A tetrahedron is the simplest Platonic solid. Mathematicians have now made one that’s stable only on one side, confirming a decades-old conjecture.

In 360 BCE, 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 tetrahedra can be arranged or rearranged, but also in how they balance. In 1966, he and the mathematician Richard Guy asked whether it was possible to construct a tetrahedron made of a uniform material — with its weight evenly distributed — that can only sit on one of its faces. If you were to place such a “monostable” shape on any of its other faces, it would always flip to its stable side.

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