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单词 DehnsTheorem
释义

Dehn’s theorem


We all know the elementary formulaMathworldPlanetmathPlanetmath to compute the area of a triangleMathworldPlanetmath: basis times height divided by two. This formula can be justified with a scissor type argument:one divides the triangle into smaller polygonsMathworldPlanetmathPlanetmath and rearranges these polygons to obtain a rectangleMathworldPlanetmathPlanetmath which should have the same area.

Can we use the same argument to compute the volume of a pyramidMathworldPlanetmath? This isthe third Hilbert’s problem. Quite surprisingly the answer is negative, as states the theorem below. This means that the formulae to compute the volume of polyhedra cannot be proved without a limiting process (for example using integrals).

Definition 1.

We say that two polyhedra P and Q are scissor-equivalent if there exists a finite number P1,,PN of polyhedra and θ1,,θN isometries such that

  1. 1.

    P=k=1NPk and Q=k=1Nθk(Pk);

  2. 2.

    PjPk and θj(Pj)θk(Pk) have empty interiorfor every kj

The properties given above assure that two scissor-equivalent polyhedramust have the same volume. It is also simple to prove that the scissor-equivalence is indeed an equivalence relationMathworldPlanetmath.

Theorem 1.

The regular tetrahedronMathworldPlanetmathPlanetmathPlanetmath is not scissor-equivalent to any parallelepipedMathworldPlanetmath.

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更新时间:2025/5/4 2:53:23