Polygons I: Criteria
Mar. 5th, 2012 07:30 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI've been promising to talk a little about my research. If you're interested, follow me under the cut.
The story begins last summer. For some reason, I looked up "Quadrilateral" on Wikipedia. If you go there, you will see a number of different kinds of quadrilaterals: trapezoids, parallelograms, kites, orthodiagonals, and so on, neatly laid out so that you can tell which kinds are subsets of which other kinds. Gazing at that diagram, I began to wonder: why is it that list? Are there other kinds of quadrilaterals that we should also be looking at? Is there some systematic way of identifying candidates?
Why are those kinds of quadrilaterals on the list? Obviously, because we can prove theorems about them. But how can we prove theorems about a class of quadrilaterals, if we haven't identified the class? So.... Long story short, I began hunting for a procedure for identifying possibly-interesting classes, not just of quadrilaterals but of n-gons, for any value of n greater than 2.
What shall we look for? First, I'm not interested in classes that are defined negatively - e.g., scalene triangles - or by means of inequalities - e.g., convex quadrilaterals. (For the latter, the precise description is that each pair of opposite vertices is on opposite sides of the other diagonal, but that involves qualities like "negative/positive" - i.e., inequalities.) For the most part, there's very little that can be said to distinguish a member of that kind of class from a non-member; and in any case, my inclination towards algebra leads me to look at equations, not inequalities.
Second - wait, let's define what an n-gon is. Let Zn be the integers mod n; think of them as {0,1,...,n-1}, with the proviso that the next number after n-1 is 0 again. Let E be the Euclidean plane. Then an n-gon is simply a function from Zn to E. The image of 0 is the zeroth vertex; the image of 1 is the first; and so on. There are a couple of things I should draw attention to. First, I make no assumptions about how the vertices are arranged; in particular, different vertices could wind up as the same point. I'll even allow n-gons all of whose vertices coincide! (They're not very interesting, but for completeness' sake I permit them.) Second, each of these n-gons has a specified zeroth - "starting" - vertex, and a specified order in which the vertices proceed. In other words, the quadrilateral ABCD is not the same as the quadrilateral BCDA (different starting vertex), or ADCB either (same starting vertex, but traced in the opposite order). It may seem silly to distinguish things that finely, but it turns out to be worth it.
So, second. If a class contains a particular n-gon, then it ought to contain every n-gon that's similar to it. In other words, if you translate, rotate or reflect the n-gon, that shouldn't make any difference; nor should it matter if you scale it up (or down). Technically, we say that the class must be "closed under similarity".
Third - well, besides mucking around with the location and size of the n-gon, we can also change the starting vertex or reverse the order of the vertices. Shouldn't our classes be unaffected by this as well? Shouldn't they be "dihedrally closed", as the phrase goes? Well... no. Think, for instance, of isosceles triangles. If I tell you that ABC is an isosceles triangle, precisely what have I told you? That two of its sides are the same length, but which two? There are three different possibilities, and it turns out to be simpler if we keep them separate. (I've gotten into the habit of saying "A-isosceles", for instance, to mean that the sides AB and AC have the same length.) So I'm not going to ask for dihedral closure. Instead, I'll make a weaker demand: that the class be closed under some "flip". A flip is the act of reversing the order, possibly changing the starting vertex: ABC becomes ACB (the "A-flip"), CBA (the "B-flip"), or BAC (the "C-flip"). There will turn out to be two kinds of classes: "Type I" classes (like parallelograms or equilateral triangles) are dihedrally closed; "Type II" classes (A-isosceles triangles, or one of the two kinds of trapezoid - I'll leave it to you to figure out what those two kinds are!) are only closed under a flip.
Fourth: I'm an algebraist as well as a geometer, and so I make an algebraic condition. The class must be defined by a polynomial in the vertices. More precisely: there has to be a polynomial, depending on the coordinates of the vertices, with the property that the n-gons in the class are those for which the polynomial works out to be zero. For instance, the lengths a,b,c of the sides of a triangle are not polynomials in the vertices (the distance formula involves square roots), but their squares are, and the A-isosceles triangles are those for which b2-c2=0. Trapezoids have two sides parallel, say AB and CD; take the vector from A to B, and the one from C to D, and they must be proportional - in other words, e.g., the determinant of the matrix whose rows are those two vectors must be zero; and that, again, is a polynomial in the vertices.
Now, my final condition. Type I classes are OK as they stand, but I need something special for Type II classes. The polynomials for A-isosceles, B-isosceles, and C-isosceles triangles are b2-c2, c2-a2, and a2-b2 respectively. If you take these three polynomials and add them up, you get 0; that implies that, if two of them are zero, so is the third. That's my condition: the polynomial may change when you reorder the vertices, but if you take all of the results, then if any two of them are zero, they all are. That's kind of complicated-sounding, but from an algebraic point of view it's perfectly natural. (The technical language is that the polynomial must belong to an "irreducible subspace".)
So those are my conditions. Almost every class of triangles, quadrilaterals, or higher n-gons that I know of satisfies them, with the exceptions mentioned in the first criterion.
