Author |
Topic: Shortest curve (Read 915 times) |
|
JocK
Uberpuzzler
Gender: 
Posts: 877
|
 |
Shortest curve
« on: Dec 12th, 2004, 10:23am » |
 Quote  Modify
|
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
|
| « Last Edit: Dec 12th, 2004, 1:58pm by JocK » |
 IP Logged |
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.
|
|
|
John_Gaughan
Uberpuzzler
Behold, the power of cheese!
    
 
Gender:
Posts: 767
|
|
Re: Shortest curve
« Reply #1 on: Dec 13th, 2004, 11:43am » |
Quote Modify
|
Maybe I am thinking a bit overly simple, but wouldn't this be just a hair over 1 unit long?
|
|
IP Logged |
x = (0x2B | ~0x2B)
x == the_question
|
|
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: Shortest curve
« Reply #2 on: Dec 13th, 2004, 12:23pm » |
Quote Modify
|
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? |
|
But my curve (part of a circle of infinite radius) is only half a hair over 1 unit long. 
|
| « Last Edit: Dec 13th, 2004, 12:27pm by ThudnBlunder » |
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
John_Gaughan
Uberpuzzler
Behold, the power of cheese!
Gender:
Posts: 767
|
|
Re: Shortest curve
« Reply #3 on: Dec 13th, 2004, 1:13pm » |
Quote Modify
|
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...
|
|
IP Logged |
x = (0x2B | ~0x2B)
x == the_question
|
|
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: Shortest curve
« Reply #4 on: Dec 13th, 2004, 1:25pm » |
Quote Modify
|
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... = /
|
|
Another cosmological conundrum thus bites the dust. 
|
| « Last Edit: Dec 13th, 2004, 1:31pm by ThudnBlunder » |
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
JocK
Uberpuzzler
Gender:
Posts: 877
|
|
Re: Shortest curve
« Reply #5 on: Dec 13th, 2004, 2:23pm » |
Quote Modify
|
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? |
|
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...
|
|
IP Logged |
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.
|
|
|
rmsgrey
Uberpuzzler
 
Gender:
Posts: 2873
|
|
Re: Shortest curve
« Reply #6 on: Dec 13th, 2004, 2:29pm » |
Quote Modify
|
on Dec 13th, 2004, 2:23pm, JocK wrote:| [...] a unilateral triangle of unit side width [...] |
|
|
|
IP Logged |
|
|
|
JocK
Uberpuzzler
Gender:
Posts: 877
|
|
Re: Shortest curve
« Reply #7 on: Dec 13th, 2004, 3:00pm » |
Quote Modify
|
sorry... I'm obviously not a native speaker.... 
Of course I mean: "an equilateral triangle of unit side length" ...
|
|
IP Logged |
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.
|
|
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: Shortest curve
« Reply #8 on: Dec 13th, 2004, 8:17pm » |
Quote Modify
|
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... |
|
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.
|
| « Last Edit: Dec 13th, 2004, 10:40pm by ThudnBlunder » |
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
John_Gaughan
Uberpuzzler
Behold, the power of cheese!
Gender:
Posts: 767
|
|
Re: Shortest curve
« Reply #9 on: Dec 13th, 2004, 9:37pm » |
Quote Modify
|
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. |
|
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.
|
|
IP Logged |
x = (0x2B | ~0x2B)
x == the_question
|
|
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: Shortest curve
« Reply #10 on: Dec 13th, 2004, 10:59pm » |
Quote Modify
|
Quote:| I think a parabola or two might be what we are looking for. |
|
I prefer a cycloid.
|
|
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Barukh
Uberpuzzler
Gender:
Posts: 2276
|
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.
|
| « Last Edit: Dec 16th, 2004, 4:21am by Barukh » |
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print