The property I'm talking about is:There is some partition of the plane figure P into n congruent figures for any n.Is it true that only discs, sectors of discs, annuli, sectors of annuli and parallelograms have this property?

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The property I&#039;m talking about is:There is some partition of the plane figure P into   n congruent figures for any   n.Is it true that only discs, sectors of discs, annuli, sectors of annuli and parallelograms have this property?

Leslie Rollins · Accepted Answer

The answer to your question is &quot;no&quot;.Let   a,  b be positive real numbers with   b  &amp;gt;  a, and   f be a continuous function mapping [a,b] to       R  . Let   X  =  {  (  x  ,  f  (  x  )  )  :  x  ∈  [  a  ,  b  ]  }.Now sweep out a planar area by rotating   X around the origin through some angle   θ in   (  0  ,  2  π  ].When   f  (  x  )  =  0, this yields your (sectors of, when   θ  &amp;lt;  2  π) annuli (or disks, when   a  =  0).We can think of your case of parallelograms as being the limit when   a is very, very large compared to   b  −  a; and where   f  (  x  )  =  (  x  −  a  )  c; we&#039;re (very loosely speaking!) rotating the figure around &quot;a point at infinity&quot;.But we aren&#039;t limited to a linear   f in either case; the only requirement is that there be no &quot;self-intersections&quot; as we sweep out the figure. And that can be satisfied if   d  (  x  ,  f  (  x  )  ) (the distance from the point   (  x  ,  f  (  x  )  ) to the origin) is injective; i.e., for all       x    0    ,      x    1    ∈  [  a  ,  b  ],   d  (      x    0    ,  f  (      x    0    )  )  =  d  (      x    1    ,  f  (      x    1    )  )  ↔      x    0    =      x    1  .So for example: Take a triangle shaped wedge out of one end of your sector-of-an-annulus, and slap it onto the other end. Assuming the triangle isn&#039;t too acute, the resulting figure will have the property you describe.Lots of much more complicated possibilities can be constructed, as long as we obey the constraint of &quot;distance from the origin is injective&quot;.(Edit: replace &quot;strictly increasing&quot; with &quot;injective&quot;.)

The property I'm talking about is: There is some partition of

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