stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2004-12-06 05:41 pm
stoutfellow) wrote2004-12-06 05:41 pm
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On Parametrizing Polygons, 1
This is the first of at least three posts discussing the content of the research paper I'm currently writing.
Some of the oldest mathematical theorems known discuss the question of determining when two geometric figures have the same size and shape - that is, when they are congruent. For example, the eighth proposition of Book I of Euclid presents the SSS theorem: if the sides of one triangle are equal in length to the sides of another, then the two triangles are congruent. In physical terms, this says that a triangular frame is stable - you can't change the shape without changing the lengths of the sides. There is no corresponding SSSS theorem for quadrilaterals. A four-sided frame isn't stable, but can flex at the joints - the shape can change, even though the side-lengths aren't changing. Bracing it across one diagonal stabilizes it, though, so an SSSSD theorem would be a possibility. Considering these two examples, we can see that it takes three numbers to describe a triangle, if congruent triangles are considered the same; we say that they determine the triangle up to congruence. But it seems to take five to determine a quadrilateral up to congruence. (It turns out in general that it takes 2n-3 numbers to determine an n-sided figure up to congruence.)
In Book VI, Euclid presents a somewhat looser relation, of similarity, where all that is demanded is that the figures have the same shape (but not necessarily the same size), and gives the AAA theorem: if two triangles have equal angles, they are similar. Note, though, that the angles of a triangle are not independent; they add up to 180 degrees. So, if you know two of the angles, you know the third, and any two of the angles are enough to determine the triangle up to similarity. (It takes 2n-4 numbers to determine an n-sided figure up to similarity.) Again, though, there is no AAAA theorem for quadrilaterals; any two rectangles, regardless of their precise shape, have the same angles - they're all 90 degrees.
This all leads to the problem that started me on this line of research, that of parametrization. Given any polygon, there are various numbers that can be associated with it - the area, the perimeter, the various sides and angles - and the problem is to find a list of numbers that determine the polygon up to congruence, or up to similarity. The lengths of the sides of a triangle might seem to do this, or the sides plus one diagonal of a quadrilateral, but there's one more catch. You might give the sides of a particular triangle as having lengths 3,4,5, but you might give them as 4,5,3, or as 4,3,5, and so on; I want the numbers - the parameters - to be such that two polygons are congruent (or similar) if they have the same list of numbers in the same order. There are ways of getting around that; for triangles, you might insist that the sides be listed shortest to longest, for example, but that's kind of arbitrary.
So here's the problem: fix the number of sides - call it n. Then I want a list of parameters that I can associate with any n-sided figure, so that two such figures are congruent (or similar) if and only if the parameters have the same values on both; and I'd like to have a generic way of setting up the list of parameters, regardless of the value of n.
There's one more thing I should point out. At best, I'd like the parameters to be completely independent - in other words, the values that each parameter can take should be absolutely unaffected by the values that the others take. Unfortunately, this isn't generally achievable. To fall back to SSS as an example, the sides of a triangle are related by an inequality: the sum of any two sides of a triangle has to be at least equal to the length of the third side. (Also, of course, all three sides must be greater than or equal to zero.) This sort of dependence is tolerable. There's a somewhat worse problem that sometimes arises, where there's actually an equation - not just an inequality - connecting parameters, but which doesn't allow you to drop any of the parameters. (For example, if two parameters were connected by an equation like x^2+y^2=1, each value of x would permit two values of y, so we couldn't just "solve for y" and be done with it.) Sadly, that sort of problem doesn't seem to be avoidable.
So, to end this section, I'll just announce the result: I have a systematic way of constructing a list of parameters for n-sided polygons, for any value of n. It's not pretty, and for various reasons it's not entirely satisfactory, but it works. In later posts in this series, I'll discuss, not the method, but some of the results in special cases.
Oh, a postscript: this is not exactly cutting-edge mathematics, and it has, as far as I know, no particular applications, either to the mundane world or in other branches of science or mathematics. I just like parametrizing things. (It probably has something to do with the love of lists I alluded to a while ago...)
Some of the oldest mathematical theorems known discuss the question of determining when two geometric figures have the same size and shape - that is, when they are congruent. For example, the eighth proposition of Book I of Euclid presents the SSS theorem: if the sides of one triangle are equal in length to the sides of another, then the two triangles are congruent. In physical terms, this says that a triangular frame is stable - you can't change the shape without changing the lengths of the sides. There is no corresponding SSSS theorem for quadrilaterals. A four-sided frame isn't stable, but can flex at the joints - the shape can change, even though the side-lengths aren't changing. Bracing it across one diagonal stabilizes it, though, so an SSSSD theorem would be a possibility. Considering these two examples, we can see that it takes three numbers to describe a triangle, if congruent triangles are considered the same; we say that they determine the triangle up to congruence. But it seems to take five to determine a quadrilateral up to congruence. (It turns out in general that it takes 2n-3 numbers to determine an n-sided figure up to congruence.)
