### Quote of the Month

"Now the elapsed time, even if it is really something different, is certainly measured most easily (if not most perfectly) by the plane area circumscribed by the planet's path", Johannes Kepler* (1571-1630)

Taking a value $e^{2 \pi i r}$ on the unit circle, for rational $r$, the area of the segment is $r \pi$. But taking an angle in rational radians, such as $e^{2 i r}$, one obtains an area of $r$. Just a thought.

* New Astronomy, translated by W. H. Donohue (C.U.P. 1992)

Taking a value $e^{2 \pi i r}$ on the unit circle, for rational $r$, the area of the segment is $r \pi$. But taking an angle in rational radians, such as $e^{2 i r}$, one obtains an area of $r$. Just a thought.

* New Astronomy, translated by W. H. Donohue (C.U.P. 1992)

## 4 Comments:

This post gets one thinking. "Just a thought" can lead to some new principle. Kepler's life including friendship with Galileo is fascinating.

Hi Louise! Yes, your version of Kepler's law is very appealing.

Kea,

I've been studying localic topoi recently and find the process of recovering a topological space from a locale to be quite powerful. A point p: 1 -> X of a locale X can be expressed as a frame morphism to the initial frame {0,1}. This standard definition is reminiscent of Connes' use of the two point space in the NCG Standard Model.

For M-theory, one may have to extend the initial frame to {0,1,2}, so that any point in the M-theory locale is defined to be a morphism of frames p^{-1}: O(X) -> {0,1,2}.

Hi kneemo. Yes, take a look at Mac Lane and Moerdijk's text, for example. This is a very important point for understanding what is generically called Stone Duality, after Marshall Stone. (For some really advanced papers on this, check out Paul Taylor's work). The idea of self-dual (often called schizophrenic) object in this context picks out, for example, U(1) for Pontrjagin duality.

I was motivated for quite a while by the search for non-abelian analogues of the locale case. Check out the (physics) papers by the Isham group on this idea - eg. by Raptis.

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