Visualizing the Fourth Dimension

[Sisyphus]

Interesting comment.

The curve is one dimensional.

The graph is 2 dimensional.

And if you put units on X (say meters), then Y are square meters: 2 dimensional. I guess the growing graph of increasing surfaces can be represented 3D (a pyramid).

 

Oh, I'm sure it can be represented by objects of any number of dimensions, but the point is all you need is two. You also only need two to represent y=x^3, or y=x^1000, since they're all just representing the relationship between 2 variables. Acceleration is not adding a dimension, it is just a non-linear relationship between two dimensions.

[michel123456]

Wait.

When you have meters (m) you get one dimension.

When you have meters by meters (m^2) you get 2 dimensions.

But when you have seconds by seconds (s^2) you don't? Where is the difference?

[Sisyphus]

It's not different from meters. In fact, that's the example I used. Graphs of y=2x, y=x^2, or y=x^37 are all showing a relationship between two variables, the horizontal and vertical dimensions. There's no reason they can't be in units of meters. "The vertical height is proportional to the 37th power of the horizontal distance." No 37-dimensional object need be involved.

 

Now, if you were going to graph y=x+z, you would need three dimensions.

[dalgoma]

Joining the 8 vertices of one cube on top of another has got a certain logic :-

 

View a square to a corner you see 2 lines

 

View a cube to an edge you see 2 squares

 

View a 4D cube from a chosen location you see 2 cubes.

 

By choosing the viewpoint we seem to be able to obscure a dimension.

 

 

[Moontanman]

I've often thought if dimensions this way. A line is one dimension a square is two dimensions but how many lines does the square contain? An infinite number of 1D lines side by side compose the 2D square. A cube is an infinite number of 2D squares all packed on top of each other, so this would mean a 4D cube equivalent would contain an infinite number of 3D cubes.

Edited February 11, 2011 by Moontanman

[Gozonji]

I've often thought if dimensions this way. A line is one dimension a square is two dimensions but how many lines does the square contain? An infinite number of 1D lines side by side compose the 2D square. A cube is an infinite number of 2D squares all packed on top of each other, so this would mean a 4D cube equivalent would contain an infinite number of 3D cubes.

 

By that logic, one could also purport that a square is an "infinite number of smaller squares stacked together" as well.

[khaled]

In highschool, my math teacher told me that rotation around origin, is the motion

that moves one object from one dimension to a higher one, with an exception of the origin ...

 

in 1D, a point can rotate, to form a circle in 2D

 

the circle in 2D can rotate, to form a ball in 3D

 

the ball in 3D can rotate, to form a rotating ball in 4D

 

By that logic, one could also purport that a square is an "infinite number of smaller squares stacked together" as well.

 

it can be stacked of any shapes that can uniform at some point into your shape or a uniform cut of it,

going down at the level, one shape is a set of points, in discrete geometry ...

[Moontanman]

By that logic, one could also purport that a square is an "infinite number of smaller squares stacked together" as well.

 

 

No, that's not what i am saying, the 4D equivalent of a cube should hold an infinite number of full sized 3D cubes, maybe an infinite number of infinitely large 3D cubes.

[michel123456]

No, that's not what i am saying, the 4D equivalent of a cube should hold an infinite number of full sized 3D cubes, maybe an infinite number of infinitely large 3D cubes.

 

Perhaps something like Time ?

[Moontanman]

Perhaps something like Time ?

 

 

That is precisely what I am saying!

[michel123456]

That is precisely what I am saying!

 

The peculiar thing is that we are convinced that there is only one cube, that we are living in it. That we are traveling in time in this single cube, making stupid comments like "how wonderful is our cube!" or asking stupid questions like "how is it possible that we have such a chance to live in this cube?", forgetting completely the infinity of other cubes.