Is a Tesseract the only model we have for 4D?

[JamieLT]

I posted this in yahoo questions, but perhaps this is a better place for it.

The Tesseract assumes that in order to add another Euclidean dimension another orthogonal plane is added into the mix, right? That is the pattern when going from 1D to 2D to 3D, but it seems like something slightly different is going on from 0d to 1D. From a point to a line - a radial line is orthogonal to a circle, but 0D is not a really a circle, it's a point, so from 0D to 1D, it doesn't really seem like adding on an orthogonal plane (what is orthogonal to a point? everything? nothing?) instead, it makes sense to me that a line is created by an infinite number of points. Following that pattern -
0D-1D: line is an infinite number of points
1D-2D: plane is an infinite number of lines
2D-3D: 3D is an infinite number of planes
3D-4D: 4D is an infinite number of 3D spaces? like a bunch of cubes stacked on top of one another or something?

Has anyone else submitted anything other than a Tesseract to represent 4D?
Also, how would you describe going from 0D to 1D?
Thanks!

(sorry if this is the wrong place to post this, I thought I saw another higher dims thread over here, so hopefully this is ok!)

 

[phinds]

It has always been my impression that a tesseract is not intended as any kind of generic representation of 4D but rather just an attempt to show what the 3D "shadow" of a 4D "cube" would be.

 

[JamieLT]

Thanks, yea I've watched through vids like this:
http://www.dimensions-math.org/Dim_reg_E.htm
that try to teach you how to see higher dims from their distorted shadows.

The problem with shadows is they are distorted... but cross sections are not. (In 2D you can walk around an undistorted cross section of a 3D object) Rather than trying to model a distorted shadow, I'd rather see a bunch of cross sections. Has anyone made a bunch of consecutive undistorted 3D cross-sections of a 4D object? Would that just consist of a small square, then a larger square, then a larger square... or small sphere, then larger sphere etc. etc.?

 

[micromass]

Thanks, yea I've watched through vids like this:
http://www.dimensions-math.org/Dim_reg_E.htm
that try to teach you how to see higher dims from their distorted shadows.

The problem with shadows is they are distorted... but cross sections are not. (In 2D you can walk around an undistorted cross section of a 3D object) Rather than trying to model a distorted shadow, I'd rather see a bunch of cross sections. Has anyone made a bunch of consecutive undistorted 3D cross-sections of a 4D object? Would that just consist of a small square, then a larger square, then a larger square... or small sphere, then larger sphere etc. etc.?

Depends. If you take slices (= cross-sections) of a hypersphere, then you indeed get a point, then a small sphere, then a larger sphere, then a smaller sphere and a point.
But with the hypercube, things are different. It depends in what direction you slice things. This is also true with the normal cube. You don't necessarily have square slices, but you can have quite complicated figures like this: http://www.crme.soton.ac.uk/publications/gdpubs/S&Dfigure6.gif

Anyway, for some movies about slices of hypercubes, see http://www.math.union.edu/~dpvc/math/4D/hcube-slices/welcome.html

 

[JamieLT]

Anyway, for some movies about slices of hypercubes, see http://www.math.union.edu/~dpvc/math/4D/hcube-slices/welcome.html

Thank you for that.

Are there any vids for rotating a 3D object into a 4D object? Like you can rotate a line to get a circle, then rotate a circle to get a sphere, seems like there would be some sort of rotation to get a 4D object out of a 3D one too, but I can't figure out the rotational axes for it.

For a line to a sphere, you can't rotate it around the longitudinal axis, you have to go down a dimension, and pick a point on the line, and then rotate it around that point. 1D line rotated around 0D point = 2D circle.

For a circle to a sphere, you can't rotate it like a saw blade, you have to draw a 1D line on the 2D circle, and rotate the 2D object around the 1D line to make the 3D sphere. 2D circle around 1D line = 3D sphere

Following the same pattern, seems like you would rotate a 3D object around a 2D plane to make the 4D object...
3D rotated around 2D = 4D?

but how do you rotate something around a 2D plane? Did that make sense?

 

[Alfi]

Another question to add to yours.

Can all the Platonic solids be made to show in 4d like the cube ?

 

[TGlad]

The Tesseract assumes that in order to add another Euclidean dimension another orthogonal plane is added into the mix, right?

no, the volume of a cube is extruded in the 4th dimension to produce a tesseract.

Following that pattern -
0D-1D: line is an infinite number of points
1D-2D: plane is an infinite number of lines
2D-3D: 3D is an infinite number of planes
3D-4D: 4D is an infinite number of 3D spaces? like a bunch of cubes stacked on top of one another or something?

Exactly.

Has anyone else submitted anything other than a Tesseract to represent 4D?

