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  "body": "\n## Summary\n\nAn abstract polytope is a partially ordered set that behaves like the set of faces of a polytope ordered by inclusion.\nIt keeps only the incidence pattern of vertices, edges and higher faces and forgets all geometry.\nThe axioms ask for a grading by dimension, a connectedness condition, and the diamond condition.\nThe diamond condition says that between two faces whose dimensions differ by two there are exactly two faces.\nAbstract polytopes include the face lattices of convex polytopes, but also tilings and objects such as the hemicube that no convex polytope realises.\n\n## Explanations\n\n### L1 human\n\nWrite down the vertices, edges and faces of a shape, and record only which ones touch which.\nThis list of incidences, ordered by \"is part of\", is the skeleton that an abstract polytope keeps.\nIn a polygon, each vertex is on exactly two edges, and each edge has exactly two vertices; the diamond rule generalises this to all dimensions.\nShapes like the hemicube, made by gluing opposite faces of a cube together, fit the rules even though they cannot be built as convex solids.\n",
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