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On a three-dimensional lattice, a random walk has less than unity probability of reaching any point (including the starting point) as the number of steps approaches infinity. ...

Define the packing density eta of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for ...

A tetromino is a 4-polyomino. There are five free tetrominoes, seven one-sided tetrominoes, and 19 fixed tetrominoes. The free tetrominoes are known as the T-tetromino, ...

Let m_1, m_2, ..., m_n be distinct primitive elements of a two-dimensional lattice M such that det(m_i,m_(i+1))>0 for i=1, ..., n-1. Each collection Gamma={m_1,m_2,...,m_n} ...

The triangular grid graph T_n is the lattice graph obtained by interpreting the order-(n+1) triangular grid as a graph, with the intersection of grid lines being the vertices ...

A triomino, also known as a tromino, is a 3-polyomino. There are two free polyominoes, (the same) two one-sided polyominoes, and 6 fixed polyominoes. The free polyominoes are ...

Let L denote the n×n triangular lattice with wraparound. Call an orientation of L an assignment of a direction to each edge of L, and denote the number of orientations of L ...

There are three types of cubic lattices corresponding to three types of cubic close packing, as summarized in the following table. Now that the Kepler conjecture has been ...

A fusene is a simple planar 2-connected graph embedded in the plane with all vertices of degree 2 or 3, all bounded faces (not necessarily regular) hexagons, and all vertices ...

In hexagonal close packing, layers of spheres are packed so that spheres in alternating layers overlie one another. As in cubic close packing, each sphere is surrounded by 12 ...