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Volumes of projections of unit cubes, Peter McMullen, Bull LMS 1984, doi.org/10.1112/blms/16.3.278

A cute theorem that deserves to be better known: if you hold a unit cube in the noonday sun, at any angle, its shadow's area equals its height (elevation difference between lowest and highest point). It follows immediately that the biggest possible shadow is a hexagon with area = long diagonal length = \(\sqrt{3}\), and the smallest shadow is a unit square. Similar things happen in higher dimensions.

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