Isomorphism

[wendy]

quickfur was asking on how isomorphism works, eg {3,5} to {3,5/2}

The process is not dissimilar to complex conjucation, but produces real results on both sides. The process replaces in the main, sqrt(x) with -sqrt(x) for class-2 systems.

The simplest form is found in the octagonal (Z4, Q4), and hexagonal (Z6) systems, there are others, like pentagonal (Z5, Q5), and even (J10).

Let's look at Z4. If you start with a square lattice, and then produce an image by rotating it at every vertex of {4,4}, by 45°, you get the vertices of an {8,8/3}. If you number the rays around a point 0 to 7, you can make an octagon by going in steps of 0, 1, 2, 3, ..., 7, and an octagram, by going in steps of 5, ie 0,5,2,7,4,1,6,3. The tiling is of course infinitely dense. You can get to a point near (0,0) by going a distance out one {4,4} and back on the other.

The isomorphism is produced by rotating one {4,4} by 180°, but leaving the other alone. The points are still the same, but if you follow the steps by numbers, the 0,1,2,... now gives an octagram, and 0,5,2,7.. gives an octagon. The process of rotation has turned sqrt(2) into -sqrt(2). It's still q^2=2, but q is now -1.414. And this is consistantly done over the plane. The point near (0,0) is now reflected deep out into the plane: you stoll go out one {4,4}, but continue further out on the (negative) other.

One can then imagine that there is a kind of 2d number, which we can only test the x-real and y-real. The over-all density of numbers is sparse in this 2-space, but we can only see the projection on the single axis.

If you know complex numbers, imagine a value j, that j^2=-1. The process then amounts to writing a+bj by a-bj. The usual tricks work with these numbers, eg the modulus is (a+jb)(a-jb), but instead of falling on circles, these fall on hyperbolae. cis becomes cosh + j sinh or cish. This is the domain of the hypercomplex number, and Z4, Z5, and Z6 are integer systems in the same way that the gaussian and eisenstein integers are on the complex plane.

The difference is that we can see only the value of a+jb, and the value of a-jb. Even though there is a good deal of fuzz around these points, the plot onto the hypercomplex plane gives a fuzzy area around a definite point: we can use that definite point.

So the rules of isomorphism is more like seeing the shadows produced by a cats-cradle, rather than the strings that make it up.

In the hypercomplex plane, it is possible for a possitive number to be conjucate to a negative one. Even among polytopes, {3,5/2,3} and {3,5,3} are isomorphic. This information tells us interesting things about both of them. For example, because the former has a definite radius, it tells us that the edges of the latter can not be set to less than some value. We see that {3,5/2,3} can not contain the vertices, of for example, {5,3,3}, because {3,5,3} can't.

Isomorphisms work over any number. In effect, it amounts to rolling through the solutions of some integer-polynomial. But it gets a bit harder further up, because you have to work out exactly what follows on from what. The class-3 systems Z7 and Z9, solve a family of cubics in the same way that the class-2 equations solve quadratics. Instead of mapping a hedrix lattice onto a line, you are mapping 3d lattices onto a line. Just as you could peek at two different values, in a class-3 system, you can peek at three different values (7, 7/2, 7/3 or 14, 14/5, 14/3).