From Pentagons to Pentagrams

10 hours ago

Galois conjugation, replacing √5 with -√5, interchanges pairs of polyhedra within a family of six, including the regular icosahedron and great icosahedron, regular dodecahedron and great stellated dodecahedron, and great dodecahedron and small stellated dodecahedron.
This operation maps regular pentagons to regular pentagrams when applied to vertices in the golden field (ℚ[√5]), due to the effect on the exterior turning angle's cosine, and generalizes to transformations of polyhedra with coordinates in that field.
The transformation extends to other shapes like the rhombicosidodecahedron, converting its pentagons into pentagrams while leaving squares and triangles unchanged if their edge squared lengths are rational.

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