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  "body": "\n## Description\n\nA *Kepler–Poinsot polyhedron* is a regular star [[polyhedron|0325SGSJAR]].\nIt is non-convex, its faces are congruent regular polygons or regular star polygons, and the same number of faces meet at each vertex.\nIts symmetries act transitively on its flags.\nThere are exactly four, given here with their [[Schläfli symbols|MAE6BF4DFK]], where $5/2$ denotes the pentagram:\n\n1. the small stellated dodecahedron $\\set{5/2, 5}$;\n2. the great dodecahedron $\\set{5, 5/2}$;\n3. the great stellated dodecahedron $\\set{5/2, 3}$;\n4. the great icosahedron $\\set{3, 5/2}$.\n\n## Remarks\n\nTheir vertices, edges and faces number $(V, E, F) = (12, 30, 12)$, $(12, 30, 12)$, $(20, 30, 12)$ and $(12, 30, 20)$ in this order.\nSo $V - E + F$ equals $-6$ for the first two and $2$ for the last two.\nThe first two do not satisfy Euler's formula $V - E + F = 2$, since their pentagram or crossing pentagon faces do not fit together into a surface shaped like a sphere.\nAll four share the icosahedral symmetry group of the icosahedron and dodecahedron.\nThey are regular in the flag sense of a [[regular polytope|F60PD912D5]], but not convex.\nKepler described the two stellated dodecahedra, Poinsot the other two, and Cauchy showed that there are no other regular star polyhedra.\n\n## Explanations\n\n### L1 human\n\nExtend the faces of a dodecahedron or an icosahedron until they meet again, and spiky star-shaped solids appear.\nFour of these are as regular as the Platonic solids: every face, edge and corner looks like every other.\nTheir faces pass through each other, which is why they are not convex.\n",
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