A strange look here that I find as interesting as to start another series which I am working on now. The effect can be generated by bringing the film to constant temperatures of about 30-32°C (86-90°F) for about a week. it seems that the layer of photo-sensitive particles starts undergoing some decay process and by controlling this process one could achieve a more or less strong "inking-like" look, or whatever you would name that.
The idea for the image was that kind of π as an irrational number that those branches formed. (Actually it is even worse than simply irrational but let's stay at normal levels of insanity for now.) The mathematical meaning of irrationality is much more surprising than all its "magical" interpretations that have to do with the "illogical" and similar. It simply says that some given quantity cannot be represented as a *ratio* of integers. It sounds like "so what?" at first, but think of it. The set of rational numbers like 3/4 or 1/7 is dense, which means that its members are "infinitely" near each other on the straight line of numbers. OK, then there is... no space left between them for other numbers, right? Wrong! There are even more infinite many numbers between the members of the dense set of rationals. (Thus, not rational = irrational.) And this set of the irreational numbers is so unbelievably dense, so infinite, that the dense infinite set of rationals has the weight of zero.nada compared to it. You see, that's mathematics! We find something even more infinite than simply.. infinite!!!
And don't think that this is only for guys that have nothing better to do. No, actually you also mess up with these strange numbers each and every day. Ask your camera, how much π does it contain? You will be surprised that it works at all though many of the numbers involved in its design and construction are irrational and thus not exactly representable or calculable. ;-)
Mathematics is not free in rigorosity of proof. It demands flawless, "gapless" logical concequence. It is free in mapping ideas to the world. If you think of "shapes" then there are shapes provided you proved them (And also provided you can accept their unexpected properties ;-))
As von Neumann once said: With mathematics you don't understand the world. You get used to it! ;-)
It exists in our minds and this is enough existence for me. You know, mathematics is the the science that you can still do when you wake up in the morning and you see that the universe is gone. ;-)
Another good example. You take a sphere with a volume of 1 liter. OK, that's it. You can't squeeze more volume into that. Think of having a spherical bottle with a volume of 1 liter. When we finished that 1 liter of whisky, that was it! (Unfortunately!!! ;-)) It was the finite volume of just and exactly one damned liter. Well, a guy named Tarski proved that it is possible to purely geometrically take such a sphere, divide it into pieces, turn them and shuffle them around, and then... construct two such spheres out of the pieces of one sphere! Don't be happy, this is purely geometrically! ;-) But what does that mean? It means that the notion of shape and volume is a bit more complex that initially thought. One of my favorites, the Banach-Tarski paradoxon, which is not a paradoxon at all. It only says: "Watch out when you cope with such things. They are a not as perfect as they seem."
Still strange to see that the mathematical method does offer all good tools for grasping the universe. All and the only good tools. All others are presenting me unholdable nonsence but mathematics gives us good means in quietness and unspectacularity. Einstein himself was wondering why it is mathematics that allows some understaning of the physical world though it was never developed for that purpose. And this is also the reason for my deep love for that science. It is free. It is "useless" if you wish, since it was not meant for calculating things and making cameras and the like. And then in some very very strange way, it proved to be the unspectacular queen in rags.
So, to your question about shape and form. It seems that such notions are inherently connected to the notion of "smootheness" in the sence of continuity. When you follow some line with your finger you tend to think that your finger "was over" all points between the start and the end. Is it true in the physical way? This is irrelevant to mathematics. It can only tell you what the consequences are, once you believed that it is true. (I.e. once you accepted the axiom of continuous real space.) It can also tell you what follows if you don't believe that. But it's up to you to decide which idea of yours best represents "reality".
