4 dimensional objects

[MacGyver1968]

My best friend asked me to post this here. I have, on many occasion, told him about this board, and it's numerous experts...just hoping to get a little feedback.

My friend has started dating a real cool, very talented artist. She's a world class flutist, plays with the "Polyphonic Spree", has her own "chamber music" group that plays classical music while wearing powder wigs, created her own electronic wind instrument...she's a really cool chick...here's a vid of her performing.
http://www.youtube.com/watch?v=gjbWvFzKEEc

Anyway...she is putting on a half art exhibit/ half party soon in a 20,000 sq ft. warehouse. The theme is "4D"...she basically wants to visually represent 4d objects in a 3d world. One example...one of her computer artist friend is going to take this animation of a tesseract and add video clips to the squares:

and project it on a screen. 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]

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.

 

[Pete]

I love that tesseract animation. Lovely.
Maybe other regular polychora? The 600-cell with an image on each of the 1200 faces would be phenomenal!
Perhaps a klein bottle (kind of like a 4D mobius strip)?
What's the 4D equivalent of a torus? (edit - Google knows all. Some great images and videos.)

 

[James R]

The animation there is actually a 3 dimensional projection of a 4-dimensional hypercube (tesseract) into 3 dimensional space (and then, obviously, further projected onto a 2-d computer screen, but we're familiar with interpreting 2-d projections/pictures of 3-d objects).

The tesseract itself has 8 "faces", each of which is a 3-d cube. In the animation, some of those cubical faces appear distorted; this is due to the projection into 3-d. To compare, it's like looking at the shadow of a 3-d wire cube on a sheet of flat paper. The shadowed edges of the cube don't all seem to join up at 90 degrees on the paper, even though they all do in the "real" 3-d cube. Similarly, for the tesseract, all the edges of the "real" 4-d cube meet at right angles.

The other bizarre thing to get your head around with the tesseract is this: opposite squares on the outside of the tesseract are actually the same square in 4-d space. In other words, the tesseract is "folded up" in 4-d space.

 

[Pete]

The other bizarre thing to get your head around with the tesseract is this: opposite squares on the outside of the tesseract are actually the same square in 4-d space. In other words, the tesseract is "folded up" in 4-d space.

I don't think that's right... I think that opposite squares on the outside of the tesseract are opposite squares of one cubical cell.

 

[GeoffP]

Wait - how is this four dimensional? It's a cyclical 3D object. It's moving, and the edges aren't static, but so what?

 

[AlexG]

Read the Robert A. Heinlein story, ""—And He Built a Crooked House—".

 

[Pete]

Wait - how is this four dimensional? It's a cyclical 3D object. It's moving, and the edges aren't static, but so what?

This is a cyclical 2D object:

It's the projection of a 3D object onto a 2D plane. The 3D object is a cube/hexahedron, with 6 square faces meeting each other at right angles.
Notice that although each face of the 3D cube is square, and each angle is 90 degrees, that in the 2D representation some faces are not square, and some angles are not 90 degrees.

This is a cyclical 2D object:

It is the projection of a 4D object onto a 2D plane. The 4D object is a hypercube/tesseract/octachora/8-cell, with 24 square faces meeting each other at right angles.
Notice that although each face of the hypercube is square, and each angle is 90 degrees, that in the 2D representation some faces are not square, and some angles are not 90 degrees.

Read the Robert A. Heinlein story, ""—And He Built a Crooked House—".

One of my favourites!

 

[GeoffP]

I must be missing something here: aren't the faces of the hypercube changing as it cycles? I'm seeing this as a 3-dimensional object that's "stretching". is it that the square that the internal transition is "passing through" that's always square? Because that always looks square. Is it 4D because it's moving: the "hyper" in hypercube? Are the 24 square faces meeting each other at right angles at some point in its cycling? Is that the 'hyper' as well? That I can see.

Sorry to be a pain; I just want to understand this. 8 cells - check. Confused on some of the rest of it.

 

[James R]

I don't think that's right... I think that opposite squares on the outside of the tesseract are opposite squares of one cubical cell.

Hmm.... yes, I think you're right.

In the projection, you can see all 8 cubes of the hypercube (with some distorted).

I was thinking of the hypercube "unfolded" into a "flat" 3-d structure, like when you unfold a 3-d cube into a 2-d "mat". (There's a technical word for this which escapes me for the moment.)

Wait - how is this four dimensional? It's a cyclical 3D object. It's moving, and the edges aren't static, but so what?

Read post #4 and Pete's post.