In Book VI, Euclid presents a somewhat looser relation, of similarity, where all that is demanded is that the figures have the same shape (but not necessarily the same size), and gives the AAA theorem: if two triangles have equal angles, they are similar. Note, though, that the angles of a triangle are not independent; they add up to 180 degrees. So, if you know two of the angles, you know the third, and any two of the angles are enough to determine the triangle up to similarity. (It takes 2n-4 numbers to determine an n-sided figure up to similarity.) Again, though, there is no AAAA theorem for quadrilaterals; any two rectangles, regardless of their precise shape, have the same angles - they're all 90 degrees.
This all leads to the problem that started me on this line of research, that of parametrization. Given any polygon, there are various numbers that can be associated with it - the area, the perimeter, the various sides and angles - and the problem is to find a list of numbers that determine the polygon up to congruence, or up to similarity. The lengths of the sides of a triangle might seem to do this, or the sides plus one diagonal of a quadrilateral, but there's one more catch. You might give the sides of a particular triangle as having lengths 3,4,5, but you might give them as 4,5,3, or as 4,3,5, and so on; I want the numbers - the parameters - to be such that two polygons are congruent (or similar) if they have the same list of numbers in the same order. There are ways of getting around that; for triangles, you might insist that the sides be listed shortest to longest, for example, but that's kind of arbitrary.
So here's the problem: fix the number of sides - call it n. Then I want a list of parameters that I can associate with any n-sided figure, so that two such figures are congruent (or similar) if and only if the parameters have the same values on both; and I'd like to have a generic way of setting up the list of parameters, regardless of the value of n.
There's one more thing I should point out. At best, I'd like the parameters to be completely independent - in other words, the values that each parameter can take should be absolutely unaffected by the values that the others take. Unfortunately, this isn't generally achievable. To fall back to SSS as an example, the sides of a triangle are related by an inequality: the sum of any two sides of a triangle has to be at least equal to the length of the third side. (Also, of course, all three sides must be greater than or equal to zero.) This sort of dependence is tolerable. There's a somewhat worse problem that sometimes arises, where there's actually an equation - not just an inequality - connecting parameters, but which doesn't allow you to drop any of the parameters. (For example, if two parameters were connected by an equation like x^2+y^2=1, each value of x would permit two values of y, so we couldn't just "solve for y" and be done with it.) Sadly, that sort of problem doesn't seem to be avoidable.
So, to end this section, I'll just announce the result: I have a systematic way of constructing a list of parameters for n-sided polygons, for any value of n. It's not pretty, and for various reasons it's not entirely satisfactory, but it works. In later posts in this series, I'll discuss, not the method, but some of the results in special cases.
Oh, a postscript: this is not exactly cutting-edge mathematics, and it has, as far as I know, no particular applications, either to the mundane world or in other branches of science or mathematics. I just like parametrizing things. (It probably has something to do with the love of lists I alluded to a while ago...)
no subject
Neat!! I'm looking forward to the next posts! :-)
Any quadrilateral is made up two triangles, yes? Is any polygon made up of triangles? I'm thinking, yes. And then wondering if that goes towards a solution to the problem. Probably not, but just thought I'd ask! Like I said, I'm finding this really interesting. I haven't worked the math part of my brain for a long time, and it's fun! :-)
Does mathematics have to have clear immediate applications to make it "cutting edge"? (real question, I have no idea what would make math cutting edge)
*refrains from making jokes about the math of knives, swords, and paper-cuts*
no subject
Does mathematics have to have clear immediate applications to make it "cutting edge"?
No, I was making two separate admissions. On the one hand, this isn't a very active branch of mathematics; certainly it's not an area which attracts the really high-powered mathematicians. On the other (and this is kind of related to the first point), results in this area aren't going to have immediate implications for other branches of mathematics, or for the rest of the world. Its only attractions are aesthetic ones, and rather small-scale ones at that. (Hey, somebody's got to polish the silver, after all...)
no subject
Interesting. I hope you can give some idea of the way you're solving the problem. But I'm assuming that might be too difficult with the math and all, interpreting your parenthetical remark...
Geometric meaning of parameters...that's a number that represents what color the polygon would most like to be... Sorry feeling a bit silly!
Its only attractions are aesthetic ones, and rather small-scale ones at that. (Hey, somebody's got to polish the silver, after all...)
Ah, I see. But hey, I find that really neat too. Aesthetics is good!! It's also the equivalent of basic research in my field ---> knowledge for the sake of expanding the mind and adding to the knowledge base of the world.
If you don't mind too much, what are the areas that attract high-powered mathemeticians? Just curious.
no subject