Re: Stupid question: what space was Euclid working in? (almost)

Re: Stupid question: what space was Euclid working in? (almost)

[tporter]

Dear All,

Although not strictly relevant to the discussion here, I recall once
looking at some of the theory of the `region connection calculus' and the
work of the Qualitative Spatial and Spatio-temporal Reasoning group at
Leeds. This is concerned with the interrelationships between regions of a
space.  The logical models vary from biHeyting algebras to various
multimodal logics. There was some discussion about trying to detect
dimension within such RCC systems.

If one assumes that  `region' is a more basic notion of position than
`point' a lot of Euclid still goes through but dimension seems very hard
to handle.  (I recall that Tarski worked on this area at one time.)  Can
any one tell me more as it seems of use in Geographic Information Systems=
,
and other models of qualitative or descriptive spatial information.  I
know of John Stell's work on this and he has clarified things from a
somewhat categorical viewpoint.

Perhaps the old puns about `pointless' arguments need revisiting!

Tim




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Re: Stupid question: what space was Euclid working in? (almost)

[Robert J. MacG. Dawson]

tporter@informatics.bangor.ac.uk wrote:

> If one assumes that  `region' is a more basic notion of position than
> `point' a lot of Euclid still goes through but dimension seems very hard
> to handle.

	The topological form of Helly's theorem might be a place to start.
However, defining dimension in terms of _convex_ structure is very
tricky once you get into general spaces. For instance, I showed in my
thesis (in a section eventualLy rewritten for _Cahiers_) that if you
define a "convex set" on S^1 to be an arc shorter than an open
semicircle, you get the obvious homology. However, if closed semicircles
and their intersections are convex, the homology becomes that of the
2-sphere.

	-Robert