Visualizing Four Dimensions - How do we envisage a 4-dimensional space? Not easy is it?(adsbygoogle = window.adsbygoogle || []).push({});

Let us take a simple approach and identify how the fourth dimension would connect or interface with the three dimensions we are so familiar with. The first principle that most people would identify is that it must be at right angles (normal) to our three existing dimensions, which are usually viewed as the axes of a Cartesian system of coordinates; not easy is it to imagine an axis normal to the existing three, certainly not without involving 4 dimensional geometry - or is it?

http://img717.imageshack.us/img717/4246/fig15fourdimensionalaxe.jpg [Broken]

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Fig. 15 Four Dimensional axes

The only way that I can envisage it is by a sphere centred on the origin of three equally scaled axes that cuts each axis at the equivalent points, for then the surface of the sphere will be to all intents and purposes, in the limiting case, a flat surface normal to each axis.

But where then is the fourth axis? Which direction does it lie in?

Well it doesn't; because it cannot lie in any mapped orientation within our three existing axes.

So let us say that it has no direction but lies in all directions, and that therefore, it may be represented by any line drawn from the origin to the surface of our sphere. And that if we say its coordinate scale is ct, light-seconds for example, then we have made time our fourth dimension, which fits well, as time can/cannot have any orientation with respect to the three spatial dimensions.

How though can we mark the passage of time, our movement along this fourth dimension, or even denote a specific point on that coordinate in relation to our other three coordinates?

In the first place, if we draw only 1 or two spatial dimensions then we can, as is the current practice, draw time as one of the dimensions represented in that diagram. For then, any particular time is represented by a line or a plane that denotes that time at every point on the other dimension(s).

The difficulty comes with trying to envisage how to draw it as the fourth dimension, for then we have to be able to represent a point in time across the whole of a 3D space. We cannot merely add more lines as they would merely represent vectors in the 3D space. No it has to be something that defines all the 3D space at a particular time. I would suggest that colour could fit the bill.

Let us say that as time passes it is represented by a changing colour of our three dimensional drawing of space, so a particular time and the associated spatial 3D diagram would be given a specific colour.

http://img846.imageshack.us/img846/3895/fig164ddiagrams.jpg [Broken]

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Then we would have the time axis that could be drawn anywhere on the diagram as a line from the origin to a particular point in a particular colour and we would have:

c²t²=x²+y²+z²

or

c²t²-x²-y²-z²=0

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# Pondering time

Can you offer guidance or do you also need help?

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