Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?

share|cite|improve this question
what do you mean by 'similar disjoint pieces'? do you maybe just mean 'disjoint pieces'? or do you mean disjoint pieces that also have the same shape? If you cut the circle (disk) up into pieces by a straight lines for example - i think there is only one possiblity to make those pieces the same shape (namely the lines all pass through the center of the disk and always halve the remaining pieces). – Peter Sheldrick Oct 12 '11 at 10:55
I mean "disjoint pieces that also have the same shape". But not necessarily the same size or rotation. I suspect its not possible but I'd like a proof. – user17497 Oct 12 '11 at 11:37
up vote 8 down vote accepted

A version of this question was asked on MO a while back: "Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?." It was determined that that particular problem (which I now think is identical to yours as currently posted) is open, and appears in Unsolved Problems in Geometry.

There Anton Geraschenko posted this interesting dissection:
which answers another version (perhaps the first version?) of your question. In the comments another dissection was given by 'sobe86':

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.