The original version of this story appeared in Quanta Magazine.
In 360 BC, Plato envisioned the cosmos as an arrangement of five geometric shapes: flat-sided solids called polyhedra. These immediately became important objects of mathematical study. So it might be surprising that, millennia later, mysteries still surround even the simplest shape in Plato’s polyhedral universe: the tetrahedron, which has just four triangular faces.
One major open problem, for instance, asks how densely you can pack “regular” tetrahedra, which have identical faces. Another asks which kinds of tetrahedra can be sliced into pieces that can then be reassembled to form a cube.
The great mathematician John Conway was interested not only in how
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