 

[James R]

I must be missing something here: aren't the faces of the hypercube changing as it cycles?

No. What you're seeing is the projection of a rotating hypercube. The hypercube's 3-d cubic "faces" don't change shape, but the hypercube's "shadow" in the 3-d space has 3-d cubes that appear to change shape as the hypercube rotates in 4-d space.

Compare Pete's one-dimension-down analogue, where you see the projection of a 3-d cube onto a 2-d screen. All the faces of a 3-d cube are squares that do not change shape as the 3-d cube is rotated. But when you look at the 2-d projection of the 3-d cube on the screen, you see the cube's faces not as squares but as parallelograms that appear to change shape.

I'm seeing this as a 3-dimensional object that's "stretching". is it that the square that the internal transition is "passing through" that's always square?

Yes. And so are the shapes connecting the "inner" cube to the "outer" one.

Are the 24 square faces meeting each other at right angles at some point in its cycling?

In the 4-d space all 24 square faces always meet at right angles.

 

[Pete]

The tesseract projection is a bit more complicated because it is in perspective, which is why the "inner" cube is smaller than the "outer" cube.

The 3D analog is like looking hoizontally at a cube rotating horizontally, so that the back face looks smaller than the front face, and appears to pass inside the front face as the cube rotates.

 

[arfa brane]

...opposite squares on the outside of the tesseract are opposite squares of one cubical cell.

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"?

 

[Emil]

I am convinced that the human mind can not understand the four-dimensional space.
How many squares are in a cube? An infinite number of squares.
So how many cubes are in a hypercube? An infinite number of cubes.
What is the dimension which allows the existence of an infinite number of cubes and still be an defined object?
Or,
Two two-dimensional objects can "see" only the perimeter each other.
We see (from the third dimension )not only the perimeter we see and the object surface.We see "inside" the object.
So what is the dimension that allows to see "inside" us?
Weird...

 

[GeoffP]

No. What you're seeing is the projection of a rotating hypercube. The hypercube's 3-d cubic "faces" don't change shape, but the hypercube's "shadow" in the 3-d space has 3-d cubes that appear to change shape as the hypercube rotates in 4-d space.

Compare Pete's one-dimension-down analogue, where you see the projection of a 3-d cube onto a 2-d screen. All the faces of a 3-d cube are squares that do not change shape as the 3-d cube is rotated. But when you look at the 2-d projection of the 3-d cube on the screen, you see the cube's faces not as squares but as parallelograms that appear to change shape.



Yes. And so are the shapes connecting the "inner" cube to the "outer" one.



In the 4-d space all 24 square faces always meet at right angles.

Okaaay, I think I'm beginning to see how this works. Interesting. To really get it I assume Id' have to know the math a little better. I'll check out the wiki page. Although I do have questions about that as well.

 

[James R]

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".

I'm not so sure.

I think you can see all 8 cells. There is one in the centre, six around it and one on the outside.

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"?

Yeah, I think that's right. The 8th, "outside" cell in the projection is "inside-out" in the projection. Or something like that...

 

[GeoffP]

Ahhhhh....

 

[GeoffP]

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

Sorry: non-physicist, non-real-mathematician.

 

[arfa brane]

The geometric series goes: polygon, polyhedron, polychoron, ...

You don't get a polygon with less than 3 sides, you don't get a polyhedron with less than 4 faces, or a polychoron with less than 5 cells.

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?

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)

I also think it's probably safe to say, given the comments made about the tesseract, that it's bloody hard to think in more than 3 dimensions.

 

[James R]

How many squares are in a cube? An infinite number of squares.

No. Just six.

So how many cubes are in a hypercube? An infinite number of cubes.

No. Just eight.

What is the dimension which allows the existence of an infinite number of cubes and still be an defined object?

Well, in terms of regular polytopes, I think that in all dimension higher than 4 there are only 3 regular polytopes. There are hypercube analogues in all higher dimensions, though. So, if you want an infinite number of cubes in a regular polytope you need to go to an infinite-dimensional space.

Two two-dimensional objects can "see" only the perimeter each other.
We see (from the third dimension )not only the perimeter we see and the object surface.We see "inside" the object.
So what is the dimension that allows to see "inside" us?

A 4th-dimensional being can see the inside and outside of you at the same time. In fact, such a being would be able to remove an organ from your body without breaking the skin, just by pulling it out in a 4th-dimensional direction. It would be like you reaching into a square to pull up a dot in the centre, without having to remove the dot through one of the walls of the square.