A "Virtual 4-space"

Recall, if you will, the definition of our 'working model' of 4-space: the combined empty space formed by an array / 'group' of 3-planes. Recall also, however, the restrictions involved in our experience of 4-space: we could experience only one 3-plane of it at a time. Furthermore, the 3-planes dealt with here were an array of planes arranged in a linear fashion: every 3-plane, therefore, possessed 2 "sides". A line extended perpendicular to the array, though hard to conceive of as of our own dimension, would represent the 2 directions in which all travel through the array is done - direction 'A' and direction 'B'. This was hard to visualize, however, because a subject confined to a 3-plane in an array of 3-planes can at best explain the location of the remaining planes of the array as occupying the "same space" as his own 3-plane. By means of the stack-diagram, however, we could overcome this awkward situation by mentally 'lining up' the 3-planes into distinct 'rows' that emphasize the spatial separateness of each plane, allowing us to manage the 3-planes in an organized fashion. What is the next step?

Learning that to a subject in a 3-plane array, the 'best guess' concerning the location of surrounding 3-planes would be that they occupy the "same space" as his own 3-plane gave us an idea of how we directly experience a 3-plane array. The use of a stack-diagram to conceive of a 3-plane array, on the other hand, gave us an intellectual knowledge of a 3-plane array. The next step is to combine these two elements to form a version of 4-space that is more elaborate than ever: a model of 4-space that one can actually 'move around' in. It is a space within which the location of a point can be designated by means of 4 spatial coordinates: the first 3 coordinates refer to the 3 dimensions of any given 3-plane. The fourth coordinate designates which 3-plane - out of an array of 3-planes - that this 3-plane is. Because we experience 4-space 'one 3-plane at a time', however, we must 'advance' through it on a plane-by-plane basis.

Think of your room as a "stage" - upon which all 'advancement' from plane to plane will take place. To enable the visualization of this advancement, picture an array of 3-planes by imagining multiple 3-planes to occupy the "same space" as your own room - your own room being, as you may recall, the central 3-plane in an array of 3-planes. To advance forward through the array, first decide which 'direction' you want to move in - direction 'A' or direction 'B'. Next, with your room as the "stage", imagine the 3-plane you are leaving, to 'disappear' behind you - as the new 3-plane to which you are addvancing 'aligns' itself with your room (there will never be more than one 3-plane on the 'stage' at any given time). You have just 'transferred' from one 3-plane of the array to another. By keeping track of which way you go when 'shifting' from 3-plane to 3-plane, you can create and manage your own 'virtual 4-space' - a stack-diagram that you are inside of!

For example, to plot a point by means of 4 spatial coordinates, plot the first 3 coordinates by visualizing the point in the specific 3-dimensional location within your room that it should be. To plot the fourth coordinate, 'shift over', by means of the technique described above, to the 3-plane that corresponds to the fourth coordinate. Take note that a Flatlander could use this same technique to create a 'virtual 3-space' by imagining movement from plane to plane within a 2-plane array. Equally so, a Linelander could create a 'virtual 2-space' by imagining movement through a 1-plane array.

It is in 'moving around' within a 'virtual 4-space' like this that you can begin to get ideas as to what the actual 4-space would be like (that is, as you may recall, the 'empty space' that is perceived by experiencing more than one 3-plane simultaneously - something that we can't do). And this 'virtual 4-space' - as you will see later - will prove invaluable in its use for the visualization of upcoming 4-dimensional structures (in that a 4-dimensional structure, by its very definition, is a structure that requires multiple 3-planes to be constructed). This concept will be demonstrated next as we attempt to construct a hypercube.