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The kindest (shortest) cut of all


Best Answer superprismatic, 27 February 2014 - 08:56 PM

Spoiler for that nice hint makes it look like

Go to the full post


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6 replies to this topic

#1 bonanova

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Posted 25 February 2014 - 10:39 PM

Here is a piece of plywood in the shape of an isosceles triangle.

The side lengths are 1, 1, sqrt(2) units.

 

Quick and dirty representation:

 

A

|  \

|     \

|        \

|           \       

B--------------C

 

The angle at B is a right angle.

 

We'd like to cut this into two pieces of equal area.

There are many ways to do this with a single cut.

Which cut has the shortest distance?


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#2 TimeSpaceLightForce

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Posted 26 February 2014 - 10:50 PM

Spoiler for cutting lines


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#3 bonanova

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Posted 27 February 2014 - 10:11 AM

Spoiler for cutting lines

 

With a careful reading of the OP can you do slightly better?


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#4 superprismatic

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Posted 27 February 2014 - 08:56 PM   Best Answer

Spoiler for that nice hint makes it look like


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#5 bonanova

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Posted 27 February 2014 - 09:01 PM

Kudos to both TSLF and SP.


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#6 superprismatic

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Posted 27 February 2014 - 10:57 PM

@Bonanova or anyone else who may have an interest in this:

Do you have a proof that that circular arc is the shortest?  Conceivably, a piece of a trigonometric curve, or exponential curve, or another conic section may be shorter.  The possibilities are endless. 


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#7 bonanova

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Posted 27 February 2014 - 11:41 PM

@Bonanova or anyone else who may have an interest in this:

Do you have a proof that that circular arc is the shortest?  Conceivably, a piece of a trigonometric curve, or exponential curve, or another conic section may be shorter.  The possibilities are endless. 

 

Yes, it's called the isoperimetric problem, generally stated,

what is the curve of constant length that encloses the greatest area?

Equivalently, what shape of constant area has the smallest perimeter?

The answer is a circle. A 45o arc can be reflected 7 times into a full circle.

Symmetry demands this answer, although rigorous proofs abound.

Here is a general discussion.

 

In three dimensions (think soap bubble) a sphere encloses more volume

than any other closed surface of the same area.

 

isoperimetric = iso (same) perimetric (perimeter).


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- Bertrand Russell




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