osman@sprite.DEC (Eric, 617 273-7484, Burlington, Ma. 01803, USA) (04/23/85)
I recall being told that the comparison of the size of two infinities is resolved by finding a mapping function from one to the other. If the mapping function can be demonstrated to map EVERY element from the first set onto EVERY element of the second, then the infinities are the same. I suspect that the number of points on a line segment and a unit square are the same. 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. /Eric Osman, Digital, Burlington, Ma, 01803, USA
albert@harvard.ARPA (David Albert) (04/23/85)
> 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)
ldenenbe@bbnccv.UUCP (Larry Denenberg) (04/24/85)
>> 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
bill@utastro.UUCP (William H. Jefferys) (04/24/85)
> > 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) {allegra,ihnp4}!{ut-sally,noao}!utastro!bill (uucp) bill%utastro.UTEXAS@ut-sally.ARPA (ARPANET)
bruce@bnr-vpa.UUCP (Bruce Townsend) (04/24/85)
----------------- > 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. Mail path: {utzoo, utcs, bnr-di, bnr-mtl}!bnr-vpa!bruce
hen@bu-cs.UUCP (Bill Henneman) (04/29/85)
The map which takes alternate digits of a point on the line into points in the plane isn't 1-1: consider the two numbers 13.194979493959.... and 14.104070403050... Both points map into (1.147435... , 4).