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Saturday, August 21, 2010
Monday, August 2, 2010
Three Sticks, Mathematics, and "the Real World"
The point of this post is twofold: First, I want to demonstrate the surprising applicability of mathematics to the physical world. Second, I want to convince the reader that most anybody is capable of doing mathematical research. We will attempt to accomplish both goals with the aid of an extremely simple experiment requiring only three sticks (or three pieces of uncooked spaghetti, three wires, three bottles, etc.). Here is the problem:
Let's address the most obvious question first, mainly, "Why is this stick problem supposed to be interesting?". Well, exactly because it represents a type of question that humans have been tackling since the dawn of scientific thought. It is related to questions like: "This stone is a mixture of an equal number of three different types of atoms. The total mass of the stone is two kilograms. What does this tell me about the masses of the different types of atoms mixed in the stone?", or "Given the total energy of a burst of light, what can we say about the energies of each individual wavelength in the light burst?". Indeed, in the case of a question about the weight of a stone, we can translate the question into one about 3 sticks by considering the stone to be the collection of all 3 sticks laid end to end, and the three different types of atoms of the stone to be the three individual sticks. We may now simply translate all questions about the weight of the stone's atoms into questions about the length of our 3 sticks by replacing "weight" with "length" everywhere in the question. Thus, we can see that any techniques we develop for telling us about individual sticks can be used to tell us about individual types of atoms in a stone. All we must do is (i) translate the question about types of atoms in a stone to be one about sticks laid end to end, (ii) find out what we can about the length of any individual sticks, and, finally, (iii) translate the answers about the length of individual sticks back into statements about the mass of individual types of atoms. Similarly, we could use methods for finding out about the lengths of our 3 sticks to find out about the energies of 3 individual wavelengths making up a light burst when given only a single collective energy measurement. Furthermore, there is nothing really special about the number "3" in this post. There could be any number of sticks, types of atoms, or wavelengths, as the case may be, and the same ideas would still work. This is the beauty of mathematical abstraction!
Having decided that there is indeed a point to contemplating our simple 3-sticks problem, we are free to labor on without fear of wasted effort. More specifically, we are now free to consider the previously stated question: "What can we tell about the individual lengths of our 3 sticks given that their total combined length is 3 feet?". As we shall see, the answer is "Quite a bit, assuming that you ask about the right individual sticks."
Let's begin by asking how long the shortest of our three sticks can be. We can help answer this question by imagining an extreme situation. In the most extreme case the shortest stick will have length almost 0 while the other two sticks make up the entire collective 3 foot length of the three sticks laid end to end. Clearly, then, the shortest of the three sticks can be arbitrarily small!
Perhaps the most natural next question is, "How long can the shortest stick be?". This question can again be answered with the help of an example situation. Certainly it is possible that all the sticks are 1 foot long. In this case any of the sticks could be considered the shortest, so clearly the shortest stick can be as long as 1 foot. That's OK, but is it possible for the shortest stick to be more than one foot long? Well, if the shortest stick is longer than a foot, then both of the other two sticks also have to be longer than a foot. However, if all three sticks are longer than a foot then their total length when laid end to end in a straight line must be longer than three feet! This is impossible since the collective end to end length of the 3 sticks is three feet (and no longer). We are forced to the conclusion, then, that the shortest stick can not be more than one foot long because if it where, the total length of the three sticks laid end to end would have to be more than three feet.
What about the longest of the three sticks? How long can it be? Well, similar to above, we can imagine an extreme example where two of the three sticks are mere specks. In such a situation the third stick must by nearly three feet long on its own. Therefore, the longest stick can be arbitrarily close to three feet long. How short can the longest stick be? Well, here we can again use the extreme case where all sticks have the same length of 1 foot. When this happens any of the three sticks can be considered the longest. Hence, the longest stick can be as short as 1 foot. Any shorter, though, and the three sticks together would have to measure less than three feet when laid end to end. Therefore, the longest stick must be at least one foot long.
This last realization brings us to our first point: Having performed only a few thought experiments we have to come to a concrete conclusion about the real world. Any time three sticks laid end to end in straight line collectively measure 3 feet in length, at least one of three sticks (i.e., the longest) MUST be at least one foot long! There is no getting around it -- I dare the reader to try to violate this mental discovery in their kitchen with three pieces of uncooked spaghetti. You will fail! Mathematics -- and, more generally, thinking -- can tell us about "the real world"!
The second point is this: mathematics and science are largely conducted in a highly abstracted language which can be difficult to learn. However, much of the real thought behind what mathematicians and scientists do is no more complicated than what we have just been using in this post to discuss our three sticks. If you can read and understand everything written here, you can -- with enough hard work and study -- be a competent and productive scientist.
