The Mystery of 355/113
I wrote a quick script to walk through Q, spitting out fractions with high "quality". Here are some early values:<p><pre><code>  22/7: 3.42928833728
  355/113: 3.20195874233
  710/226: 2.79251047296
</code></pre>
A quick search through 1/1 - 100,000/100,000 shows no other interesting values. My algorithm is extremely naive though (brute force), a faster search could potentially be used to find other high quality values much more quickly.<p><i>Update</i> - A slightly optimized algorithm, and nothing special found with denominators up to the hundreds of millions. I know this means nothing wrt actual mathematics though...