The same set of vertices can be connected by edges in different ways. For example the pentagon and pentagram have the same vertex arrangement, while the second connects alternate vertices.
pentagon |
pentagram |
A vertex arrangement is often described by the convex hull polytope which contains it. For example, the regular pentagram can be said to have a (regular) pentagonal vertex arrangement.
| ABCD is a concave quadrilateral (green). Its vertex arrangement is the set {A, B, C, D}. Its convex hull is the triangle ABC (blue). The vertex arrangement of the convex hull is the set {A, B, C}, which is not the same as that of the quadrilateral; so here, the convex hull is not a way to describe the vertex arrangement. |
Infinite tilings can also share common vertex arrangements.
For example, this triangular lattice of points can be connected to form either isosceles triangles or rhombic faces.
Lattice points |
Triangular tiling |
rhombic tiling |
Zig-zag rhombic tiling |
Rhombille tiling |
Read more about this topic: Vertex Arrangement
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