Now what? How do I use these conditions to find interesting classes? I'll talk about that in the next post on this subject.
The story begins last summer. For some reason, I looked up "Quadrilateral" on Wikipedia. If you go there, you will see a number of different kinds of quadrilaterals: trapezoids, parallelograms, kites, orthodiagonals, and so on, neatly laid out so that you can tell which kinds are subsets of which other kinds. Gazing at that diagram, I began to wonder: why is it that list? Are there other kinds of quadrilaterals that we should also be looking at? Is there some systematic way of identifying candidates?
Why are those kinds of quadrilaterals on the list? Obviously, because we can prove theorems about them. But how can we prove theorems about a class of quadrilaterals, if we haven't identified the class? So.... Long story short, I began hunting for a procedure for identifying possibly-interesting classes, not just of quadrilaterals but of n-gons, for any value of n greater than 2.
What shall we look for? First, I'm not interested in classes that are defined negatively - e.g., scalene triangles - or by means of inequalities - e.g., convex quadrilaterals. (For the latter, the precise description is that each pair of opposite vertices is on opposite sides of the other diagonal, but that involves qualities like "negative/positive" - i.e., inequalities.) For the most part, there's very little that can be said to distinguish a member of that kind of class from a non-member; and in any case, my inclination towards algebra leads me to look at equations, not inequalities.
Second - wait, let's define what an n-gon is. Let Zn be the integers mod n; think of them as {0,1,...,n-1}, with the proviso that the next number after n-1 is 0 again. Let E be the Euclidean plane. Then an n-gon is simply a function from Zn to E. The image of 0 is the zeroth vertex; the image of 1 is the first; and so on. There are a couple of things I should draw attention to. First, I make no assumptions about how the vertices are arranged; in particular, different vertices could wind up as the same point. I'll even allow n-gons all of whose vertices coincide! (They're not very interesting, but for completeness' sake I permit them.) Second, each of these n-gons has a specified zeroth - "starting" - vertex, and a specified order in which the vertices proceed. In other words, the quadrilateral ABCD is not the same as the quadrilateral BCDA (different starting vertex), or ADCB either (same starting vertex, but traced in the opposite order). It may seem silly to distinguish things that finely, but it turns out to be worth it.
So, second. If a class contains a particular n-gon, then it ought to contain every n-gon that's similar to it. In other words, if you translate, rotate or reflect the n-gon, that shouldn't make any difference; nor should it matter if you scale it up (or down). Technically, we say that the class must be "closed under similarity".
Third - well, besides mucking around with the location and size of the n-gon, we can also change the starting vertex or reverse the order of the vertices. Shouldn't our classes be unaffected by this as well? Shouldn't they be "dihedrally closed", as the phrase goes? Well... no. Think, for instance, of isosceles triangles. If I tell you that ABC is an isosceles triangle, precisely what have I told you? That two of its sides are the same length, but which two? There are three different possibilities, and it turns out to be simpler if we keep them separate. (I've gotten into the habit of saying "A-isosceles", for instance, to mean that the sides AB and AC have the same length.) So I'm not going to ask for dihedral closure. Instead, I'll make a weaker demand: that the class be closed under some "flip". A flip is the act of reversing the order, possibly changing the starting vertex: ABC becomes ACB (the "A-flip"), CBA (the "B-flip"), or BAC (the "C-flip"). There will turn out to be two kinds of classes: "Type I" classes (like parallelograms or equilateral triangles) are dihedrally closed; "Type II" classes (A-isosceles triangles, or one of the two kinds of trapezoid - I'll leave it to you to figure out what those two kinds are!) are only closed under a flip.
Fourth: I'm an algebraist as well as a geometer, and so I make an algebraic condition. The class must be defined by a polynomial in the vertices. More precisely: there has to be a polynomial, depending on the coordinates of the vertices, with the property that the n-gons in the class are those for which the polynomial works out to be zero. For instance, the lengths a,b,c of the sides of a triangle are not polynomials in the vertices (the distance formula involves square roots), but their squares are, and the A-isosceles triangles are those for which b2-c2=0. Trapezoids have two sides parallel, say AB and CD; take the vector from A to B, and the one from C to D, and they must be proportional - in other words, e.g., the determinant of the matrix whose rows are those two vectors must be zero; and that, again, is a polynomial in the vertices.
Now, my final condition. Type I classes are OK as they stand, but I need something special for Type II classes. The polynomials for A-isosceles, B-isosceles, and C-isosceles triangles are b2-c2, c2-a2, and a2-b2 respectively. If you take these three polynomials and add them up, you get 0; that implies that, if two of them are zero, so is the third. That's my condition: the polynomial may change when you reorder the vertices, but if you take all of the results, then if any two of them are zero, they all are. That's kind of complicated-sounding, but from an algebraic point of view it's perfectly natural. (The technical language is that the polynomial must belong to an "irreducible subspace".)
So those are my conditions. Almost every class of triangles, quadrilaterals, or higher n-gons that I know of satisfies them, with the exceptions mentioned in the first criterion.
Now what? How do I use these conditions to find interesting classes? I'll talk about that in the next post on this subject.
