Polygons II: Transformations
Mar. 8th, 2012 02:30 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI have to go into the weeds for this next bit. Buckle up.
Suppose you have a set S. Maybe it's a set with some structure - a vector space, a topological space, the Euclidean plane. A transformation on S is simply a one-to-one and onto function from S to itself; if S has structure, the function should "respect" that structure. (If you've taken linear algebra, think of linear transformations; if topology, continuous functions; etc.) A transformation group on S is a set G of transformations, with the following properties.
Now, recall that an n-gon is a function from Zn to the Euclidean plane E. We have two transformation groups to work with. Sim is a transformation group on E, containing all the similarity transformations: translations, rotations, reflections, and dilations, plus a couple of other kinds derived from these. Dn is a transformation group of Zn, containing the transformation r, which takes any element j to j+1 (remembering that we're working mod n, so (n-1)+1=0), and f, which takes j to n-j (and 0 to 0). It also contains powers of r, taking j to j+k mod n, for some k, and other flips besides f, taking j to k-j mod n for some k.
The groups Sim and Dn can be regarded as transformation groups on the set of n-gons as well. Suppose that T is in Sim - let's say, it's rotation by ninety degrees around a point P - and p is an n-gon. Then we can compose them: Tp is the n-gon p, rotated by ninety degrees around P, and similarly for any other element of Sim. Likewise, if we take an element of Dn - say r - we define rp to be the composition of p with r-1. (We use the inverse for technical reasons; things work better if we do it that way.) So, now, we're regarding similarity transformations, not as taking points of E to points of E, but n-gons to n-gons, and likewise for the "dihedral" transformations in Dn.
Now we have to go one step further. We're interested in polynomials in the vertices of n-gons, which we'll think of as functions from the set Pn of all n-gons to the complex numbers C. (The functions we're really interested in are real-valued, but framing things in the complexes is, heh, simpler.) We play the same trick as before: if F is one of our polynomials and T is a similarity transformation, we define TF to be the composition of F with T-1, and likewise for dihedral transformations.
Let's do an example. Take n=3, so we're talking about triangles, and let F be the function b2-c2. To be precise, F is the square of the distance from vertex 0 to vertex 2, minus the square of the distance from vertex 0 to vertex 1. Now, let p=ABC be a triangle - A is vertex 0, B is vertex 1, and C is vertex 2 - and let T be a dilation, expanding everything by a factor of 2. Then T-1p is ABC scaled down by a factor of 2, and TF = 1/4 F. (It's 1/4, because F involves the squares of the lengths of the sides.) Meanwhile, f-1p is the triangle ACB; vertex 1 is now C, and vertex 2 is B. Therefore fF is c2-b2, or -F.
We've set up the machinery we need in order to start; we'll start using them next time. Hope you aren't too lost....
Suppose you have a set S. Maybe it's a set with some structure - a vector space, a topological space, the Euclidean plane. A transformation on S is simply a one-to-one and onto function from S to itself; if S has structure, the function should "respect" that structure. (If you've taken linear algebra, think of linear transformations; if topology, continuous functions; etc.) A transformation group on S is a set G of transformations, with the following properties.
1) The identity function, which takes every element to itself, is in G.
2) If f and g are in G, so is their composition.
3) If f is in G, so is the inverse transformation f-1
Now, recall that an n-gon is a function from Zn to the Euclidean plane E. We have two transformation groups to work with. Sim is a transformation group on E, containing all the similarity transformations: translations, rotations, reflections, and dilations, plus a couple of other kinds derived from these. Dn is a transformation group of Zn, containing the transformation r, which takes any element j to j+1 (remembering that we're working mod n, so (n-1)+1=0), and f, which takes j to n-j (and 0 to 0). It also contains powers of r, taking j to j+k mod n, for some k, and other flips besides f, taking j to k-j mod n for some k.
The groups Sim and Dn can be regarded as transformation groups on the set of n-gons as well. Suppose that T is in Sim - let's say, it's rotation by ninety degrees around a point P - and p is an n-gon. Then we can compose them: Tp is the n-gon p, rotated by ninety degrees around P, and similarly for any other element of Sim. Likewise, if we take an element of Dn - say r - we define rp to be the composition of p with r-1. (We use the inverse for technical reasons; things work better if we do it that way.) So, now, we're regarding similarity transformations, not as taking points of E to points of E, but n-gons to n-gons, and likewise for the "dihedral" transformations in Dn.
Now we have to go one step further. We're interested in polynomials in the vertices of n-gons, which we'll think of as functions from the set Pn of all n-gons to the complex numbers C. (The functions we're really interested in are real-valued, but framing things in the complexes is, heh, simpler.) We play the same trick as before: if F is one of our polynomials and T is a similarity transformation, we define TF to be the composition of F with T-1, and likewise for dihedral transformations.
Let's do an example. Take n=3, so we're talking about triangles, and let F be the function b2-c2. To be precise, F is the square of the distance from vertex 0 to vertex 2, minus the square of the distance from vertex 0 to vertex 1. Now, let p=ABC be a triangle - A is vertex 0, B is vertex 1, and C is vertex 2 - and let T be a dilation, expanding everything by a factor of 2. Then T-1p is ABC scaled down by a factor of 2, and TF = 1/4 F. (It's 1/4, because F involves the squares of the lengths of the sides.) Meanwhile, f-1p is the triangle ACB; vertex 1 is now C, and vertex 2 is B. Therefore fF is c2-b2, or -F.
We've set up the machinery we need in order to start; we'll start using them next time. Hope you aren't too lost....
