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One suspects, then, that The Obvious Necessary Condition is Also Sufficient. There are no graphs (yet) known which satisfy this condition but do not tile $Q_n$ if $n$ is sufficiently large.There is an obvious necessary condition: each of the $2^n$ vertices in $Q_n$ is supposed to be used exactly once, which means that the number of vertices in $G$ had better divide $2^n$ in other words, $V(G)=2^k$ for some $k$.In both cases $G$ is the graph on two vertices with an edge between them.) (In this way, tiling is a generalization of finding maximal matchings, and packing is a generalization of finding perfect matchings. When we say “packing” or “tiling”, we mean that we want to cover as many vertices as possible such that each edge and each vertex is used at most once- and for tiling we demand vertices are used exactly once. This talk focused on the case where the larger graph was the $n$-dimensional hypercube $Q_n$,Īnd the smaller graph $G$ was allowed to be arbitrary. Moreover, one may discretize the problem: instead of packing a shape with copies of a smaller shape, you can instead try to pack a graph with copies of a smaller graph. The “packing” variant naturally extends to any shape there’s no obvious reason to choose spheres except that they seem pretty simple.įor both problems, there’s no reason to choose the plane except that you then don’t have to worry about what happens near the “edges”: you could in principle try to pack spheres into a cube, or a larger sphere or you can ask what shapes tile an octagon etc. However, one can ask the question of how well it is possible to do: how should we put spheres in space so they cover as much space as possible? Can you cover at least 95%? (no) Can you cover at least 75%? (yes in the plane, but no in space) Sphere Packing: It’s probably obvious that spheres of any dimension, if they are all of the same size, will never fill up space: they are round and therefore just don’t fit together nicely.This problem isn’t easy, but it’s nothing compared to… A newer variation, but not actually very different, is to extend the question to higher dimensions. Without too much issue, you can extend the question to 3D space instead. Tiling: figuring out what sort of shapes, if you have infinitely many copies of them, can cover an entire plane without gaps or overlap.I’ll begin by talking about two very classical (read: old) problems in geometry: (I looked around for how I usually introduced JMM posts, and I discovered that I don’t, unless to highlight that it was a “big name” talk… I’m just normally writing close enough to the meetings that you were supposed to understand, I guess.) I’ll spend most of the post on setting up the problem, and present the part of the talk related to the title briefly at the end.
![hypercube in mathematica hypercube in mathematica](http://i.stack.imgur.com/QkgTY.png)
He cited Griggs, Milans, and Stoner as collaborators. This talk was given by David Offner at the 2016 JMM.
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