most recent 30 from http://mathoverflow.net 2013-05-20T06:52:52Z http://mathoverflow.net/feeds/user/15381 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66053/dissecting-a-square Colin D Wright 2011-05-26T13:10:00Z 2011-05-29T06:03:32Z

<p>Edited - some comments may now be out-of-date.</p> <p>I thought I had a complete set of solutions to this:</p> <pre><code>Cut a square into identical pieces so that they all touch the center point. </code></pre> <p>It became clear after some discussions that I was very, very wrong.</p> <p>There are infinite families of solutions, and a sporadic. So I have two questions:</p> <ol> <li><p>What do you think is a complete set of solutions?</p></li> <li><p>What techniques and approaches can I use to prove that the ones I have are all there are?</p></li> </ol> <p>Hope that's clearer. Thanks.</p>

http://mathoverflow.net/questions/66053/dissecting-a-square/66058#66058 Colin D Wright 2011-05-30T08:08:58Z 2011-05-30T08:08:58Z

@Aaron Re your comment and Beni's situation: noted &amp; understood. Re the original question, also understood. That's the culture of the place, and that's what I didn't know. My original cases didn't include a figure of 16 cells in 4 left justified rows of 4,3,5,4 - and I can say that without actually understanding your description. My original three infinite families were very naive, and only ever had 2, 4 or 8 pieces. I've tried your 128 solution but can't make it work, but that's no indication either way.

http://mathoverflow.net/questions/66053/dissecting-a-square/66058#66058 Colin D Wright 2011-05-29T11:30:29Z 2011-05-29T11:30:29Z

@Aaron - this is why I'm pleased (in retrospect) that I asked the question in the vague way I did - the answers that I've had are incredibly illuminating. Thank you for your time - I hope you've found it interesting. It's taught me a lot. @Daniel - I was offended by the way it was phrased, but I understand that offence often arises from a mismatch of culture. I need to become enculturated, so I swallowed hard and reacted in what I hope is a positive and appropriate manner.

http://mathoverflow.net/questions/66053/dissecting-a-square Colin D Wright 2011-05-26T21:44:16Z 2011-05-26T21:44:16Z

Have done already, as soon as I saw your posting there. I don't have much time at the moment, but I'm certainly interested. Currently trying to work out how to DM you there, but I have an early start to a long day tomorrow, so I might have to try another time. Thanks again.

http://mathoverflow.net/questions/66053/dissecting-a-square/66058#66058 Colin D Wright 2011-05-26T20:56:54Z 2011-05-26T20:56:54Z

My infinite families are as follows. Take any non self-intersecting path from the center to the border such that a 180 degree rotation does not intersect. That divides the square into two. Similarly for a 90 degree rotation, which divides the square into 4. Now divide into four squares, and for each small square, divide diagonally with a curve that has 180 degree symmetry. That divides the square into 8. The sporadic is the trivial case - one piece. But fedja has just demonstrated a 16 piece dissection, so now I know I know nothing.

http://mathoverflow.net/questions/66053/dissecting-a-square Colin D Wright 2011-05-26T20:47:46Z 2011-05-26T20:47:46Z

@fedja That's wonderful. No, it's not a solution I had, so that partly answers my intended question. Thank you. I have much to think about, and much to work on. And now my brain hurts.

http://mathoverflow.net/questions/66053/dissecting-a-square/66082#66082 Colin D Wright 2011-05-26T20:22:37Z 2011-05-26T20:22:37Z

Or define two partitions to be equivalent if they agree on all but a nowhere dense set, which is probably the same, but I'd need to work hard to think about that clearly enough. Very helpful - thank you.

http://mathoverflow.net/questions/66053/dissecting-a-square/66058#66058 Colin D Wright 2011-05-26T19:34:20Z 2011-05-26T19:34:20Z

@Douglas Zare Noted. Thank you.

http://mathoverflow.net/questions/66053/dissecting-a-square/66058#66058 Colin D Wright 2011-05-26T19:29:09Z 2011-05-26T19:29:09Z

So thank you for your comprehensive reply, and for the references. They are most valuable. I'll go away and reconsider my approach. Next time I have a question I'll consider my phrasing more carefully. I will certainly question whether this is the right forum.

http://mathoverflow.net/questions/66053/dissecting-a-square/66058#66058 Colin D Wright 2011-05-26T19:26:14Z 2011-05-26T19:26:14Z

&quot;If you want respect on this site then stop hinting ...&quot; - forgive me, but I'm accustomed to sites where one creates discussion so people can contribute to finding a solution. This site feels different, and I'm certainly learning quickly that it has a different ethos. My apologies. I've been intrigued to see the different results where the partitioning of the square has not been into sets homeomorphic to a disk up to a set of Lebesgue measure zero. Seeing the speculation has been of use to me, and would have been missed had I been more specific.

http://mathoverflow.net/questions/66053/dissecting-a-square/66082#66082 Colin D Wright 2011-05-26T18:48:46Z 2011-05-26T18:48:46Z

Yes, but it's not all the solutions. Agreed that more solutions won't increase the cardinality, but not including all the solutions seems &quot;sub-optimal.&quot; What if you additionally require the &quot;pieces&quot; to be connected (in some sense)?

http://mathoverflow.net/questions/66053/dissecting-a-square Colin D Wright 2011-05-26T17:53:43Z 2011-05-26T17:53:43Z

@fedja Nice exploration of the pathological - thank you. Perhaps we should say that the sets are connected, or have non-empty interior, or that the interiors of their closures are disjoint, or something similar. Something to better capture the &quot;intuitive&quot; concept of piece. But I like the pathological, and will think on it further.

http://mathoverflow.net/questions/66053/dissecting-a-square Colin D Wright 2011-05-26T17:40:12Z 2011-05-26T17:40:12Z

Congruent - reflections allowed. It's interesting to separate out those dissections that use reflection from those that don't. Again, I believe they can be classified.

http://mathoverflow.net/questions/66053/dissecting-a-square Colin D Wright 2011-05-26T15:03:49Z 2011-05-26T15:03:49Z

@fedja I'd be interested to see the assumptions that would let you get arbitrarily many pieces.

http://mathoverflow.net/questions/66053/dissecting-a-square Colin D Wright 2011-05-26T14:30:39Z 2011-05-26T14:30:39Z

Do I need to re-write the question again to make it clear what I mean by &quot;cut&quot; and &quot;piece&quot;? People are producing incomplete sets of dissections, but no one is talking about proofs of completeness. Is that just because it's hard?

http://mathoverflow.net/questions/66053/dissecting-a-square Colin D Wright 2011-05-26T14:01:46Z 2011-05-26T14:01:46Z

@Tapio we should probably add &quot;connected,&quot; and perhaps strengthen that to &quot;pathwise connected&quot;