# Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?

Problem: is it possible to dissect the interior of a circle into a finite number of congruent pieces (mirror images are fine) such that some neighbourhood of the origin is contained in just one of the pieces?

It may be conceivable that there is some dissection into immeasurable sets that does this. So a possible additional constraint would be that the pieces are connected, or at least the union of connected spaces.

A weaker statement, also unresolved : is it possible to dissect a circle into congruent pieces such that a union of some of the pieces is a connected neighbourhood of the origin that contains no points of the boundary of the circle?

This is doing the rounds amongst the grads in my department. So far no one has had anything particularly enlightening to say - a proof/counterexample of any of these statements, or any other partial result in the right direction would be much obliged!

Edit: Kevin Buzzard points out in the comments that this is listed as an open problem in Croft, Falconer, and Guy's Unsolved Problems in Geometry (see the bottom of page 87).

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The first question is listed as an unsolved problem in "Unsolved problems in geometry" by Croft, Falconer and Guy (section C6). So unless anyone has any recent news, you're probably not going to see a solution posted here. PS I am not an "unsolved problems in geometry" guru---I just googled for 'cutting a circle into congruent pieces' (without quotes) to see if there were any interesting examples on the web and it led me straight to the book. – Kevin Buzzard Mar 6 '10 at 20:44
I should probably also share this link: forums.xkcd.com/viewtopic.php?f=3&t=27963 the users there decided the first problem was 'probably impossible', but no major progress was made on a proof... This diagram is an example of a near miss for the second problem: i.imgur.com/iOfRI.png . This is why I think it may be possible... – sobe86 Mar 6 '10 at 20:51
Please edit the title to reflect the question. "Is it possible to dissect the circle into congruent pieces, so that a neighborhood of the origin is contained within a single piece?" is plenty short enough to fit in a title. – Theo Johnson-Freyd Mar 7 '10 at 1:02
Actually, "disk" would be a much better word that "circle". – Theo Johnson-Freyd Mar 7 '10 at 1:03
I think the question should be edited to incorporate Kevins comments about the first question being listed as an open problem in Croft, Falconer and Guy. – Grétar Amazeen Mar 7 '10 at 1:16

Given that this is an open problem, I figured I may as well make these comments an "answer".

Here's a related problem that I don't know how to answer:

1. Is any dissection of the disk into a finite number of congruent pieces rotationally symmetric?

It seems likely (to me) that any example of a dissection into a finite number of congruent pieces so that the center is in the interior of one of them is going to fail to be rotationally symmetric.

Actually, there's an easier problem that I don't know the answer to:

2. Is every dissection of the disk into a finite number of congruent pieces one of the following?

• Slices of pizza. (i.e. every two pieces are congruent via a rotation around the center of the circle)
• This one:
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There's a trivial negative answer to the last question, in that if you take two adjoining pieces in this 12-piece dissection you get a dissection into 6 copies. – Gerry Myerson Mar 8 '10 at 5:33
@Gerry: Unless I've misunderstood what you mean, that would give you a dissection into 6 pizza slices. – Anton Geraschenko Mar 8 '10 at 5:46
The answer to both of these questions is no. See the following example: imgur.com/bxtbB.png I think a harder task would be to find a dissection such that any of the pieces do not touch the boundary at all... – sobe86 Mar 8 '10 at 16:27
@sobe86: very nice! – Anton Geraschenko Mar 8 '10 at 17:35
I retract my comment, which was based on a misunderstanding of the concept of pizza slices. – Gerry Myerson Mar 8 '10 at 23:03

A new paper was posted to the arXiv on related questions: "Infinite families of monohedral disk tilings," by Joel Haddley and Stephen Worsley (arXiv abs.). Here are tilings of the disk into congruent pieces where at least one piece does not touch the center (a result mentioned by Anton Geraschenko):

They conjecture (with extensive support) that,

"for any monohedral tiling of the disk, the centre may only intersect a tile at a vertex."

This would answer the original posed question in the negative.

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So the second tiling ($D_3^1$) answers the weaker question in the affirmative, right? – Fan Zheng yesterday
@FanZheng: The weaker question: "Is it possible to dissect a circle into congruent pieces such that a union of some of the pieces is a connected neighbourhood of the origin that contains no points of the boundary of the circle?" Excellent observation, Fan: Indeed you are correct. – Joseph O'Rourke yesterday