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 Posted: Mon Nov 12, 2007 9:52 am
I know the most fundamental basic graph like the Cartesian graph can explain (x,y,z) though when we add a fourth we add t also its sketchy. So my overall question is how would one repsents the axis t in a graph if it can be done or how would that consider with the other three axis. I know that it is orthaginal of the z though what is troubling is its almost unknown were at a certain point.
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Posted: Tue Nov 13, 2007 2:40 am
The best way to visualise a fourth dimension is using colour. Its done very well here.
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Posted: Thu Nov 15, 2007 4:30 pm
jestingrabbit The best way to visualise a fourth dimension is using colour. Its done very well here.I see...Humm it seems you can only represent a dimensions that you are in to yours...
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Posted: Mon Dec 03, 2007 4:02 pm
I find drawing in three dimensions hard enough without trying to bring in a fourth.
Have fun!
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Posted: Sat Dec 29, 2007 1:05 pm
Thanks all I been self teaching myself Euclidean geometry, very nice this subject. Makes me like spaces even more. Wanted to show you something cool I found while looking this up so here it is.. http://unfolding.apperceptual.com/
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Posted: Sat Dec 29, 2007 11:40 pm
You all are absolutely silly ^^;; Of course you can represent the t axis on a graph. And you don't have to use silly colors or remove x,y,and z to do it. Simply think about how t affects our every-day life.
The t allows two objects to occupy the same space, because they do so at different times. This basically means that t is a factor that causes x, y, and z to change. The problem is that you're trying to imagine a graph that does not change. If you graph the simple x, y, and z coordinates on a moving graph, then t is represented by the rate at which x, y, and z are changing.
Elementary, my dear Watson rolleyes The problem is, you were trying to display 4 dimensions using 3. To display 4 dimensions, you need 4 or more. Almost any graph we use in mathematics uses only 3 dimensions simply because if our graph changed it would be difficult to show it to friends and have them looking at the same data we witnessed. Thus, we can accurately graph any 3 dimensions on our 3-dimensional graph. Since we live in a 3 spatial-dimension world, we cannot physically create a 4-dimensional graph purely out of spatial dimensions. Thus, for a fourth-dimension to the graph we must utilize our dimension of time and create a graph with different values at different times.
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Posted: Mon Dec 31, 2007 8:11 pm
spider_desu You all are absolutely silly ^^;; Of course you can represent the t axis on a graph. And you don't have to use silly colors or remove x,y,and z to do it. Simply think about how t affects our every-day life. The t allows two objects to occupy the same space, because they do so at different times. This basically means that t is a factor that causes x, y, and z to change. The problem is that you're trying to imagine a graph that does not change. If you graph the simple x, y, and z coordinates on a moving graph, then t is represented by the rate at which x, y, and z are changing. Elementary, my dear Watson rolleyes The problem is, you were trying to display 4 dimensions using 3. To display 4 dimensions, you need 4 or more. Almost any graph we use in mathematics uses only 3 dimensions simply because if our graph changed it would be difficult to show it to friends and have them looking at the same data we witnessed. Thus, we can accurately graph any 3 dimensions on our 3-dimensional graph. Since we live in a 3 spatial-dimension world, we cannot physically create a 4-dimensional graph purely out of spatial dimensions. Thus, for a fourth-dimension to the graph we must utilize our dimension of time and create a graph with different values at different times. But you'd still basically be getting only a piece of the puzzle at any given time. You can represent 3-dimensional surfaces in 2 dimensions by looking at level curves of the graph, i.e. the curves that intersect the plane z=0, 1, 2, etc. Similarly,you could represent 3-dimensional surfaces (4-d "graphs") in 3-space by looking at the 3-d cutouts at different times. It still only gives you a piece of the puzzle, but doesn't reqire being able to animate your graphs. (The other problem is that this approach isn't...complete in a sense. It can be phenomenally hard at times to understand what surfaces look like just from their level curves. I imagine looking at "time-slices" only exacerbates that problem)
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Posted: Tue Jan 01, 2008 12:47 pm
Swordmaster Dragon (The other problem is that this approach isn't...complete in a sense. It can be phenomenally hard at times to understand what surfaces look like just from their level curves. I imagine looking at "time-slices" only exacerbates that problem) Oh god, yes. It's terrible, enough so that a large branch of differential geometry was created just to look at it. Damn you, Morse Theory as applied to height functions, damn you and your multiply redundant meta-homotopy levels.
