4 dimensional objects

[James R]

OK, so wait, then the 4D is the 'transient state'. The shape 'lives' in the 4D by moving through it.

I'm not quite sure what you're saying here. There are two ways you can think about the animation above. One way is that you have a fixed light source in the 4-d space and the 4-d hypercube is rotating. You're looking at a 3-d "screen" onto which the shadow of the rotating hypercube is projected.

The other way you can think of it is that you have a stationary 4-d cube that isn't moving at all. Then you walk "around" it in the 4-d space so that you can see it from different sides.


---

If it's true that a 3-cube is the 'face' of a 4-cube, is a 4-cube the face of a 5-cube, etc?

Yes.

Is it generally true that a n-cube is the face of a (n+1)-cube, and if so, how large can n be?

Yes. And n can be as large as you like.

 

[GeoffP]

Is it generally true that a n-cube is the face of a (n+1)-cube, and if so, how large can n be?
(I've been wondering about that question for awhile)

Presumably. What would a 5D construct look like, connection-wise?

Edit: http://www.youtube.com/watch?v=lEiNsrtuGcc

 

[GeoffP]

Oookay: so this is really a flat space - a shadow as you say. I was interpreting it as a 3D object that wasn't rotating, but rather 'squeezing' itself through the centre.

 

[arfa brane]

Be careful about rotations in a 2d projection. The 3d cube is rotating around a one-dimensional axis;

the 4d cube is rotating around a two-dimensional "axis".

Yes, it's bloody hard to think in more than three of those dimension whatsits.

 

[Emil]

James R,
you do not understand me,the cube has six sides.
Hypercube has cubes as sides.(I do not know how many.)
A point has no dmensiune.
An infinity number of points form a line and has one dimension.
An infinite number of lines form an surface and has two dimensions.
An infinite number of areas form a space and has three dimensions.
An infinite number of space form a hyperspace and has four dimension.
Similarly for a segment, square, cube, hypercube.

 

[Emil]

Yes, hypercube has eight sides each a cube.

 

[James R]

Oookay: so this is really a flat space - a shadow as you say. I was interpreting it as a 3D object that wasn't rotating, but rather 'squeezing' itself through the centre.

That's a correct interpretation of what you see. But you're seeing the 3-d shadow of the full 4-d object. As Pete said, the small cube that you see passing though the larger cube appears smaller because it is "further away" in some 4-d sense. The analogous thing is if you look at a 3-d cube the square face furthest away from you looks smaller than the face closest to you because it is futher away in the 3rd dimension. In fact, all squares on a 3-d cube are the same size, and all 3-d cubical faces of a tesseract are the same size.

James R,
you do not understand me,the cube has six sides.
Hypercube has cubes as sides.(I do not know how many.)

I already told you: eight.

A point has no dmensiune.
An infinity number of points form a line and has one dimension.
An infinite number of lines form an surface and has two dimensions.
An infinite number of areas form a space and has three dimensions.
An infinite number of space form a hyperspace and has four dimension.

Yes.

Similarly for a segment, square, cube, hypercube.

I think you'll agree that a square has four sides, and not an infinite number of sides. Right? And a cube has six faces, not an infinite number. So, a tesseract has 8 "faces".

 

[Emil]

So we can conclude that the hypercube is a four dimensional object that has eight sides,each cube,and which is formed from an infinite number of cubes?

 

[John99]

That figure is also known as a polychoron. Also goes by the name of regular 4-polytope; it's one of 6 convex types in four dimensions. You can consider the tesseract or hypercube as a 4d cube with cubic 'faces' or chora.

What your friend is considering is called a tesselation.

Its still in three dimensions though.

http://en.wikipedia.org/wiki/Fourth_dimension

There really is no fourth dimension in terms of visualization. Three covers everything...or we can just call it a sphere.

http://en.wikipedia.org/wiki/Sphere

 

[arfa brane]

There really is no fourth dimension in terms of visualization. Three covers everything...aka a sphere.

I'm not sure I understand. A sphere isn't a regular polytope. What exactly does "visualization" mean, and how is a 4-polytope "still in three dimensions"?

What about a n-polytope?

 

[John99]

Visualization is what we see. includes DOF. three dimension on three axis including\encompassing radiuses.

 

[John99]

I say DOF because what we see in the image in op is a representation of artificial DOF aka deliberate DOF. suffice it to say merely an optical illusion to represent actual DOF as seen through the human eye. If you can tell us what the 4th dimension is then by all means do so.

