Lesson plan
Coordination sequence
In crystallography and graph theory, the coordination sequence of a vertex v is an integer sequence that counts how many vertices are at each possible distance from v . That is, it is a sequence n_0, n_1, n_2,\dots where each n_i is the number of vertices that are i steps away from v . If the graph is vertex-transitive, then the sequence is an invariant of the graph that does not depend on the specific choice of v . Coordination sequences can also be defined for sphere packings, by using either the contact graph of the spheres or the Delaunay triangulation of their centers, but these two choices may give rise to different sequences. As an example, in a square grid, for each positive integer i , there are 4i grid points that are i steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence 1,4,8,12,16,20,\dots\ . in which, except for the initial value of one, each number is a multiple of four. The concept was proposed by Georg O. Brunner and Fritz Laves and later developed by Michael O'Keefe. The coordination sequences of many low-dimensional lattices and uniform tilings are known. The coordination sequences of periodic structures are known to be quasi-polynomial.
Article summary
In crystallography and graph theory, the coordination sequence of a vertex v is an integer sequence that counts how many vertices are at each possible distance from v . That is, it is a sequence n_0, n_1, n_2,\dots where each n_i is the number of vertices that are i steps away from v . If the graph is vertex-transitive, then the sequence is an invariant of the graph that does not depend on the specific choice of v . Coordination sequences can also be defined for sphere packings, by using either the contact graph of the spheres or the Delaunay triangulation of their centers, but these two choices may give rise to different sequences. As an example, in a square grid, for each positive integer i , there are 4i grid points that are i steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence 1,4,8,12,16,20,\dots\ . in which, except for the initial value of one, each number is a multiple of four. The concept was proposed by Georg O. Brunner and Fritz Laves and later developed by Michael O'Keefe. The coordination sequences of many low-dimensional lattices and uniform tilings are known. The coordination sequences of periodic structures are known to be quasi-polynomial.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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