Yes, there are plenty of 4d objects, for example a Pentachoron is the 4d equivalent of a tetrahedron.

seems like you would rotate a 3D object around a 2D plane to make the 4D object...
3D rotated around 2D = 4D? but how do you rotate something around a 2D plane?

Yes, that's right. You can rotate around a 2d plane because if those 2 dimensions are fixed you still have two spare to rotate around. Just multiply by a 4d matrix, e.g.
cos(x) sin(x) 0 0
-sin(x) cos(x) 0 0
0 0 1 0
0 0 0 1

Can all the Platonic solids be made to show in 4d like the cube ?

You can render any 4d object you like... but there isn't a direct equivalent to each platonic solid in 4d, just as there is no 3d version of an octagon. But you could make each into a 4d prism. The triangle and the square can both be extended to any number of dimensions.

 

[phinds]

TGlad, you should learn to use the quote feature on the forum

 

[TGlad]

Done :)

 

[JamieLT]

Thanks TGlad! I guess I need to start playing around with a 4d matrix, that's a good tip.

 

[Tinyboss]

Instead of thinking about rotating lower-dimensional objects to get higher-dimensional ones, it's probably better to think about extrusion, as explained in the video posted to this thread.

The reason is that, qualitatively, rotations behave very differently in different dimensions. What TGlad said about multiplying by (orthogonal, determinant-one) matrices of the appropriate size is exactly right and always works, but doesn't really give you a feel for what's happening. If you're looking for an extension of your 3D intuition to higher dimensions, rotations aren't what you want. They get weird. However, the operation of extrusion, or making "prisms" into higher and higher dimensions, always works as you'd expect.

An example of how rotations are not so intuitive is the idea of the axis of a rotation--by axis we mean the set of points that remain still during the rotation. In 3D, every rotation can be described as spinning around some line. But in 2D, the axis is just a point. Okay, not so bad, at least every rotation still has an axis, and the axis for every rotation is the same kind of object. But these are low-dimensional coincidences that don't hold up as the dimension increases. As soon as we get to 4D, we get different kinds of rotations--some of them have a single point as their axis, and others have a whole plane. In 5D, the axis could be a line or a three-dimensional space. In general, a rotation in N dimensions can have as its "axis" any subspace with dimension N-K, where K is an even number.

 

[lavinia]

one can use color as a fourth dimension overlayed onto 3 spatial dimensions. this is commonly done in computer graphics.

Time is another easily conceived fourth dimension.

One can understand four dimensions by breaking down 3 dimensional objects inside of it into pieces that can be visualized then describing how they are put back together.

An easy example is the sphere in four dimensions.

 

[Zula110100100]

0D-1D: line is an infinite number of points
1D-2D: plane is an infinite number of lines
2D-3D: 3D is an infinite number of planes
3D-4D: 4D is an infinite number of 3D spaces? like a bunch of cubes stacked on top of one another or something?

That is why time seems so obvious as a 4th dimension, if you have a point, stretched over instances of time it become a line, a line a plane, and a plane becomes a cube. If you want cross-sections of a 4d, you can watch a movie.

Another thing I have wondered is how long we have considered time a dimension, and do phrases like "some point in time" and "time-line" predate that?

 

[Alfi]

Another question to add to yours.

Can all the Platonic solids be made to show in 4d like the cube ?

Thank you for your answer.


I will continue with my toothpick models.

 

[laserblue]

I posted this in yahoo questions, but perhaps this is a better place for it.

The Tesseract assumes that in order to add another Euclidean dimension another orthogonal plane is added into the mix, right? That is the pattern when going from 1D to 2D to 3D, but it seems like something slightly different is going on from 0d to 1D. From a point to a line - a radial line is orthogonal to a circle, but 0D is not a really a circle, it's a point, so from 0D to 1D, it doesn't really seem like adding on an orthogonal plane (what is orthogonal to a point? everything? nothing?) instead, it makes sense to me that a line is created by an infinite number of points. Following that pattern -
0D-1D: line is an infinite number of points
1D-2D: plane is an infinite number of lines
2D-3D: 3D is an infinite number of planes
3D-4D: 4D is an infinite number of 3D spaces? like a bunch of cubes stacked on top of one another or something?

Has anyone else submitted anything other than a Tesseract to represent 4D?
Also, how would you describe going from 0D to 1D?
Thanks!

(sorry if this is the wrong place to post this, I thought I saw another higher dims thread over here, so hopefully this is ok!)

Buckminster Fuller questioned the orthogonal assumptions of multidimensional drawing and suggested instead, using 60 degree base vectors for x,y,z on one plane, with the fourth dimension as the altitude of a tetrahedron. He also preferred to refer to the second power of a number as 'triangling' rather than 'squaring'.
Also note that the cross product of vectors is not defined for euclidean spaces greater than 3 dimensions but the outer product is multidimensional.