I searched for some more of an explanation and came up with some pretty good links. Merely what i see, instantaneously, of exactly what is transpiring in the image in post #1: Is an extrapolation of the perception of distance and forming, albeit erroneously, a relationship to time but in reality all it is is the object moving within the confines of the three dimensional world in which we inhabit. I am assuming fourht dimension is time, for some reason, which i dont agree with, lest someone can articulate why it is. IF it is time you just need to explain why. To associate time with distance is it not still residing in three dimensions. Of course i think it is\does.

http://www.sciencedirect.com/scienc...6000610f91e469209bc782fed35ef8d2&searchtype=a

http://www.nvnews.net/articles/3dimagery/humaneye1.shtml

 

[James R]

So we can conclude that the hypercube is a four dimensional object that has eight sides,each cube,and which is formed from an infinite number of cubes?

No. A hypercube is formed from 8 cubes.

Visualization is what we see. includes DOF. three dimension on three axis including\encompassing radiuses.

What's DOF?

I say DOF because what we see in the image in op is a representation of artificial DOF aka deliberate DOF. suffice it to say merely an optical illusion to represent actual DOF as seen through the human eye.

:crazy: No need to read further.

 

[Pete]

Opposite squares on the outside (external faces) are from opposite cells. Each cell is a hexahedron = a cube. A closer look shows there are actually 7 cells in the diagram--one at the center and 6 around it--the 8th is "in the 4th dimension". Or if you like, you can say the 6 external faces "are" the 8th cell, because the tesseract "lives" in the 4th dimension; perhaps that's what James meant with "opposite squares"?

External squares are definitely faces of one cell.

It's somewhat like this perspective view of a cube:

Note that the cube has six faces. The external edges are the edges of one face.

Note also that just as every edge on a cube is shared by two faces, every face on a tesseract is shared by two cells.

 

[John99]

what's dof?
.

depth of field.

 

[GeoffP]

Be careful about rotations in a 2d projection. The 3d cube is rotating around a one-dimensional axis;

OK, then this is what I thought it was doing.

That's a correct interpretation of what you see. But you're seeing the 3-d shadow of the full 4-d object. As Pete said, the small cube that you see passing though the larger cube appears smaller because it is "further away" in some 4-d sense. The analogous thing is if you look at a 3-d cube the square face furthest away from you looks smaller than the face closest to you because it is futher away in the 3rd dimension. In fact, all squares on a 3-d cube are the same size, and all 3-d cubical faces of a tesseract are the same size.

Ahhhhh. I begin to see.

 

[Captain Kremmen]

Not sure about the four dimensions, but is is somehow very satisfying to watch.
Are there any more you can show?

 

[arfa brane]

About those animations: the first one is the transformation in two dimensions of a 3d object, the second is transforming in the same two dimensions of projective space.

The cube only has one dimension left to transform "in", rotations are more interesting than just scaling so that's what the animation does.

But the hypercube has two left, and so on.

 

[MacGyver1968]

She's looking for more ideas. I was just hoping some of you "math folk" might direct me to other 4d objects, or other ways of expressing 4D in an artistic way.

Thanks in advance.

 

[arfa brane]

One example...one of her computer artist friend is going to take this animation of a tesseract and add video clips to the squares:

Ok, I said this (adding video to the squares) was an example of a tesselation. If you start in 3d instead, you can use a bit of graph theory to see that each face of a cube is the area between adjacent edges of a trivalent graph, and if you disconnect each face so it becomes an 'edge-disjoint' figure, you get a different kind of graph.

Start with a stereographic projection of 5 of the 6 faces of a cube, as posted by Pete:

.
If you make each face edge-disjoint you get five 2d quadrilaterals projected on to two dimensions, so the 6th face, which is nominally the outer square when the faces are connected, is no longer 'seen'. This is what happens when you make each of the faces into frames of a movie--you can only see 5 at a time.

Likewise when you reduce the faces of a hypercube into eight disjoint 3d volumes, each a rhomboid or a cube, you can only arrange 7 at a time in three dimensions. The edges of a hypercube are the faces of each of these volumes or cells.

So now instead of considering frames of a movie pasted to each 'square', consider each square face as a single pixel of a larger image which is 4-dimensional, and there you go. Slice up each hypersurface (edges of a hypercube are 2-dimensional, remember) into smaller 'hypersquares' or each 'hyperface' into 'hypercubelets' as pixels. I can't say that this will somehow overcome the stubbornly persistent tendency for humans to think a 2d image is a projection of a measly 3--what can I say, that's biology.