This mental limit of not being able to think about even more numbers between the rational was also a mistake that ancient Greeks did. When Pythagoras formulated the well known theorem it came out that there exist also numbers that are... different. The hypothenuse of the rightangled triangle with two perpendicular sides of equal length 1 is then sqrt(2) - sqrt denotes the square root. Now, if the rational numbers completely cover the line of numbers and leave no "unoccupied gaps" between them, then that sqrt(2) has to correspond, to be equal with some rational number, ey? It must fall onto some distinct rational. But it was shown that there is no rational number that is equal to sqrt(2). Sqrt(2) is... irrational. So... there are still other numbers between thos rationals that can completely cover the line of numbers? The ancient Greeks almost lynched Pythagoras for that! ;-)
But so it is. Starting from quite distinct, easily understandable, almost "feelable" counting quantities you end up like that. For remember: The sides of that rightangled triangle were... plain wonderful integers. Nothing unsual. The length of the perpendicular sides was simply 1. And the hypothenuse proves to be impossible to grasp.
Some time later another very very big one, Cantor, proved that... there are infinitely more numbers (and thus quantiny labels) right between those that apparently cover the "whole range". You could keep up magnifying and magnifying some given part of the line of numbers on the previous attachment, somewhere between, say 7/5 (=1.4) and 3/2 (=1.5). You would never find a rational equal to sqrt(2). But if the sequence of those rational points covers that line without gaps, then... where is that damned sqrt(2)? And much more where are the even more infinite many irrationals. The answer is, you can't really locate them exactly though they *are* precisel defined. ;-) That set of *real numbers* is unbelievable more dense than anything. (They are called "real" but this is not a label for physical reality.)
Back to real world and we are counting "distances". The physical entity of space doesn't need to be such a continuum in the sense of those real numbers. It could also be dicrete, that is not containing really all *imaginable* numbers. The real number sqrt(2) could be just a consequent continuation of a logical thought, but it doesn't need to "really exist" as a quantity, a distance, a weight, whatever.
Well... hmmm, hard to summarize in just a few words, but mathematics don't care about the "physical". This is one of the big misundestandings around. Mathematics is not for "explaining" the physical world. This is the subject of physics. Mathematics is for analyzing the very ideas that *you* carry in mind, no matter if there is some "physical correspondance" to that. In other words, there sits good old Visar and thinks about the distance between two points and then... then he applies that idea to a roadmap for finding out how far two cities are from each other. Good concept, that distance, but what is that? It is the "straigh line connecting two points". Great! Only that... a line has zero thickness! You were able to think of that, but is there anything with zero thickness on this world? ;-) No, we have just wires and threads that can be really very very thin, but they do have a thickness greater than zero. So, that concept in your mind proved to be extremely powerful and useful for applications in real world, or else you wouldn't know what the distance is from, say Beograd to Ljubljana. (What a city, by the way!) It does have to do something with "reality" but it is not physical reality itself. It is some kind of "model", "idealisation", necessary for subliming the few most imporant things about it. This method, naive platonism, is an extremely powerful tool but it doesn't really care about physical reality. Strange thing is that then, the "thought out" things show an intimate relationshiop to what "is". Still they may contain also some moments of Disneyland. ;-)
About the current issue. We had to develop some kind of "counting" things. You know, 1, 2, 3... and so on. Then we found addition, subtraction, , multiplication... *division*! He, hee, a father has two apples for three kids, and so each kid gets... 2/3 of an apple. OK, well done. The idea of rational numbers was born. And somebody asked, how many rational numbers do exist? After some search it was clear that there are infinite many of them even on a finite piece of the "line of numbers". (Attachment.)
So, these funny things, the rationals, seem to completely cover any possible "distance", or quantity ey? If there are infinite many of them, then they occupy completely that line of numbers. Or don't they? For each pair of unequal rationals you can place infinite many of them between the former two. No place left for anything else?
I am in awe with your understanding of physical content of matter, Nick- and even more with the description and the cracking of it- in and through what makes, and yet, devides the physical contents- thus, shape!! but- there is something between this 'infinity'- that is the actual distance between two points, which, what i can extract from here, is infinite many numbers close and far between two objects, and that actually compose the form- or is it just relativisation of the issue?! in other words, the form that we percieve with our senses is that infinitely fluid, changable, amorphic- or, is it just that space pressed in between, infinitely...... grrrrrrr, i hope i am clear with this! :)
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