About making a Point

This thought had to be written
it had nagged me long enough

We all have done this.

According to rules of geometry, our world is three-dimensional with the following attributes: length, width and height.

Science has agreed, on what evidence I don’t know, that time is the fourth dimension and subsequently, that there also exists a fifth. Of course, why not? Just because we can’t see something, it does not disprove its existence. And why would one stop at five? Once you have started, you may as well continue up to the “n”th dimension.

However, moving in that direction turns quite tedious, one runs out of names for the next dimension, they become more and more speculative and meaningless. Or should I say pointless?  That was not my point; I got a bit sidetracked here.

Making my point, I intended to move into the other direction, not expansion but reduction. In our three-dimensional world, a simple shape defined by length, width and height could be a cube. A beautiful square cube, which is mathematically expressed as a length “x” times a width “y” and times a height “z”. And when we continue looking at our world everything has length, width and height. And that is good that way.

For interest sake, imagine a world with only two dimensions, namely length and width. In mathematics, it is termed a plane and is defined by a length “x” times a width “y”. A two-dimensional being living on this plane also would only have expansion in two directions. To us three dimensioners it may look like one of those coloured, blotted shapes that psychologists use to figure out our subconscious.

As much as in our world a confined space is defined by walls or bars limiting our moves in the three directions we could draw an encircling line around the two-dimensional being, and it would feel imprisoned. Fancy that! Does it not know that it only has to step over the line and it would be free? Would that not be the same comment a four-dimensional being would make about us?

Got a bit sidetracked again. That stuff about dimensions is fascinating. So let me emphasise again, two dimensions are defined by length and width.

Now going further down there would be the one-dimensional world, defined only by length. A line of any shape having a beginning and an end but no width and no height. A bit hard to imagine because when we draw a line, it has the width that the drawing implement traces on the paper and a thickness too, that of the layer of graphite or paint that makes the line visible to us.

The line is actually three-dimensional, defined by length, width and thickness so we can see it.  Could one conclude from that observation that if something is not composed of matter, meaning: consisting of three dimensions we cannot see it? So maybe the four-dimensional being cannot see us?

It happened again; sidetracked! Let’s summarise: the last dimension was length, but there is still one significant component in the realm of geometry, the point. Now, there are no dimensions left to describe the point. And the rules of geometry state that it has not got any, it is only defined by its position, which on paper is defined by its “x” and “y” coordinates or in a spatial arrangement, the add “z” coordinate.

Again, we could theorise, that in a four-dimensional world there would be “w” coordinate and so on. In statistics, for example, higher dimensional calculations are popular with each dimensional expansion representing a variable. The point has no dimension.

The point in our world has, it’s a dot (1). Therefore, elaborating on what the laws of geometry have already stated, a point (in geometric terms) is a fictitious expression, a definition of a position but is made up of nothing.

I feel I am getting somewhere. Referring to the laws of geometry again, a line is formed of an innumerous number of points or expressing this more mathematically correct: it is an infinite number of points that make up a line.

I remember starting a discussion with my math teacher at that point (2) in time that even if I multiply a piece of nothing an infinite number of times, it still would be nothing, not even a bigger sort of nothing because nothing has no dimension thus no size. That was half a century ago, and I am sure that mathematics has agreed on one consensus and released a new axiom (3) on that topic.

I do not believe that any sort of something can emerge out of nothing, not even a line with only one dimension, unless… the line is composed out of nothing too (4). Thus, if the point does not exist, is only an imaginary “thing” then it follows that anything made from it must be imaginary too.

Now, as long as the trail is warm, let’s step it up quickly. The one-dimensional line is made up of nothing, then the two-dimensional area, which has length and width, defined by two lengths of lines both made up of nothing; therefore, the resulting plane is imaginary too. I have suggested that it may be invisible, now we know it does not even exist.

And again let’s move up another level into our three-dimensional world with the additional dimension of height, again, which is only defined by a length of a line. And as we have agreed already before nothing times nothing still remains nothing no matter how often one multiplies it with itself.

Does that mean that our world is imaginary too? No, it can’t!



(1) see my comments on lines

(2) If a point in space is fictitious, may I suggest that a point in time, the infamous NOW is fictitious as well?

(3) Axiom . . . (in the field of logic) . . . a proposition that is not susceptible to proof or disproof; its truth is assumed to be self-evident

(4) It ‘s hard to describe nothing with words because they all refer to something material.

(5)  This was some time ago.  Now there is a whole topic on Nothing


Wolfgang Köhler
Sunday, 1 November 2009

Why ingeneer?
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