Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site bbnccv.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!ihnp4!bbnccv!ldenenbe
From: ldenenbe@bbnccv.UUCP (Larry Denenberg)
Newsgroups: net.math
Subject: Re: Which has more points, the unit square, or the unit line segment ?
Message-ID: <255@bbnccv.UUCP>
Date: Tue, 23-Apr-85 17:39:17 EST
Article-I.D.: bbnccv.255
Posted: Tue Apr 23 17:39:17 1985
Date-Received: Wed, 24-Apr-85 05:04:12 EST
References: <1767@decwrl.UUCP> <72@harvard.ARPA>
Reply-To: larry@harvard.UUCP (Larry Denenberg)
Organization: Bolt Beranek and Newman, Cambridge, MA
Lines: 17
Summary: 

>> 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.

It's onto, but not one-to-one.  For example, the number .11 maps to (.1,.1).
But so does .100909090909... .  All that has been shown is that there
are at least as many points in the unit interval as in the unit square.
Somehow I think this should satisfy the proposer of the original question.

Larry Denenberg
larry@harvard