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Posted: Wed Jan 02, 2008 5:04 pm
Swordmaster Dragon spider_desu You all are absolutely silly ^^;; Of course you can represent the t axis on a graph. And you don't have to use silly colors or remove x,y,and z to do it. Simply think about how t affects our every-day life. The t allows two objects to occupy the same space, because they do so at different times. This basically means that t is a factor that causes x, y, and z to change. The problem is that you're trying to imagine a graph that does not change. If you graph the simple x, y, and z coordinates on a moving graph, then t is represented by the rate at which x, y, and z are changing. Elementary, my dear Watson rolleyes The problem is, you were trying to display 4 dimensions using 3. To display 4 dimensions, you need 4 or more. Almost any graph we use in mathematics uses only 3 dimensions simply because if our graph changed it would be difficult to show it to friends and have them looking at the same data we witnessed. Thus, we can accurately graph any 3 dimensions on our 3-dimensional graph. Since we live in a 3 spatial-dimension world, we cannot physically create a 4-dimensional graph purely out of spatial dimensions. Thus, for a fourth-dimension to the graph we must utilize our dimension of time and create a graph with different values at different times. But you'd still basically be getting only a piece of the puzzle at any given time. You can represent 3-dimensional surfaces in 2 dimensions by looking at level curves of the graph, i.e. the curves that intersect the plane z=0, 1, 2, etc. Similarly,you could represent 3-dimensional surfaces (4-d "graphs") in 3-space by looking at the 3-d cutouts at different times. It still only gives you a piece of the puzzle, but doesn't reqire being able to animate your graphs. (The other problem is that this approach isn't...complete in a sense. It can be phenomenally hard at times to understand what surfaces look like just from their level curves. I imagine looking at "time-slices" only exacerbates that problem) Isn't this exactly what you wanted? The value of 3 variables (x,y, and z) at any given time (t)? After all- a 2-dimensional graph tells you the value of y at any given x. A 3-dimensional graph tells you the value of z at any given x and y. A 4-dimensional graph tells you the value of t at any given x, y, and z. Or the value of x given y, z, and t. Or the value of y given x, z, and t. A 4-dimensional graph merely refers to a graph with different values at different times. This, to me, means a changing graph. I can think of no continuous way to represent this other than an animated graph. Of course continuity isn't a necessity. You could display intersections of the three dimensions at different times. Show it's value at t=0 seconds, t=1 second, t=2 seconds... I don't quite understand why my solution was incorrect...
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Posted: Fri Jan 04, 2008 3:11 pm
It's not that your solution is incorrect; it's that you can never get the entire picture at once. The "graph" - for our purposes, any 3-manifold embedded in 4-space - is an object in its own right. To use a bad analogy, the whole is not the sum of its parts. Understanding what the "graph" looks like at every point in time not only requires you to have an infinite number of 3-d pictures, but there could be additional structure that simply will not reveal itself in this method.
To anyone tracking both this thread and the one on linear spaces; this method works for linear spaces because, well...they don't really have any additional interesting structure (so long as they're finite-dimensional). In those cases, you really *can't* hide any structure.
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Posted: Fri Jan 04, 2008 7:21 pm
Swordmaster Dragon It's not that your solution is incorrect; it's that you can never get the entire picture at once. The "graph" - for our purposes, any 3-manifold embedded in 4-space - is an object in its own right. To use a bad analogy, the whole is not the sum of its parts. Understanding what the "graph" looks like at every point in time not only requires you to have an infinite number of 3-d pictures, but there could be additional structure that simply will not reveal itself in this method. To anyone tracking both this thread and the one on linear spaces; this method works for linear spaces because, well...they don't really have any additional interesting structure (so long as they're finite-dimensional). In those cases, you really *can't* hide any structure. Eh, with linear spaces you don't really need to make it complicated at all, because you can reduce the entire thing to a set of basis vectors with a plus sign and a scalar field on the end.
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Posted: Wed Feb 13, 2008 3:21 pm
You could have a 3D graph inside a tessaract with different sections representing different times
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