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Winning the 1992 Pólya Award

Howard Eves
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Summary

Two planar pieces that can be placed so that they intercept chords of equal length on each member of some family of parallel lines, or two solid pieces that can be placed so that they intercept sections of equal area on each member of some family of parallel planes, are said to be Cavalieri congruent.

Some years ago I proved the following two theorems which, at first encounter scarcely seem to be true:

Though one can exhibit two tetrahedra of equal volume that are not Cavalieri congruent, any two triangles of equal area are Cavalieri congruent.

Though one can easily show that there exists no polygon Cavalieri congruent to a circle, there exists a tetrahedron Cavalieri congruent to a sphere.

It was my publishing of these two theorems that won me the 1992 Pólya Award. The certificate of the award was accompanied by the following paragraph.

This short article really packs a wallop! The author's two theorems indeed “at first encounter scarcely seem to be true,” and the proofs are excellent illustrations of the power and beauty of geometry. The Cavalieri equivalence of a sphere and a tetrahedron is truly memorable—the sort of result which geometers in ancient times would have inscribed on their tombstones. The article is graced with the author's historical scholarship and lucid prose.

Cavalieri equivalence of a sphere and a tetrahedron

Theorem.There exists a tetrahedron to which a given sphere is Cavalieri congruent.

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Mathematical Reminiscences , pp. 169 - 170
Publisher: Mathematical Association of America
Print publication year: 2001

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