The Point

“We points don't have the roundness and fullness of dots…”

— The Point —

I am a point, and contrary to what some people say I do exist — just not in the way that most people are used to thinking of existence. Except for Plato, Plato is one of those rare human beings who really “got it” - along with Euclid. However, now I am getting ahead of myself. As I said previously, I am a point and I do exist, don’t let the fact that I have to keep reminding myself of this diminish your belief in me. However, since this is a math book I really shouldn’t talk about belief, so let me attempt to define myself for you. } Geome- try is not about belief.

As I said, I am a point, which means that I am dimensionless by defini- tion, having no existence whatsoever in terms of space. Perhaps that explains why I am a little sensitive about my existence, for I do exist - just not in space. In what way do I exist then, you might be wondering? I exist in time, as an Idea like the other Geometric figures that I have come to know, love and find cama- raderie with. Being dimensionless it may appear that all points are the same, but don’t let that fool you. Us points may not have dimension in space, but we are unique, each one of us responsible for holding a different location in both space and time.

Some points prefer to keep to themselves, others are more social, hang- ing out with figures of higher dimensions. Me, I like who I am and content with my existence as a dimensionless being. Don’t let conventional depictions deceive you on this matter, by the way, all points, myself included are inherent- ly dimensionless as any good book on Geometry will take the time to carefully note. However, being dimensionless does mean that I have certain advantages that other figures don’t, most notably that I am completely indestructible. I believe that Euclid said it best when he stated that, “a point is that which has no parts”. Not only am I completely lacking in parts or bits, but am also exactly as I appear to be, or to be more accurately, exactly how I don’t appear.

Some of you might be saying to yourselves, wait, but I’ve seen plenty of points! Math books are riddled with them, as are works of prose! } points are everywhere the majority of which go completely unseen After all, what is a period, but a point that has been kicked out of the math world and decided to embark upon a career in the humanities? Typical human approach, overlaying their habits and customs on other creatures where such ideas don’t even apply - usually, except in this case, at least somewhat.

Admittedly, some of what you humans say is true, and those of us in the math world do tend to question the choices of other points, particularly those that parcel themselves out to authors and writers of books who use them to mark the ends of their sentences. Like that one there, over to the left of me, just one comma, nine words, and ten spaces over. Truth be told, humans wouldn’tknow what to do without us, either inside or outside of mathematics. However, what most people have come to think of as points are really markers for points, or “dots” as we like to refer to them. If there is one thing that I need to set straight, for the sake of all points, is this — dots are not points. Please, I beg you, no I absolutely implore you, to listen to me. This is simple fact that often goes overlooked, leading to much unnecessary confusion — dots are not points. Perhaps I should put this another way, just to make sure that things are absolutely clear — points are not dots either.

People like to use dots to symbolize points, perhaps even fantasizing that points are actually infinitesimally small dots, too small for the naked eye to see. However, that is simply not the case! Euclid, friend to all points understood this better than anyone in the West, carefully stating that, “a point is that which has no parts”, making us completely uncuttable. A dot on the other hand, no matter how people might choose to conflate the two of us use them to symbolize us, has dimension and therefore can be cut and sectioned! To think that the two of us could be confused!

In all fairness, I don’t envy the humans, and I have no idea what else one could possibly use to represent us points. So a dot it is. Even though a dot has dimensions. Still, there are drawbacks to this approach, most notably the amount of ink that many (of us) points find ourselves surrounded with, not that anyone notices, mind you. You might expect us to complain that we don’t like being covered in ink, but nothing could be further from the truth. Being di- mensionless has certain advantages, ones that creatures of “higher dimensions” such as yourselves will never be able to fully appreciate or experience. For one thing, we can never get “covered in” anything, ink or otherwise. While it is true that points can be surrounded by ink, water, (or my personal favorite chocolate syrup), these liquids can never actually cover us since we are dimensionless by definition.

What many of you have come to think of as “different colored points” are in fact “different colored dots”, for dots can come in any color you wish, except of course for the ones that are transparent, but that is the exception rather than the norm a different story altogether. Suffice it to say that as dimensionless entities, points experience an altogether different form of existence than what humans do.

However, don’t let our diminutive nature fool you into thinking that we are more delicate than others, we’re not. Having no parts makes us indestructible for all practical purposes, not to mention that we can never be grasped or picked up.

At this point (smile) some of you reading this might secretly wish that you were points instead of human beings, and overall I can’t say that I blame you. However, like any other figure, being a point is not without its drawbacks. I already mentioned the confusion that arises between points and dots. Some points, myself included just accept this as a normal outcome of our existence, due to our dimensionless nature. Others, however, are not quite so content with this situation and end up pining away, always hoping to be noticed. Still others decide to take matters into their own hands, and do everything they can to be seen by hanging out with “higher dimensional” figures.

Like the twins I used to be friends with who are always hanging out with that line. Don’t they realize that the line is no better than us and still can’t be said to actually exist in a physical sense? Personally, if I can’t be seen, I’d rather be a point than a line. While lines and points share many of the same advantages (invisibility, can never get dirty, never get wet, etc.) they can be cut. I have no idea why those lines assume that they are better than us, “higher dimensions” or not. Besides, don’t the lines realize that they are utterly depen- dent upon points? For all of the airs that they put on, I’ve never known a line that wasn’t hanging out with at least two points at the same time, sometimes incorporating many more along their length.

These are the types of things that other points don’t like to talk about. Yet, sometimes at night after a long day I find myself wondering the oddest things. Questions bubble up that aren’t normally considered proper for a point to have. Lately I have found myself wondering how many points could fit across the length of a line. Most points believe that you could naturally fit a larger number of points across a longer line, however, I’m still not convinced that this is the case.

After all, we points don’t have the roundness and fullness of dots, we’re points and dimensionless by definition. Everyone tells me that I’m crazy, but

I have a sneaking suspicion that all the points in the Universe could still fit within the smallest line. The other points tell me that I shouldn’t think of such things, and that I should be just be content with the benefits of my dimension- less existence. Truth be told, I am. I just want to understand what it actually means to be dimensionless and why lines appear to be so dependent upon points.

Such talk isn’t considered proper, after all what dimensionless entity has the right to question the nature of a being of higher dimensions (higher dimen- sional being)? Although I could get in trouble for saying this, I have a hunch that things are much more complicated than many of the other points suspect. Although I have never mentioned this to anyone I secretly wonder whether the lines depend upon us for their existence. Who knows, perhaps some day I’ll bring it up to some of the other points. After all non-dimensionality does have its advantages — it’s not like I can be erased or anything, after all, I’m not a dot. And yes, no matter what some might say I do exist.