How do you graph in four dimensions?
That is, for mathematical dimensions for things like hypercubes and hyperspheres... NOT the Einsteinian idea of Time being a fourth spatial dimention
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Physically, it can't be done! For graphing purposes, however, create line segments at right angles to a 3D object:
It is impossible (not unlikely or not yet possible, but impossible, actually shown to be never possible) to construct a higher dimension from a lower one.
A quick way to grasp this impossibility is this. A straight line segment defines one dimension. Put another line segment perpendicular to this first line segment (getting something in the form of an L) and you have defined two dimensions (length and width). A third line, sticking out from the paper at a right angle, defines depth. Now, to define a fourth dimension of measurement (as we are talking of instantaneous occurrences, we ignore time measurements in our argument) take another line segment and put it at right angles to all of the other three. This will give you a fourth dimension. This is also quite impossible in our physical universe.
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I wonder if anyone will see this?
Couldn't the 4th plot be time in this way? By representing the coordinates as themselves plus some rate of change?
That is the coordinate for time could be a set of functions that show a rate of change for the values of the three other coodinates, at some specific amount of time later, so that in order to "graph" this you would need a series of graphs plotted one after another representing the effect of time on the other variables? A cube that doesnt change, if filmed, is a four dimensional plot of a 3-D object when film is played back and viewed, right? LOL
So, if you were to take say 23 pictures a second of a 3-d plot and run through the set of time functions for the xyz coordinates for each frame you film, then you could see how the shape changes over time. Or you could superimpose all the frames on top of each other and see what it looks like (the equivalent of showing all time functions at once, plotted)
LOL using a little mental gymnastics one can do this. LOL
Actually, now you can creat 4-dimensional graphs using high-powered softwares. Some computer uses 4D Graph Visualizers to track the change of Internet over time.
This might come in really late, but might help others.I think using color coding (and splices) would help to visualize 4-d functions. There is a simple matlab file for this. visit this link. one look at the image will tell you the story :)
the line segment explanation was good. what i'd like to add is,
let the co-ordinates be (a,b,c,d)
once we get to 3-d points(a.b.c), now add a line segment in an imaginary plane, and simply visualize as if you have definite points on that segment, and when you "mouse" over each of those imaginary points, you get a 3-d graph at that point. so when you point over "d", you get (a,b,c) to give (a,b,c,d). Now color code the d value from zero to whatever value you think is the highest. hope it helps.
So, let's try to explain it in order to everybody understands.
Please, try this link! You'll see some pics and maybe it will become easier to understand.
From another theoretical sense. The fourth dimension would be time. Therefore not on paper, or any current medium available, but if an extraordinary medium was invented, it may be possible to create a 3-D object with a time aspect, therefore making it 4-D (would a shifiting hologram be a 4-D graphic).
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