .... no gaps whatsoever between 8 cubes that form a tesseract. NO GAPS: they align vertex to vertex, edge to edge and face to face. This can mean only one thing: that 3d volumes can be tiled in such a way that they fill completely a 4D space. And it does not take an infinity of them. 8 will do.
4Dspace wrote:Planes don't fill space.
ICN5D wrote:4Dspace wrote:Planes don't fill space.
Likewise, a 3D volume is not a 4D volume. A 3-volume cannot fill in a 4-volume in the same way as the 2d square --> cube analogy works. You can arrange 8 of these 3-volumes in 4D, to build up the surface of a hollow tesseract. But it will still be hollow, with no filled-in 4D stuff to speak of. A hollow tesseract is just a bunch of 3D volumes arranged in 4-space like a box. They enclose the 4D volume, but do not fill it in, like water.
There is a difference here. Surfaces are not interiors.
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