Hi John. Good to hear from you.
Motion is a very complicated idea. There must be an object which moves, an observer to register the motion, and a background against which the motion can be registered. Each of these three can be said to possesses some degree of dimensionality. The object may be thought of as a point, as, for instance, the point of a pen or pencil. Of course a pen or pencil is a three dimensional object, but we can imagine that we are only interested in the zero dimensional "point" of the instrument, the rest of it being there only to indicate just where the point of movement actually is.
We as observers generally imagine ourselves to be four dimensional, that is, we occupy and can manipulate a three dimensional space in one dimension of time. My personal goal is to see if the observer can comprehend higher dimensionalities, but let us leave that for now as a goal.
Then there must be a background against which the object is to move. We might choose to wave the pencil about in the air, or to place it against a plane sheet of paper, or even to trace it along a stretched wire. The wire would represent a one dimensional background. The paper represents a two dimensional background, and the air is generally thought to be three dimensional.
Now we may think of motion of a point as occurring in a one dimensional background, as along the wire. The wire, mathematically, is thought of as a series of points, and mathematically is infinitely divisible along any section of its length. However, we have determined that the moving point is not a mathematical point, but is more like a pencil point, and posseses some minimum measurable size. So we could make a mark on the wire, and another right next to it, and another next to that and so on, and give the points so marked names, Fred and Sam and Tom would do, but we like to put them in sequence and give them the names of our counting numbers, and this turns out to be very valuable in terms of calcualtions we may wish to make at some future time, such as distance, velocity, accelleration.
Now in a one dimensional background, moving a one dimensional point, we three dimensional beings can define two directions of movement for the pencil point. It can go from a marked point on the wire that we might call zero to a marked point on the wire that we might call one, or it can go in the other direction to the point we might call negative one.
In the case of a two dimensional background, as represented by the top side of a sheet of paper, we have more choices of which way to move the pencil. We might start by laying the wire down on the paper and drawing a line, since we already have explored how to do that in one dimension. Then with our pencil point on the paper, we see that we can still move along the line to 1 or -1, but we also can move on the background in some other directions which are not on the line. We can move our pencil point off the line, and make a mark that is no part of the one dimensional model we have now embedded in the two dimensional surface.
We could move off the line and make a mark and give it a name like Sally or Linda, but it would be nice if we could retain the advantages of sequence, which allows calcualtions of two dimensional quantities like surface area and curvature of the line. There are a number of ways to do this, and we can name the points on the plane off the line using polar coordinates or degrees of angle or something, but the main change from the one dimensional model is that we now have to use two terms to specify a point on the plane. It is no longer good enough to say that it is one (mark from the zero), because there are a circle of points that are all one from the zero. We need to say that it is one mark from the zero at some angle or longitude. Or we can do something that was given to us by Descarte, if I remember my history lessons. Anyway it is called the cartesian method, and gives us a nameing system for points on the plane.
This is done by defineing a second line, orthagonal to the first line in the plane, which just means that the angle from the new line to 1 on the old line is the same angle as from the new line to -1 on the old line. We call the first line x and the second line y, and name any point on the cartesian plane by calling it something like x1, y5. You see from this that to find the named point, you travel one mark on x in the positive direction, then make the orthagonal (90 degree) turn and move five marks on y in the positive direction, and you come to the place. To save writing, we always write the x first and the y second, and denote the place more simply as (1,5). Now we can name any point on the plane with two numbers, using the orthagonal basis lines x and y as referents.
This can easily be extended to three dimensions by adding a z line and nameing points in three dimensions something like (1,5, 7). Now we have the orthagonal cubic grid that is commonly used when comparing points in three dimensional space. Other methods are also used in special cases as in when talking about points on the surface of a sphere, but the other methods can always be converted to the cartesian coordinates without changing the physical qualities of the system under consideration.
Ok. Now when you say "If points in space can only be so close together, that produces a space in which you can only travel back and forth in no more than six directions," I think you are referring to a two dimensional background. If the points are arranged in the usual othogonal cartesian plane, one might think that there are only four options for point to point movement in the plane. One could move the point from the origin (0,0) to the point (1,0), or to the point (0,1), or to (-1,0) or to (0,-1).
There are problems with the cartesian plane method, and I think you have put your finger on the chief among them. It was designed to make calculation as easy as possible, which is a great advantage to those of us with innumeracy. However, it is an artifice, and can be misleading when trying to understand things like motion and dimensionality. For example, it is reasonable to assume that motion at its most basic level can be thought of as occurring along a line, from point to point. For motion to occur, one should not be allowed to skip points or just go at random from any point to any other point. Motion has to have some sense of continuity. We need to be able to imagine that the object that is moving is more or less the same object when we begin the move as it is when we finish moving. Motion would not be very meaningful if the object being moved were allowed to change itself along the way. We can't calculate the velocity of a jeep leaving Los Angeles and a Cadillac arriving in New York. The jeep might have a velocity and the Caddy might have a velocity but there is no meaningful way to talk about velocity when the object arriving and the object departing are two different objects.
As it happens, there are non-orthagonal methods for nameing the points on the plane, but most of them are not very useful. We could name every point on the plane Tom or Sally or something, but there would be no easy way to tell from the names just how to move to get from where Tom is to where Sally is. I believe what you have done is to look at one of these other methods. You have observed (correct me if I am wrong) that a plane can be marked with points equally spaced in triangles, making a plane that is divided up into hexagons. This effect can be seen by tiling a table top with pennies or other coins. You can lay the coins out carefully in squares, but if you push on one side of the formation, you see that the square formation quickly and naturally collapes into the triangular formation. In fact, if the table top is vibrating, you will have to exert a lot of attention to maintain the square formation, because the coins will 'naturally' tend to the more densely packed triangular formation.
Then, if you give into nature and go ahead and lay the coins out on the table in the triangular formation, and if you then move your pencil from anyone coin to any adjacent coin, you will find, as you have described, that there are six available next coins to which you can move. If I have mistated your argument, please do not be offended, but correct me, and I promise not to be offended either.
I am less certain what you mean when you state that these six directions are the six "extra" dimensions of string theory, but it is possible as far as I know that you are correct. I am not sure what it means to add the usual three dimensions to the six directions on the non-orthagonal plane, or if they can be added meaningfully. And as for string theory, the higher dimensions mentioned there remain a mystery to me.
In a three dimensional background, divided into a set of orthagonal points and lines, there are six possible directions of movement, say up down left right foreward backward. In the non-orthagonal dense packed sphere regime, variously known as the Kepler stack, the dymaxion, and the cubeoctahedron, there are twelve possible directions of movement. This seems by extension then to give a six dimensional space. I suppose one could superimpose the cartesian othagonal system on the natural Kepler stack, and come up with the additive result of nine dimensions, but this seems to be a little short of the ten or eleven dimensions I have seen mentioned in string theory.
Anyway, it is good to be able to talk with you again, and, as always,
thanks for being.
Richard
