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Topic: Irrational punching/painting (revisited) (Read 628 times) |
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Aryabhatta
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Irrational punching/painting (revisited)
« on: Jun 13th, 2008, 10:24am » |
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You have a device which is capable of painting all points at an irrational distance from a given point (in the 2D plane).
To use the device, you just have to select that point and all points at an irrational distance from that point are painted black.
Initially you start out with all points white in the 2D plane.
Prove/Disprove:
To paint the whole plane black, selecting the three points (0,0), (1,0) and (sqrt(2),0) is enough.
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TenaliRaman
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I am no special. I am only passionately curious.
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Re: Irrational punching/painting (revisited)
« Reply #1 on: Jun 13th, 2008, 2:00pm » |
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I am not for one to get any kind of solutions, but I can sure find links 
Irrational Coloring
Irrational Device
Non-integer colouring (Not same but thematically similar)
-- AI
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Aryabhatta
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Re: Irrational punching/painting (revisited)
« Reply #2 on: Jun 13th, 2008, 2:21pm » |
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I did check two of those threads and decided to use the word "revisited" because those threads already exist.
I believe the probem stated above hasn't been treated in any of those threads, but, they are all too long for me to be sure.
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Eigenray
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Re: Irrational punching/painting (revisited)
« Reply #3 on: Jun 13th, 2008, 3:28pm » |
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No I don't think this problem was considered. It was shown that (0,0), (1,0), and (t,0) suffice whenever t is not algebraic of degree 1 or 2 over the rationals. Obviously that doesn't apply here.
But it looks like t=sqrt(2) still works. Otherwise we would find positive integers with a4 - b4 = c2, which has no nontrivial solutions by descent.
I wonder, can we describe all t for which (0,0), (1,0), and (t,0) do not work?
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| « Last Edit: Jun 13th, 2008, 3:32pm by Eigenray » |
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Aryabhatta
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Re: Irrational punching/painting (revisited)
« Reply #4 on: Jun 13th, 2008, 3:35pm » |
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Yes, that is exactly what I had in mind, Eigenray!
The other problem you state is interesting...
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Aryabhatta
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Re: Irrational punching/painting (revisited)
« Reply #5 on: Jun 13th, 2008, 4:21pm » |
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t = sqrt(17) does not work. (0,4/3) is at a rational distance from (0,0), (1,0), (sqrt(17), 0).
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Eigenray
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Re: Irrational punching/painting (revisited)
« Reply #6 on: Jun 13th, 2008, 7:06pm » |
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There will be many such t. For example, if t2 = (a/b)2 - (2mn/(m2-n2))2, with a,b,m,n integers.
More generally, if r,s are rationals with r+s>1, |r-s|<1, then we can pick a point (x,y) a distance r from (0,0) and s from (1,0). Then (t,0) can be a point of intersection of the x-axis with any circle about (x,y) with sufficiently large rational radius. This leads to a parameterization with 3 rational parameters. But I don't know if there's a simpler form. In particular, is there an algorithm for deciding whether a given number works?
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