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JocK
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Shortest curve
« on: Dec 12th, 2004, 10:23am » |
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A well-known problem, but don't think it has been posed here yet. So here goes:
What is the shortest planar curve* that does not fit inside an equilateral triangle of unit side-length?
* length measured along the curve
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| « Last Edit: Dec 12th, 2004, 1:58pm by JocK » |
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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John_Gaughan
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Re: Shortest curve
« Reply #1 on: Dec 13th, 2004, 11:43am » |
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Maybe I am thinking a bit overly simple, but wouldn't this be just a hair over 1 unit long?
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ThudnBlunder
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Re: Shortest curve
« Reply #2 on: Dec 13th, 2004, 12:23pm » |
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on Dec 13th, 2004, 11:43am, John_Gaughan wrote:| Maybe I am thinking a bit overly simple, but wouldn't this be just a hair over 1 unit long? |
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But my curve (part of a circle of infinite radius) is only half a hair over 1 unit long. 
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| « Last Edit: Dec 13th, 2004, 12:27pm by ThudnBlunder » |
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John_Gaughan
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Re: Shortest curve
« Reply #3 on: Dec 13th, 2004, 1:13pm » |
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Perhaps I was not explicit enough. "Hair" is a technical term that means "such a tiny amount that there is no smaller amount." So "half a hair" cannot exist 
Anyway, how can you have a circle of infinite radius? If you define [pi] as the ratio of the circumference to the diameter, and both c and d are [infty], then:
c = [pi]d
[infty] = [pi][infty]
[pi] = [infty]/[infty]
3.14159... = [infty]/[infty]
I guess that simplifies some of the math problems I had in college.
If you want to use reckless (and wrong) application of L'Hopital's rule, you could even evaluate it like this:
[infty]/[infty] = 0/0 = 3.14159...
That simplifies some more math problems...
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ThudnBlunder
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Re: Shortest curve
« Reply #4 on: Dec 13th, 2004, 1:25pm » |
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Quote:Anyway, how can you have a circle of infinite radius? If you define as the ratio of the circumference to the diameter, and both c and d are , then:
c = d
=
= /
3.14159... = /
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Another cosmological conundrum thus bites the dust. 
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| « Last Edit: Dec 13th, 2004, 1:31pm by ThudnBlunder » |
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JocK
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Re: Shortest curve
« Reply #5 on: Dec 13th, 2004, 2:23pm » |
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on Dec 13th, 2004, 11:43am, John_Gaughan wrote:| Maybe I am thinking a bit overly simple, but wouldn't this be just a hair over 1 unit long? |
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Are you claiming that all curves shorter than unity fit in a unilateral triangle of unit side width?
I claim there are plenty of counterexamples...
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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rmsgrey
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Re: Shortest curve
« Reply #6 on: Dec 13th, 2004, 2:29pm » |
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on Dec 13th, 2004, 2:23pm, JocK wrote:| [...] a unilateral triangle of unit side width [...] |
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JocK
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Re: Shortest curve
« Reply #7 on: Dec 13th, 2004, 3:00pm » |
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sorry... I'm obviously not a native speaker.... 
Of course I mean: "an equilateral triangle of unit side length" ...
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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ThudnBlunder
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Re: Shortest curve
« Reply #8 on: Dec 13th, 2004, 8:17pm » |
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on Dec 13th, 2004, 2:23pm, JocK wrote:
Are you claiming that all curves shorter than unity fit in a unilateral triangle of unit side width?
I claim there are plenty of counterexamples... |
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I suspect that you have in mind a discontinuous counterexample consisting of 2 points of zero length but having a curvilinear distance apart of 1+[smiley=calg.gif], where [smiley=calg.gif] is a Guaghan Hair.
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| « Last Edit: Dec 13th, 2004, 10:40pm by ThudnBlunder » |
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John_Gaughan
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Re: Shortest curve
« Reply #9 on: Dec 13th, 2004, 9:37pm » |
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on Dec 13th, 2004, 8:17pm, THUDandBLUNDER wrote:| I suspect that you have in mind a discontinuous counterexample consisting of 2 points with zero length but a curvilinear distance apart of 1+[smiley=calg.gif], where [smiley=calg.gif] is a Guaghan Hair. |
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Exactly. See, we have Euclidean geometry, and we have Gaughanean geometry where everything reduces to linear equations and the most complex object is a triangle -- no curves.
Anyway, I think I begin to understand how a curve of length < 1 might not fit in the triangle. Back to Euclidean geometry, I think a parabola or two might be what we are looking for.
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ThudnBlunder
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Re: Shortest curve
« Reply #10 on: Dec 13th, 2004, 10:59pm » |
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Quote:| I think a parabola or two might be what we are looking for. |
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I prefer a cycloid.
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Barukh
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First of all, I suppose that Jock has in mind an open triangle (that is, the sides are not included) – otherwise the problem is not well defined.
Take a point D on a base AB of the triangle close to A. Construct a parallelogram DFCG as depicted so that DF = FG. The curve DFGC does not fit into the open triangle ABC (why?) and has the length less than 1 for small values of the angle ACD. The minimum 0.98198… is obtained when ACD [approx] 3.67 degrees.
This solution was proposed by A.S. Besicovitch (remember him?) in 1965. I have no idea if this was ever improved.
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| « Last Edit: Dec 16th, 2004, 4:21am by Barukh » |
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