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Dissection Recreation


Draw a rectangle...

Dissect (i.e. cut) it into two congruent pieces.

Can you do it in more than one way? That is, think about...

  • Dividing the rectangle into two congruent rectangles using a line drawn between midpoints on opposite sides (does it matter which pair of sides?)
  • Dividing the rectangle into two congruent triangles using a line drawn between opposite vertices
  • Or, can you find other ways...such as L-shaped blocks...or even, trapazoids formed by a drawing a line from 1/3-point on one side to the 2/3-point on the opposing side?
Now, change the task: Dissect a rectangle into three congruent pieces.

Can it be done in more than one way?

And finally, shift the problem using these modifications:

  • Dissect a rectangle into n congruent pieces, for n>3
  • Dissect other quadrilaterals (e.g. non-rectangular trapexoids, parallelograms, kites, general convex or concave quadrilaterals) into 2, 3, ... congruent pieces. Possible?
  • Dissect a rectangular parallelpiped (i.e. box) into 2, 3, ... congruent pieces. Possible?
  • What about dissecting other 3-dimensional and/or n-dimensional shapes into congruent pieces?

 


Hint: Hints have already been provided for the case of dissection into two congruent shapes...

 


Solution Commentary: Perhaps you arte frustrated trying the case of three congruent sections of a rectangle....but hopefully it was fun trying!

In 1994, Samuel Maltby proved: If a rectangle is dissected into three congruent pieces, the pieces themselves must be rectangles. [Journal of Combinatorial Theory, 66: 40-52].

His proof is lengthy and very technical, but uses only elementary mathematics involving generalizations akin to working with polyomino tiles.

As to the other suggested generalizations to 2-dimensional quadrilaterals and those of higher dimensions, I believe they are still unsolved problems.