Just wondering out of curiosity. For instance, this does not qualify:
Having been trying for a while, I would appreciate if someone can give a proof that it is impossible (in case it is). Thanks in advance :).
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1$\begingroup$ Are you asking whether a circle can be partitioned into several pieces, all of them congruent, where exactly one of the pieces does not touch the circumference of the circle? $\endgroup$– user326210Commented Jan 4, 2017 at 6:46
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$\begingroup$ Added a compact ppt-drawn image $\endgroup$– JoffanCommented Jan 4, 2017 at 13:24
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1 Answer
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If you mean at least one rather than exactly one, then the second tiling below qualifies:
This is from Haddley and Worsley, "Infinite families of monohedral disk tilings" (2015), which I found via the similar MathOverflow question "Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?". That question seems to still be an open problem.
