In the simple Euclidean plane, when we are faced with drawing a straight line we have the following challenge: if we are going to make a straight line with a straightedge, the straightedge itself must be straight, but what does it means we have to prove that a straightedge is straight?
Rulers are used to measure and as straightedge.
We can compare it with another ruler, or we can draw two lines one above the other turning the ruler 180 degrees and see if they match exactly. Or, we can use rigid metal straightedges, or similar methods. But what must be aware of is that even when a line is one-dimensional, to make a plain straight line we need to be working on a plane.
Simple as it may be seem, this assertion is important because it goes against the usual induction reasoning used to prove the existence of the fourth dimension . The usual reasoning is that we can generate a plane by moving a line perpendicular to itself; and that we can move the plane perpendicular to itself to generate a cube, and that we can then move the cube perpendicularly to itself —in all directions— to generate a hypercube.
The flaw in this reasoning is that we cannot move a line perpendicular to itself to generate a plane because this implies the pre-existence of a space into which we can move our straight line. But this is a contradiction because we are supposed to be generating the space, so we cannot assume it pre-existence.
The following figure, taken from the free EBook Selected Papers of Charles Hinton About the Fourth Dimension , illustrates this type of reasoning.
How to go from one dimension, to two dimensions, to three dimensions by "moving" perpendicularly to the previous dimensions.
Hinton explains the above figure as follows:
The straight line AB gives rise to the square ABCD by a movement at right angles to itself. If motion be confined to the straight line AB, a backward and forward motion is the only one possible. No sideway motion is admissible. And if we suppose a being to exist which could only move in the straight line AB, it would have no idea of any other movement than to and fro. The square ABCD is formed from the straight line by a movement in a direction entirely different from the direction which exists in AB. This motion is not expressible by means of any possible motion in AB. A being which existed in AB, and whose experience was limited to what could occur in AB, would not be able to understand the instructions we should give to make AB trace out the figure ABCD.
"If motion be confined to the straight line AB, a backward and forward motion is the only one possible." This is the principle of the straightedge,
Note this short segment: "If we suppose a being to exist which could only move in the straight line AB, it would have no idea of any other movement than to and fro", and the later one: "A being which existed in AB, and whose experience was limited to what could occur in AB, would not be able to understand the instructions we should give to make AB trace out the figure ABCD." This "we" he is talking about are the beings from superior dimensions higher from Lineland; beings not confined to the dimensions of Lineland. (Lineland is kingdom found in Flatland, I borrow the from there.)
Thus, people that live in the kingdom of Lineland can only move to and fro; they will never understand what is a plane, much less what is a cube.
The problem with this classical reasoning where one tries to go from one dimension to the fourth dimension is that activity of "moving perpendicularity" to the existent dimensions.
In the figure ABCD there is a possibility of moving in a variety of directions, so long as all these directions are confined to one plane. All directions in this plane can be considered as compounded of two, from A to B, and from A to C. Out of the infinite variety of such directions there is none which tends in a direction perpendicular to figure 2; there is none which tends upwards from the plane of the paper. Conceive a being to exist in the plane, and to move only in it. In all the movements which he went through there would be none by which he could conceive the alteration of figure 2 into what figure 3 represents in perspective. For 2 to become 3 it must be supposed to move perpendicularly to its own plane. The figure it traces out is the cube ABCDEFGH.
All the directions, manifold as they are, in which a creature existing in figure 3 could move, are compounded of three directions. From A to B, from A to C, from A to E, and there are no other directions known to it.
But if we suppose something similar to be done to figure 3, something of the same kind as was done to figure 1 to turn it into figure 2, or to figure 2 to turn it into figure 3, we must suppose the whole figure as it exists to be moved in some direction entirely different from any direction within it, and not made up of any combination of the directions in it. What is this? It is the fourth direction.
But all this is an illusion: its the opposite reasoning that is right: We can easily go from a two-dimensional plane into any straight line (or curve) within that plane.
Very long ago the ancient Greek mathematician Euclid wrote the foundations of classic geometry in the all-time reference Elements. This treatise in geometry is divided in several books. Euclid's Elements is considered the first axiomatic system in mathematics.
It was in the first of his Books where he laid the intuitive notions of points, lines, end pints, straight lines, surface, planes, etc. His first definitions are the following:
Thus, for Euclid, the plane is a special case of surface. However, since in Definition 5 he defines a surface as "that which has length and breadth only", he is thinking of the plane as a two-dimensional geometrical completely flat entity.
But we are seeking for planes in the stricter and most intuitive sense; for example, that plane that extends indefinitely from a table desktop. The Webster's definition of plane reflects what this intuitive notion is all about: "a: a surface in which if any two points are chosen a straight line joining them lies wholly in that surface b: a flat or level surface". The key phrase here is "lies wholly in that surface", and this is the impossible dream: we cannot test for the pure flatness of a plane with the tools of that "flatland".
So we are returning to our starting point: a flat surface is the one made up of straight lines only, but to be certain about the straightness of a line we need a two-dimension plane.
This seems to imply that we can "test" a dimension only from a higher one, not from the dimension itself. We need the plane to test for straightness of line; we need the 3D-space if we want to be sure of the flatness of a plane.