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 :).
Can a circle be divided into congruent shapes with one not sharing a point with circumferences?
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$\begingroup$
combinatorics
geometry
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1Are 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? – 2017-01-04
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0Added a compact ppt-drawn image – 2017-01-04
1 Answers
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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.
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0What an interesting paper! I particularly like this one [circ_tess](https://i.stack.imgur.com/KcDhd.jpg) – 2017-01-04