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Cutting a sphere into pieces of larger volume

Is it possible to cut a sphere into a finite number of pieces and reassemble into a solid of twice the volume?

This question has many variants and it is best answered explicitly.

Given two polygons of the same area, is it always possible to dissect one into a finite number of pieces which can be reassembled into a replica of the other?

Dissection theory is extensive. In such questions one needs to specify

Some dissection results

References

Boltyanskii. Equivalent and equidecomposable figures. in Topics in Mathematics published by D.C. HEATH AND CO., Boston.

Dubins, Hirsch and ? Scissor Congruence American Mathematical Monthly.

``Banach and Tarski had hoped that the physical absurdity of this theorem would encourage mathematicians to discard AC. They were dismayed when the response of the math community was `Isn't AC great? How else could we get such counterintuitive results?' ''



next up previous contents
Next: The Continuum Hypothesis Up: Axiom of Choice and Previous: The Axiom of Choice



Alex Lopez-Ortiz
Fri Feb 20 21:45:30 EST 1998