Suppose we find a flat surface, lay our three sticks end to end in a straight line, and then measure the total length spanned by the three sticks to be 3 feet. Given only this information about the total measure of the 3 sticks we want to find out everything we can about the lengths of each of the three individual sticks.
Let's address the most obvious question first, mainly, "Why is this stick problem supposed to be interesting?". Well, exactly because it represents a type of question that humans have been tackling since the dawn of scientific thought. It is related to questions like: "This stone is a mixture of an equal number of three different types of atoms. The total mass of the stone is two kilograms. What does this tell me about the masses of the different types of atoms mixed in the stone?", or "Given the total energy of a burst of light, what can we say about the energies of each individual wavelength in the light burst?". Indeed, in the case of a question about the weight of a stone, we can translate the question into one about 3 sticks by considering the stone to be the collection of all 3 sticks laid end to end, and the three different types of atoms of the stone to be the three individual sticks. We may now simply translate all questions about the weight of the stone's atoms into questions about the length of our 3 sticks by replacing "weight" with "length" everywhere in the question. Thus, we can see that any techniques we develop for telling us about individual sticks can be used to tell us about individual types of atoms in a stone. All we must do is (i) translate the question about types of atoms in a stone to be one about sticks laid end to end, (ii) find out what we can about the length of any individual sticks, and, finally, (iii) translate the answers about the length of individual sticks back into statements about the mass of individual types of atoms. Similarly, we could use methods for finding out about the lengths of our 3 sticks to find out about the energies of 3 individual wavelengths making up a light burst when given only a single collective energy measurement. Furthermore, there is nothing really special about the number "3" in this post. There could be any number of sticks, types of atoms, or wavelengths, as the case may be, and the same ideas would still work. This is the beauty of mathematical abstraction!
Having decided that there is indeed a point to contemplating our simple 3-sticks problem, we are free to labor on without fear of wasted effort. More specifically, we are now free to consider the previously stated question: "What can we tell about the individual lengths of our 3 sticks given that their total combined length is 3 feet?". As we shall see, the answer is "Quite a bit, assuming that you ask about the right individual sticks."
Let's begin by asking how long the shortest of our three sticks can be. We can help answer this question by imagining an extreme situation. In the most extreme case the shortest stick will have length almost 0 while the other two sticks make up the entire collective 3 foot length of the three sticks laid end to end. Clearly, then, the shortest of the three sticks can be arbitrarily small!
Perhaps the most natural next question is, "How long can the shortest stick be?". This question can again be answered with the help of an example situation. Certainly it is possible that all the sticks are 1 foot long. In this case any of the sticks could be considered the shortest, so clearly the shortest stick can be as long as 1 foot. That's OK, but is it possible for the shortest stick to be more than one foot long? Well, if the shortest stick is longer than a foot, then both of the other two sticks also have to be longer than a foot. However, if all three sticks are longer than a foot then their total length when laid end to end in a straight line must be longer than three feet! This is impossible since the collective end to end length of the 3 sticks is three feet (and no longer). We are forced to the conclusion, then, that the shortest stick can not be more than one foot long because if it where, the total length of the three sticks laid end to end would have to be more than three feet.
What about the longest of the three sticks? How long can it be? Well, similar to above, we can imagine an extreme example where two of the three sticks are mere specks. In such a situation the third stick must by nearly three feet long on its own. Therefore, the longest stick can be arbitrarily close to three feet long. How short can the longest stick be? Well, here we can again use the extreme case where all sticks have the same length of 1 foot. When this happens any of the three sticks can be considered the longest. Hence, the longest stick can be as short as 1 foot. Any shorter, though, and the three sticks together would have to measure less than three feet when laid end to end. Therefore, the longest stick must be at least one foot long.
This last realization brings us to our first point: Having performed only a few thought experiments we have to come to a concrete conclusion about the real world. Any time three sticks laid end to end in straight line collectively measure 3 feet in length, at least one of three sticks (i.e., the longest) MUST be at least one foot long! There is no getting around it -- I dare the reader to try to violate this mental discovery in their kitchen with three pieces of uncooked spaghetti. You will fail! Mathematics -- and, more generally, thinking -- can tell us about "the real world"!
The second point is this: mathematics and science are largely conducted in a highly abstracted language which can be difficult to learn. However, much of the real thought behind what mathematicians and scientists do is no more complicated than what we have just been using in this post to discuss our three sticks. If you can read and understand everything written here, you can -- with enough hard work and study -- be a competent and productive scientist.
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