A bit more on the Infinite Reflection Groups Project.
A curious student (a prospective student of MATH5003/4) writes:
"Hi there,
Would you be able to give me some more information about the Infinite Reflection Groups project?
Thank you
[name redacted]"
Hi [name],
Yes certainly. To cover _all_ bases things eventually get a bit technical - which is fun, but doesn't fit in
an email. So let's try and do it with an illustrative example:
I'm going to build up an infinite reflection group by taking a limit of finite ones.
First think of the real line, and a point of `reflection' (a 0-d mirror in a 1-d world) that just takes any
point to its image on the other side of the `mirror'. This world has 3 `parts': the mirror itself;
the stuff on one side of it; and then the looking-glass world on the other side.
From here we can go in two ways. We could either add another 0-d mirror and see what happens
in terms of the generated group of reflections
- can you see what happens?
Or we could go up a dimension to a 1-d mirror in a 2-d world. At first this has almost the same
decomposition into 3 parts - the mirror; one side; the other side. ...But then we could again add
another mirror. This time we have the choice of the new mirror being parallel or not to the
first one. If it is not parallel then the mirrors meet, and we have the choice of the angle...
- can you see the kind of thing that can happen. (Many things. My mum used to have a triptych
mirror with adjustable panels, which I played with quite a bit when I was a kid.)
Then finally we can consider n-dimensional mirrors in n+1-dimensional worlds... same kind of idea...
But now let n go to infinity.
How are we doing so far?
all the best
PPM