Posted by: Alexandre Borovik | October 5, 2008

## Shoot-out in the Hall of Mirrors

My answer to a traditional question: “You are a mathematician. What do you actually do?”

I study symmetries. More precisely, I study multi-mirror symmetries, where  mirrors get reflected in other mirrors, and reflections breed and multiply all over the place. I design ways to find safe paths in the maze of mirrors, and to tell real objects from sneering phantoms.

If you happen to be involved in a shoot-out in a Hall of Mirrors (like in the famous scene from Orson Wells’  The Lady from Shanghai,  see a YouTube clip, I can provide a consultancy, at UCU recommended professorial rates.

This clip also gives an answer to a question I heard from a graduate student: why all calculations in simple algebraic groups end up in manipulations with root systems? Because root system is just a convenient way to write down on paper the skeleton mirror system of the group, and calculation with roots is the final shoot-out at the end of the film.

## Responses

1. There is an intersting work (theological) by Ramon Lull who uses mirror reflections in his reasoning (and often geometrical metaphors). Of course it’s far from formal logic but some concepts seem to be quite close to modern ones (especially in Ars Magna).

http://lullianarts.net/infusa/sent54.htm

2. There is an intersting work (theological) by Ramon Lull who uses mirror reflections in his reasoning (and often geometrical metaphors). Of course it’s far from formal logic but some concepts seem to be quite close to modern ones (especially in Ars Magna).

http://lullianarts.net/infusa/sent54.htm

3. If I’m being silly (as I sometimes like to be), I would reply to that question by saying that I drink tea (or coffee if you insist). And if the question asker doesn’t know what that implies then he should consider himself very lucky!

4. Dmitry – thanks for a link to Ramon Lull.

When two round mirrors, a large one and a small one, are put face to face, and one should be large, and the other small so that the figure of the small one can be seen in the large one, then three circles appear in the large mirror. One of these is larger than the others, and this is a figure of the small circular mirror, containing in turn one circle, which is a circular figure of the larger mirror again containing a figure of the smaller mirror. Of course, someone with good eyesight can see four, five, and naturally, even more circles that are not normally visible. But in our example, we only want to deal with three circles, and we call the largest circle A., and the medium size circle B., and the smallest circle C. And thus, there follow three more circles in the smaller mirror, and the largest of these we call D., this one is from the big mirror. And we call the medium size circle E., and this one belongs to the small mirror, and the smallest circle we call F., and it is cast by the big mirror. […] And the Speech of Angels consists in the said mode, be they good or evil, as each one represents its own concept to the other, and each one also represents whatever it receives from the other, in the same way as C. does with F. and F. with C.

Was Lull the first one to describe (interpersonal) communication as synchronous reflection of interlocutors? It is commonplace now that you understand other person only if you see what kind of images (s)he is making of you, ad infinitum. I’ll try to ask experts.

5. it’s difficult to say who was the first to introduce such an approach. perhaps chinese or indian thinkers could have a similar idea before, or 20th century psychologists after and independently. lull himself could be influenced by eastern mystics since he traveled a lot as a missionary, e.g. by pythagoreans. dov gabbay refers to lull as to forerunner of cut-paste technique described in the recent book
http://www.springer.com/math/book/978-1-4020-6781-5
and it’s well known that leibniz was deeply influenced by lull, as well as he was in correspondence with chinese scientists of the time. so it’s possible that mathematics consists of a constant number of ideas (or possibly just one idea :)) and developing only its language 🙂

6. Root systems are classified by means of Dynkin diagrams, which can be found
in various parts of the math, see, for example, M. Hazewinkel, W. Hesselink,
D. Siermsa, and F. D. Veldkamp, “The ubiquity of Coxeter-Dynkin diagrams (an
introduction to the ADE problem)” available at
http://math.ucr.edu/home/baez/hazewinkel_et_al.pdf
The interesting R.Proctor theorem states that the extended Dynkin diagrams of
types ADE are the only diagrams that admit a positive labeling (labeling of the
nodes by positive real numbers) with the following property: Twice any label is
the sum of the labels on adjacent vertices, see “Two Amusing Dynkin Diagram
Graph Classifications”, http://www.jstor.org/stable/2324217 These diagrams can be
sent into space (and perhaps already sent ?) to communicate with other
civilizations.