What's the story for the sphere instead of the circle? That is, what ways are there, if any, to dissect a sphere into a finite number of congruent pieces, other than the pieces just being fundamental domains of some finite subgroup of O(3)? --Michael On Thu, Apr 19, 2012 at 1:53 PM, Michael Reid <reid@gauss.math.ucf.edu>wrote:
What a great problem! I was still working through the first few minutes of "but... how... isn't..." when Jessica said "Okay, how about this?..." and solved it.
But is there more than one solution? I know about the answer depicted in http://i.imgur.com/iOfRI.png, but is that unique?
Ah: it's not unique, in that there are solutions like http://i.imgur.com/bxtbB.png as well.
Are there any solutions that aren't a refinement of the basic dissection you get by gluing together pairs of pieces from either of the images above, though? That seems like it might be provable.
There are also dissections into 6n pieces, of which 6 border on the center, and the others do not.
By the way, this has been around for a while. I first(?) saw it in Croft, Falconer and Guy's "Unsolved Problems in Geometry". It's on page 89 of my 1991 edition, well, the first solution Michael gave. The generalization (to his second solution) I mention above is more or less "obvious". Your mileage may vary.
Michael Reid
--Michael
On Wed, Apr 18, 2012 at 10:50 PM, Bill Gosper <billgosper@gmail.com
wrote:
Colin Wright (e.g., *Colin Wright*: The Mathematics of *Juggling* on Vimeo<http://vimeo.com/27998521> )
posed this at G4GX (and G9?): Dissect a disk into congruent parts
at least one of which avoids the center by a positive distance. Neil
just beat me to a solution. Grr.
--rwg
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