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From: albert@harvard.ARPA (David Albert)
Newsgroups: net.math
Subject: Re: Which has more points, the unit square, or the unit line segment ?
Message-ID: <72@harvard.ARPA>
Date: Tue, 23-Apr-85 11:06:37 EST
Article-I.D.: harvard.72
Posted: Tue Apr 23 11:06:37 1985
Date-Received: Wed, 24-Apr-85 04:32:06 EST
References: <1767@decwrl.UUCP>
Organization: Aiken Computation Laboratory, Harvard
Lines: 24

> 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 ?

Yup.  Map 0 to (0,0) and 1 to (1,1).  Then for any number x in (0,1),
let y be the number created by taking every other digit in the decimal
exansion of x, beginning with the first, and z the number created by
taking every other digit beginning with the second, and map x to (y,z).
This map is one to one and onto, and is easily reversed as well.

For instance, the number 0.4275 maps to (0.47, 0.25), and PI-3 maps to
(0.11963..., 0.45255...).

Note that the inverse mapping of points on the upper and right boundaries
of the square requires converting 1 into 0.99999....
-- 

David Albert
ihnp4!seismo!harvard!albert (albert@harvard.ARPA)