With just six pieces we can cover 97_ o

Tim Hutton - 2013-12-03 17:31:34+0000 - Updated: 2013-12-03 17:32:41+0000

With just six pieces we can cover 97% of a circle and a square of equal area, without overlapping. Can you beat this? What about for five pieces? Join in the challenge at https://github.com/timhutton/circle-squaring

Background: https://math.stackexchange.com/questions/553571/cutting-up-a-circle-to-make-a-square

With just six pieces we can cover 97% of a circle and a square of equal area, without overlapping. Can you beat this? What about for five pieces? Join in the challenge at https://github.com/timhutton/circle-squaring Background: https://math.stackexchange.com/questions/553571/cutting-up-a-circle-to-make-a-square

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+1'd by: Irena Steiner, Nick Holder, leonhard carl, Ondřej Čertík, Paul Fisher, David Moore, Štěpán Roučka

Tim Hutton - 2013-12-04 10:27:21+0000

The records for up to 10 pieces have now been beaten a total of 213 times. The current best for N=6 is 97.95%.

Dave Greene - 2013-12-16 20:08:25+0000

Interesting optimzation problem.  Looks like it's trivial to get N=8 and above to 99%+, just by "sliver management"... but stealing the N-1 solution feels suspiciously like cheating.  Outlawing slivers altogether might produce more elegant solutions (?).

Tim Hutton - 2013-12-16 20:41:38+0000

Yeah I don't like the slivers. Tempted to ban them but it seems too arbitrary. We need a better automated search! The code I've tried so far (see the search folder) is terrible at it.

Dave Greene - 2013-12-16 21:04:53+0000

I wouldn't be surprised if there's a non-sliver layout that can beat each of the sliver-contaminated solutions.  With any luck, if the automated search gets a bit better, and it's biased against slivers, that might be enough to banish them.

I used the auto mode to clean up after readjusting slivers, but it's mighty tricky with the current GUI without a zoom mode.  As long as I was very careful not to create overlaps, no damage would be done.  If a corner of a sliver touched any other object, though, all bets were off -- auto mode would usually wreck the layout.

Dave Greene - 2013-12-18 21:31:54+0000

Never mind the high-N sliver solutions, though -- the interesting case to work on is N=2.  Why exactly can't that strange asymmetric smaller shape be improved upon?  It's certainly possible to clean up the N=2 solution, bringing the two pieces into contact along a nice simple straight line without losing area.  (I have a saved solution that's 92.87715%, but that's apparently not enough of an improvement to submit.)  But once the contact zone becomes a straight line, what is that line's ideal slope, and why?

Dave Greene - 2013-12-18 21:33:57+0000

I'd like to be able to rotate the smaller N=2 piece by one quantum clockwise or counterclockwise, and then reestablish a straight-line boundary between the pieces.  At the moment this can't be done in auto mode without causing serious damage, and in manual mode it requires moving too many nodes.

Might it be possible to have some kind of "Snap To Boundaries" function in Manual mode, lining up the selected piece's edges with adjacent boundaries and trimming off any overlaps automatically, but definitely not unleashing auto mode on all the pieces at once? 

Tim Hutton - 2013-12-18 22:35:27+0000

Would being able to run auto mode on a single piece help? It would leave the others untouched. (The N=2 has been jiggled automatically, so there shouldn't be easy improvements to find.)

Dave Greene - 2013-12-18 22:43:09+0000

Yes, I think that'd probably be the first thing on my wishlist.

But I'm also trying to figure out how to get cleaner-looking connections between pieces.  As it stands, after test-rotating a piece, you often end up with minor wobbles along edges that might as well be simple straight lines or circular arcs.

Tim Hutton - 2013-12-18 23:10:04+0000

There's a 0.01 minimum improvement threshold before the submit button lights up.

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