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From: bill@utastro.UUCP (William H. Jefferys)
Newsgroups: net.math
Subject: Re: Which has more points, the unit square, or the unit line segment ?
Message-ID: <54@utastro.UUCP>
Date: Tue, 23-Apr-85 23:48:03 EST
Article-I.D.: utastro.54
Posted: Tue Apr 23 23:48:03 1985
Date-Received: Fri, 26-Apr-85 02:20:47 EST
References: <1767@decwrl.UUCP> <72@harvard.ARPA>
Organization: U. Texas, Astronomy, Austin, TX
Lines: 23

> > I suspect that the number of points on a line segment and a unit square
> > are the same.
> 
> Yup.
> 
> > Can anyone come up with a mapping function that maps the P in segment [0,1]
> > to some (X,Y) in square [(0,0),(1,1)] such that the mapping provides
> > complete coverage ?
> 
[The standard solution was given]
 
The real trick is mapping the unit line onto the unit square *continuously*.
This was first done by Peano many years ago.  I don't think his example is
one-to-one, however.

-- 
"Men never do evil so cheerfully and so completely as when they do so from
	religious conviction."  -- Blaise Pascal

	Bill Jefferys  8-%
	Astronomy Dept, University of Texas, Austin TX 78712   (USnail)
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