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Path: utzoo!utcs!bnr-vpa!bruce
From: bruce@bnr-vpa.UUCP (Bruce Townsend)
Newsgroups: net.math
Subject: Re: Which has more points, the unit square, or the unit line segment ?
Message-ID: <52@bnr-vpa.UUCP>
Date: Wed, 24-Apr-85 10:03:48 EST
Article-I.D.: bnr-vpa.52
Posted: Wed Apr 24 10:03:48 1985
Date-Received: Wed, 24-Apr-85 11:21:14 EST
References: <1767@decwrl.UUCP>
Organization: Bell Northern Research, Ottawa, Ontario
Lines: 39

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> 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 ?

> If perhaps you have a mapping function that works on (0,1] or some other
> variation of open/closed interval combinations, that would be fine too.

Here is a one-to-one mapping that maps [0, oo) to {[0, oo), [0, oo)}:
Express a real number in decimal notation, e.g.  23.71148125891286...
This maps into a pair on the positive quarter plane by:
Make the first number by taking every second digit starting with the
ones place: e.g. 3.1415926...
Make the second number by taking every other second digit:
2.7182818....

An interesting feature of this mapping is that the unit line segment
maps to the unit square!

It seems to me that the coverage is complete, since every pair on the
unit square corresponds to a point on the unit line segment, and vice-
versa.

Now for a puzzle... Is the above mapping *really* one-to-one? Can
you find a point on the segment that maps to two different points on
the square (more points in square than in segment)? Or, can you find
two different points on the segment that maps to a single point on
the square (more points in segment than in square!)?

Have fun!
-- 
			-Bruce Townsend

			Voice Processing Applications,
			Bell-Northern Research,
			Ottawa, Ontario.

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