EdgeTwice

Scientists have designed cages made of one long protein chain. The chain runs along every edge of the cage twice, and the two pieces on each edge stick together in pairs. Can you draw a route like that without lifting your pen?

Toy model A simplified picture to show one idea. Not a measurement, and not a prediction. Of a designed protein cage: corners and edges only, where the real chain is coiled pieces that pair up along each edge.

Pick a cage:

Why it always works

Count the edge-ends at each corner. When every edge is used twice, each corner has an even number of them, so a route that comes in can always go out again. Leonhard Euler worked out this rule in 1736, and Carl Hierholzer proved in 1873 that such a route always exists. The tetrahedron has 6 edges, so the chain has 12 pieces; the square pyramid needs 16; the triangular prism 18.

These cages were designed by people and checked by experiments in the lab (a tetrahedron in 2013, then a pyramid and a prism in 2017). They are not natural proteins, and they are not in the AlphaFold Database.