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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: an 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. http://oeis.org/A186704 implies that there is one>

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2 Answers 2

up vote 13 down vote accepted

I wish I could get images to work, but here goes my poor explanation:

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.

edit: just checked the OEIS reference, and it's available on google books. The picture you want is on page 200 at http://books.google.com/books?id=cT7TB20y3A8C&printsec=frontcover&dq=Research+Problems+in+Discrete+Geometry&source=bl&ots=amqJ7zFfB4&sig=U99_5spjO8UIwbehycahkz6M2yg&hl=en&ei=Hql6TeyFKpDrrAHm7bzCBQ&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEMQ6AEwBA#v=onepage&q&f=false

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If I may supplement Logan Maingi's apposite answer with a snapshot of the page to which he refers:

p.200
(I couldn't resist including the surrounding conjecture.)

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Thanks! I tried to no avail to embed the image, but I'm still rather bad with TeX and couldn't get it to work. –  Logan Maingi Mar 14 '11 at 14:35

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