Herschel graph
Polyhedron

By Steinitz's theorem, every graph that is planar and 3-vertex-connected has a convex polyhedron with the graph as its skeleton. Because the Herschel graph has these properties, it can be represented in this way by a convex polyhedron, an enneahedron having polyhedron has nine quadrilaterals as its faces. This can be chosen so that each graph automorphism corresponds to a symmetry of the polyhedron, in which case three of the faces will be rhombi or squares, and the other six will be kites.

The dual polyhedron is a rectified triangular prism, which can be formed as the convex hull of the midpoints of the edges of a triangular prism. When constructed in this way, it has three square faces on the same planes as the square faces of the prism, two equilateral triangle faces on the planes of the triangular ends of the prism, and six more isosceles triangle faces. This polyhedron has the property that its faces cannot be numbered in such a way that consecutive numbers appear on adjacent faces, and such that the first and last numbers are also on adjacent faces, because such a numbering would necessarily correspond to a Hamiltonian cycle in the Herschel graph. Polyhedral face numberings of this type are used as "spindown life counters" in the game Magic: The Gathering, to track player lives, by turning the polyhedron to an adjacent face whenever a life is lost. A card in the game, the Lich, allows players to return from a nearly-lost state with a single life to their initial number of lives. Because the dual polyhedron for the Herschel graph cannot be numbered in such a way that this step connects adjacent faces, Constantinides & Constantinides (2018) name the canonical polyhedron realization of this dual polyhedron as "the Lich's nemesis".
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