Four Dimensions – How do we envisage a 4-dimensional space? Not easy is it?
Let us take a simple approach and identify how the fourth dimension would connect or interface with three the dimensions that 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, the axes of a Cartesian diagram; and the only way that can be done is by a sphere centred on the origin of three equal scaled axes that cuts each axis at the equivalent points, for then the surface of the sphere will be to all intents and puposes, 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, that 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 added time as our fourth dimension, which fits quite well as time has no direction.
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? I would suggest that it must be something other than by adding new lines yet it must be visible across the whole three dimensional space; and so I would turn to colour.
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.
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