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Topic: reversing a needle in a plane (Read 1678 times) |
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JocK
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reversing a needle in a plane
« on: Jul 22nd, 2004, 10:43am » |
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What area do you need to turn round a needle of unit length that is restricted to move in a plane?
Of course one can turn this needle inside an area of size 0.785398 (a circular disk with unit diameter). A smaller area (0.577350) is obtained when selecting an equilateral triangle of unit height. Who comes up with a still smaller area? What is the smallest area possible? What shape does this area take?
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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Jack Huizenga
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Re: reversing a needle in a plane
« Reply #1 on: Jul 22nd, 2004, 11:06am » |
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There is no smallest possible area. That is, given any epsilon>0, there exists a shape with area<epsilon in which a unit needle can be reversed. The shapes which prove this are the deltoid (a sort of pointy concave triangle), star shaped 5-oid (a pointy concave 5-pointed star), a star shaped 7-oid, etc. This is known as the Kakeya Needle problem.
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JocK
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Re: reversing a needle in a plane
« Reply #3 on: Jul 22nd, 2004, 12:03pm » |
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on Jul 22nd, 2004, 11:06am, Jack Huizenga wrote:| <..> The shapes which prove this are the deltoid (a sort of pointy concave triangle), star shaped 5-oid (a pointy concave 5-pointed star), a star shaped 7-oid, etc. <..> |
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Can you demonstrate that a set of star-shaped (2n+1)-oids exists in which a unit needle can be reversed, and that have areas that approach zero for n[mapsto][infty] ?
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| « Last Edit: Jul 22nd, 2004, 12:04pm 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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Jack Huizenga
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Re: reversing a needle in a plane
« Reply #4 on: Jul 22nd, 2004, 1:13pm » |
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It's a lot harder to prove that the shapes actually work then to just think about the problem and justify it in your head. If you make an odd number of spikes about the origin, each of almost unit length and very thin, you can shift the needle from one spike to one of the two opposite spikes using very little area; this rotates the needle by an angle of 2 pi/n, where n is the number of spikes.
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Grimbal
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Re: reversing a needle in a plane
« Reply #5 on: Jul 22nd, 2004, 4:57pm » |
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Yes, but as the spikes become thinner, they increase in number, so it is not clear if the area tends to zero.
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Speaker
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Re: reversing a needle in a plane
« Reply #6 on: Jul 22nd, 2004, 5:31pm » |
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Just a little not math input.
I can picture the circular area, where the needle spins around its middle.
I can picture the triangle area, where the needle falls over, then goes up then finally is reversed.
With all the polygons (five pointed etc.) does the needle move back and forth between points? Like a car doing a three point turn. Back and forth. If this is the case, then the more points you have, it seems the space comes closer to forming the original circle. So, you must start gaining space after the triangle.
Just my two cents.
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Jack Huizenga
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Re: reversing a needle in a plane
« Reply #7 on: Jul 22nd, 2004, 10:32pm » |
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Yes, the needle moves like in a 3-point turn. But you don't gain extra space when doing this if you define the n-oid properly.
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Speaker
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Re: reversing a needle in a plane
« Reply #8 on: Jul 22nd, 2004, 10:36pm » |
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So, it would look like a spiral staircase, but sort of squished over. So, it would be reversed, but not in its original space. Or, can you get it back to the original space?
Or, maybe like a boat backing out of a berth, then swinging its bow around in an arc, while its back stays put. Like the "Y" in the New York Mets logo?
Or, maybe backing around in a big U. Would changing the radius of the U change the area?
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| « Last Edit: Jul 22nd, 2004, 10:48pm by Speaker » |
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They that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety. <Ben Franklin>
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Barukh
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Re: reversing a needle in a plane
« Reply #9 on: Jul 23rd, 2004, 12:01am » |
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When I was yet a guest here, I posted that question on apparently unrelated thread (reply #18 ) – just to show what Besikovitch has done.
There is plenty of material about this problem on the web, with various generalizations, almost all rather technical. A good one is Terry Tao’s webpage. There is also an interactive guide to Besikovitch’s construction.
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JocK
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Re: reversing a needle in a plane
« Reply #10 on: Jul 23rd, 2004, 2:21am » |
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Can you specify the kite-type shape with top angle 2[pi]/(2n+1) (n = 1, 2, 3, ...) which, when added together into a (2n+1)-pointed star, allows a unit needle to be reversed?
(see attachment for the case n=2)
Does the area A[subn] of such an individual 'kite' satisfy lim n[mapsto][infty] (2n+1)A[subn] = 0 ? Is it necessary that the 'kite' has a convex shape? Would a 'kite' with straight edges do the job?
J CK
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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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JocK
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and the attachment...
(the red, blue and green line-segments indicate subsequent needle positions)
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| « Last Edit: Jul 23rd, 2004, 11:38am 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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Grimbal
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Re: reversing a needle in a plane
« Reply #12 on: Jul 23rd, 2004, 4:07am » |
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This reminds me of the sofa in a corridor problem.
You live at the end of a long, 1m wide corridor that turns with a 90° angle. You want to buy a sofa, but you will have to carry it around the corner. So you wonder what is the maximum area the sofa can have (the shape is free) and still be able to pass the corner?
For instance, a 1/2 disk with radius 1m would be fine. But not optimal.
Note: the problem is completely 2D. There is no tilting of the sofa.
Here also, you end up with funny clever shapes.
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| « Last Edit: Jul 23rd, 2004, 4:08am by Grimbal » |
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JocK
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Re: reversing a needle in a plane
« Reply #13 on: Jul 23rd, 2004, 4:59am » |
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on Jul 23rd, 2004, 4:07am, Grimbal wrote:| This reminds me of the sofa in a corridor problem. |
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Indeed.. but in this case, given the 'sofa' (okay, for simplicity a very simple and perhaps less comfortable sofa!  ), one needs to determine the smallest corridor...
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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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Grimbal
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Re: reversing a needle in a plane
« Reply #14 on: Jul 23rd, 2004, 1:44pm » |
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No, no, it is the same, just inside out. The sofa is outside and the corridor is inside.
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Eigenray
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Re: reversing a needle in a plane
« Reply #15 on: Jul 23rd, 2004, 2:27pm » |
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Regarding Besicovitch's construction (shown in the interactive guide Barukh linked to):
Unless I'm mistaken, the needle cannot be rotated entirely within the region shown. After rotating each 45o/2n, you need to transport the needle to a parallel position. But this can be done in arbitrarily small extra area as well.
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