return to homepage
If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many noms de plume. If you are a Paypal user, there is no fee; so it might be worth your while to become one. Otherwise they will rob us 33 cents for each transaction.
Does Size Matter?
by Miles Mathis
In an email from one of my readers, I was asked to explain the ratio of dimensions varying with size, specifically the ratio of volume to surface area. I have stated in various papers that size is relative. But the way volume changes in relation to surface area as an object gets larger appears to conflict with this statement. We know that small objects freeze faster than large objects, for instance, so the difference is not only an abstract or mathematical difference. It is real. This is my resolution of the apparent contradiction.
I think I have the answer you were looking for. Start with your elephant/mouse example. You said the elephant had more heat to start with. Well, that is exactly the reason he freezes slower. His volume to surface area ratio is greater than the mouse, so he doesn't have as much area to dissipate the heat with, compared to the mouse. So the standard theory is correct. However, your question goes beyond that. Your question is why? If it is true empirically, how is it true logically? How do we explain it? We are taught that size is relative, so how can the ratio change? I have showed in my papers that size is not an absolute thing. You can expand or shrink things indefinitely. How can we believe both these things at the same time? It is a good question, and I don't know that anyone has ever given it a thorough answer. You say that no one has ever given you a thorough answer, at any rate, so I will attempt to do so.
In one sense, size is completely relative and the ratio of volume to surface never changes. In another sense, size does make a difference, as we have seen with the mouse and the elephant. It turns out that whether you get the ratio to stay the same or not depends on how you measure it. That is to say, it is an operational difference. And the operational difference is determined by the facts at hand. You use different operations to describe different physical circumstances.
Let's go back to the elephant to see what I mean. The entire reason that the heat problem works with the elephant and the mouse is that heat is a function of molecular motion. Molecules happen to be a certain size. Meaning, an elephant does not have larger molecules than a mouse. Now, if we let the molecules be our measuring rod, then they will act somewhat differently in a larger volume than a smaller one. The ratio of volume to surface area changes because we are increasing the size of the object—from mouse to elephant—but we are not increasing the size of our measuring rod, the molecule. But, if molecules got bigger as the object got bigger, then the ratio would stay the same. If molecules got bigger as the object they made up got bigger, then smaller objects would not freeze faster.
The key to the problem above is that molecules have an absolute size. You cannot increase or decrease the size of molecules. They are bigger than atoms and smaller than mice, and that is that. This fact determines the operation of measurement and therefore determines the answer to the problem. But this fact is peculiar to the problem and may not be generalized. Which is just to say that in some problems you may be dealing with objects that can theoretically be increased or decreased in size, unlike molecules.
We can’t generalize, but we can extend the finding above to many (or most) physical problems. We can do this for two reasons. One: most physical objects have a standard size that does not vary. Elephants and mice come in one size (with minor variations), like molecules. Two, numbers themselves work just like molecules. Numbers are a constant size. Yes, numbers get bigger as you go from 1 to 10, say. But the number 1 is always the same size, and so is the number ten. The number 1 does not get bigger. It is a constant. A giant mathematician and a tiny mathematician will assign the same value to 1. The number 1 is your measuring rod, just like the molecule was the measuring rod before. Because the number 1 stays the same size, the ratio of volume to surface area changes. But if the number 1 got larger as the object got larger, then the ratio would stay the same.
This is a bit hard to grasp at first, so try thinking of it like this. In all these problems with elephants and mice, we have been doing measurements. This is applied math, not pure math. In applied math you must define your dimensions. If you use the number 1, you must assign it to a dimension, as you pointed out. It must be one meter or one inch or one hand or one foot. "One" by itself in applied math is pretty much meaningless. So let us apply the number 1 in a non-standard way, and then follow it through a problem. This will keep us from making the same old assumptions unconsciously. Let us say we have a tiny elephant the size of a mouse. We are going to blow him up to a large size, the size of a normal elephant, and see if the ratio of surface area to volume changes. To do this, we assign the number one to the little elephant's foot. The diameter of one of his feet is 1 foot. Now we blow him up to the size of a real elephant. Has the ratio changed or not? Well, it depends if we measure using the tiny elephant's foot or the new big elephant's foot. When we blew the elephant up, his foot got larger too, obviously. So in a sense, our number 1 got larger. The length 1 foot got much larger. If we measure using the new big foot, and the new big number 1, then the ratio is not changed. In fact, there is no way to know the elephant got larger at all, just by measuring the elephant.
This is what I mean when I say that size is relative. I mean that size is impossible to determine except relative to outside considerations. We need something that did not get bigger in order to measure the thing that did get bigger. We can go back and use the tiny elephant's foot, in which case the ratio of surface area to volume will have changed. The ratio has changed because we have used a dimension that is in some sense external to our dimensions in question. The dimensions in question are the surface area and volume of the large elephant. But the length of the foot of the tiny elephant is external to this object. It is not even guaranteed to exist.
What I mean by that last sentence is that if everything in the universe were expanding, you could not know it by direct measurement. Why? Because if your elephant is expanding, then at any time after t0 the tiny elephant no longer exists. You can't use his tiny foot to measure anything since the tiny foot has become the big foot. The tiny foot is gone. If you have it marked on a ruler, the ruler is bigger. If you took a picture of it, the picture has gotten bigger. If you have it on a computer chip, the chip has gotten bigger. You could only know about the expansion by side-effects. I have shown that gravity is one of these side-effects. Apparent bending of starlight is another.
I hope I have been of some help to you and your students.
To see how this applies to a real problem in contemporary physics, go to my paper on Gravity at the Quantum Level, where I show that a mistake in scaling accelerations led to an error of 1022 in quantum electrodynamics.