Erdos distance problem n=12 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:53:08Z http://mathoverflow.net/feeds/question/58203 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58203/erdos-distance-problem-n12 Erdos distance problem n=12 jkruso 2011-03-11T22:25:46Z 2011-03-12T01:46:35Z <p>The recent paper On the Erdos distinct distance problem in the plane Authors: Larry Guth, Nets Hawk Katz prodded me to get a non-trivial example. Here is what I cannot find: and example of 12 distinct points in the plane with only 5 different distances between points. The regular 12-polygon has 6 different lengths but I cannot do better. <a href="http://oeis.org/A186704" rel="nofollow">http://oeis.org/A186704</a> implies that there is one></p> http://mathoverflow.net/questions/58203/erdos-distance-problem-n12/58207#58207 Answer by The Cheese Stands Alone for Erdos distance problem n=12 The Cheese Stands Alone 2011-03-11T23:07:24Z 2011-03-11T23:07:24Z <p>I wish I could get images to work, but here goes my poor explanation:</p> <p>Take a regular hexagonal lattice with distance 1 between nearest neighbors, and choose a 15-point equilateral triangle in this lattice (15 is a triangular number). Remove the 3 verticies of the triangle. You'll be left with 12 points and 5 distinct distances.</p> <p>edit: just checked the OEIS reference, and it's available on google books. The picture you want is on page 200 at <a href="http://books.google.com/books?id=cT7TB20y3A8C&amp;printsec=frontcover&amp;dq=Research+Problems+in+Discrete+Geometry&amp;source=bl&amp;ots=amqJ7zFfB4&amp;sig=U99_5spjO8UIwbehycahkz6M2yg&amp;hl=en&amp;ei=Hql6TeyFKpDrrAHm7bzCBQ&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=5&amp;ved=0CEMQ6AEwBA#v=onepage&amp;q&amp;f=false" rel="nofollow">http://books.google.com/books?id=cT7TB20y3A8C&amp;printsec=frontcover&amp;dq=Research+Problems+in+Discrete+Geometry&amp;source=bl&amp;ots=amqJ7zFfB4&amp;sig=U99_5spjO8UIwbehycahkz6M2yg&amp;hl=en&amp;ei=Hql6TeyFKpDrrAHm7bzCBQ&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=5&amp;ved=0CEMQ6AEwBA#v=onepage&amp;q&amp;f=false</a></p> http://mathoverflow.net/questions/58203/erdos-distance-problem-n12/58209#58209 Answer by jkruso for Erdos distance problem n=12 jkruso 2011-03-11T23:24:54Z 2011-03-11T23:24:54Z <p>Thanks this was driving me crazy.</p> http://mathoverflow.net/questions/58203/erdos-distance-problem-n12/58213#58213 Answer by Joseph O'Rourke for Erdos distance problem n=12 Joseph O'Rourke 2011-03-12T00:05:17Z 2011-03-12T01:46:35Z <p>If I may supplement Logan Maingi's apposite answer with a snapshot of the page to which he refers:</p> <p><img src="http://cs.smith.edu/~orourke/MathOverflow/Erdos12.jpg" alt="p.200"> <br /> (I couldn't resist including the surrounding conjecture.)</p>