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I think I read somewhere that at one time it was thought the only way to lay tiles that would fill a circle with no overlap of the tiles and no exposed space in the cirlce, was to lay pieces that would converge at the center of the cirlce and go around the circle, the simplest example being a pie sliced in, say, eight even pieces. This was demonstrated to be false by a circle with congruent tiles that weren't all rotated about the center. What did this pattern look like.

Sorry if I'm not being clear or if I'm not remembering correctly.

Thanks for your help.

RobPratt
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Peter
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    See here for some examples of rather "nontrivial" dissections: https://cp4space.hatsya.com/2012/12/20/dissecting-the-disc/ – Wojowu Feb 12 '23 at 05:13
  • With the pie slices, what happens to the center of the circle? – Gerry Myerson Feb 12 '23 at 22:31
  • @GerryMeyerson The interiors of the tiles are required to be disjoint, not the boundaries. Otherwise none of these tilings would work. – Robert Israel Feb 13 '23 at 00:55
  • See also https://mathoverflow.net/questions/17313/is-it-possible-to-dissect-a-disk-into-congruent-pieces-so-that-a-neighborhood-o (although that url is currently giving me an error message, but Google has the page cached so you can see it deals with essentially the same question as this one). – Gerry Myerson Feb 15 '23 at 23:58

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