Maximize the Number of Vertices in an Equilateral Triangular Lattice
An equilateral triangular lattice is parameterized by four variables: an and offset for the initial vertex, the angle of the grid with respect to the horizontal, and a scale parameter for the edge lengths. In this Demonstration, you can vary these parameters and see how many vertices and edges fit in the open workspace (white). Drag the locator to create a "narrow passage," while keeping the area of the workspace constant.
Let if is even, and 0 otherwise. The positions of the vertices in an equilateral triangular lattice indexed by are .
These positions can be modified by a scale parameter , a rotation , and the offset :
Narrow passages are an important concept in motion planning. A narrow passage is a small region whose removal changes the connectivity of the configuration space.