# Forming a polytope from only its combinatorial data

I would like to visualize a polytope given only its face lattice, i.e. something like

{1}, {2}, {3}, {4}, {1, 2}, {2, 3}, {3, 4}, {4, 1}, {1, 2, 3, 4}

for a square.

Is that possible in sage?

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From Sage version 8.9+ there will be CombinatorialPolyhedron, which can be initialized with a list of facets, as

CombinatorialPolyhedron([[1,2], [2,3], [3,4], [4,1]])


With this object you have a face iterator, the face lattice and some other methods. I don't know if that helps.

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If by "visualize" you mean "give a geometric realization of" then I think that the answer should be "no" as I rather suspect that there are high-dimensional combinatorial polytopes that have no geometric realization. Even for three-dimensional polytopes the realization problem is delicate - there are even several layout algorithms for planar graphs. (There is at least one such implemented in sage - see:

http --- fix --- doc.sagemath.org/html/en/reference/plotting/sage/graphs/graph_plot.html

). On the other hand, if you want to see the face poset in a graphical way, this is much easier. For example see this post:

https --- fix --- sheaves.github.io/Subgroup-Lattice/

and pay attention to the "plot" method of a Poset.

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