-How do we envisage a 4-dimensional space? Not easy is it?VisualizingFour Dimensions

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

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Fig.15 Four Dimensional axes

Theonly way that I can envisage it is by a sphere centred on the originof three equally scaled axes that cuts each axis at the equivalentpoints, for then the surface of the sphere will be to all intents andpurposes, in the limiting case, a flat surface normal to each axis.

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

Wellit doesn't; because it cannot lie in any mapped orientation withinour three existing axes.

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

Howthough can we mark the passage of time, our movement along thisfourth dimension, or even denote a specific point on that coordinatein relation to our other three coordinates?

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

Thedifficulty comes with trying to envisage how to draw it as the fourthdimension, for then we have to be able to represent a point in timeacross the whole of a 3D space. We cannot merely add more lines asthey would merely represent vectors in the 3D space. No it has to besomething that defines all the 3D space at a particular time. I wouldsuggest that colour could fit the bill.

Letus say that as time passes it is represented by a changing colour ofour three dimensional drawing of space, so a particular time and theassociated spatial 3D diagram would be given a specific colour.

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

c²t²=x²+y²+z² or

c²t²-x²-y²-z²=0