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        riddles >> hard >> reversing a needle in a plane
        
(Message started by: JocK on Jul 22^nd, 2004, 10:43am)

Title: reversing a needle in a plane
Post by JocK on Jul 22^nd, 2004, 10:43am

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

Title: Re: reversing a needle in a plane
Post by Jack Huizenga on Jul 22^nd, 2004, 11:06am

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.

Title: Re: reversing a needle in a plane
Post by towr on Jul 22^nd, 2004, 11:33am

here's a link for the interested:
http://mathworld.wolfram.com/KakeyaNeedleProblem.html

Title: Re: reversing a needle in a plane
Post by JocK on Jul 22^nd, 2004, 12:03pm

on 07/22/04 at 11:06:09, 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. <..>

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] ?

Title: Re: reversing a needle in a plane
Post by Jack Huizenga on Jul 22^nd, 2004, 1:13pm

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.

Title: Re: reversing a needle in a plane
Post by Grimbal on Jul 22^nd, 2004, 4:57pm

Yes, but as the spikes become thinner, they increase in number, so it is not clear if the area tends to zero.

Title: Re: reversing a needle in a plane
Post by Speaker on Jul 22^nd, 2004, 5:31pm

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.

Title: Re: reversing a needle in a plane
Post by Jack Huizenga on Jul 22^nd, 2004, 10:32pm

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.

Title: Re: reversing a needle in a plane
Post by Speaker on Jul 22^nd, 2004, 10:36pm

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?

Title: Re: reversing a needle in a plane
Post by Barukh on Jul 23^rd, 2004, 12:01am

When I was yet a guest here, I posted that question on apparently unrelated thread (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action=display;num=1028613864) (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 (http://www.math.ucla.edu/~tao/kakeya.html). There is also an interactive guide (http://www.math.ucla.edu/~tao/java/Besicovitch.html
) to Besikovitch’s construction.

Title: Re: reversing a needle in a plane
Post by JocK on Jul 23^rd, 2004, 2:21am

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?

J8)CK

Title: Re: reversing a needle in a plane
Post by JocK on Jul 23^rd, 2004, 2:29am

and the attachment...

(the red, blue and green line-segments indicate subsequent needle positions)

Title: Re: reversing a needle in a plane
Post by Grimbal on Jul 23^rd, 2004, 4:07am

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.

Title: Re: reversing a needle in a plane
Post by JocK on Jul 23^rd, 2004, 4:59am

on 07/23/04 at 04:07:19, Grimbal wrote:

This reminds me of the sofa in a corridor problem.

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...

Title: Re: reversing a needle in a plane
Post by Grimbal on Jul 23^rd, 2004, 1:44pm

No, no, it is the same, just inside out. The sofa is outside and the corridor is inside.

Title: Re: reversing a needle in a plane
Post by Eigenray on Jul 23^rd, 2004, 2:27pm

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 45^o/2ⁿ, you need to transport the needle to a parallel position. But this can be done in arbitrarily small extra area as well.