Hilbert's Third Problem - Further Information

Further Information

In light of Dehn's theorem above, one might ask "which polyhedra are scissors-congruent"? Sydler (1965) showed that two polyhedra are scissors-congruent if and only if they have the same volume and the same Dehn invariant. Børge Jessen later extended Sydler's results to four dimensions. In 1990, Dupont and Sah provided a simpler proof of Sydler's result by reinterpreting it as a theorem about the homology of certain classical groups.

Debrunner showed in 1980 that the Dehn invariant of any polyhedron with which all of three-dimensional space can be tiled periodically is zero.

Read more about this topic:  Hilbert's Third Problem

Famous quotes containing the word information:

    I believe it has been said that one copy of The Times contains more useful information than the whole of the historical works of Thucydides.
    Richard Cobden (1804–1865)

    Rejecting all organs of information ... but my senses, I rid myself of the Pyrrhonisms with which an indulgence in speculations hyperphysical and antiphysical so uselessly occupy and disquiet the mind.
    Thomas Jefferson (1743–1826)