Tripod packing

rdf:langString Tripod packing

rdf:langString In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube, the union of the grid cubes along three positive axis-aligned rays with a shared apex. Several problems of tiling and packing tripods and related shapes were formulated in 1967 by Sherman K. Stein. Stein originally called the tripods of this problem "semicrosses", and they were also called Stein corners by Solomon W. Golomb. A collection of disjoint tripods can be represented compactly as a monotonic matrix, a square matrix whose nonzero entries increase along each row and column and whose equal nonzero entries are placed in a monotonic sequence of cells, and the problem can also be formulated in terms of finding sets of triples satisfying a compatibility condition called "2-comparability", or of finding compatible sets of triangles in a convex polygon. The best lower bound known for the number of tripods that can have their apexes packed into an grid is , and the best upper bound is , both expressed in big Omega notation.