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I am trying to get VertexCoordinates of this picture of the problem https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_21?fbclid=IwAR2bdHwJEpjarZEwope1hYOcnnXqnvzae38Y8ZrfWKCKQunfAuT9s30q32o

I tried with syntax PolyhedronData["Dodecahedron", "VertexCoordinates"] but I do not know the name of Polyhedron. How can I get VertexCoordinates of this polyhedron?

enter image description here

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That looks like half of a truncated octahedron.

truncOctVertices = PolyhedronData["TruncatedOctahedron", "VertexCoordinates"]

will get you all of the vertex coordinates.

We can filter to get just the lower half (z value nonpositive):

lowHalfVertices = Select[truncOctVertices, NonPositive@*Last]

We can visualize to confirm:

ConvexHullMesh[lowHalfVertices]

enter image description here

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  • $\begingroup$ How can I get FaceIndices by usingPolyhedronData["lowHalfVertices", "FaceIndices"]? $\endgroup$ Commented Dec 22, 2022 at 1:48
  • $\begingroup$ What are you trying to do? FaceIndices would give you insight into Mathematica's representation of a polyhedron, and that seems sort of out of left field based on your original question. Also, once we've turned it into a BoundaryMeshRegion, we have a different representation. There are functions that allow you to extract data from mesh regions: MeshPrimitives, MeshCells, MeshCoordinates, and yes, MeshCellIndex. $\endgroup$ Commented Dec 22, 2022 at 2:08

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