Polylines

Connect the dots is a simple game we play with pencil on paper to outline a hidden image that emerges.  We connect numbered dots in order, with line segments, and most often, these connected sticks form a non-intersecting loop that can be called a simple closed polyline path, a polygonal circuit, or just a polygon.    The game can be played in 3-dimensional space as well, where the numbered dots can be chosen from an infinite regular array of points called a lattice.   A familiar lattice is the cubic lattice, a 3-d version of square graph paper:  its points are at the corners of neatly stacked cubes that fill space.

Instead of numbering dots to be connected, a polyline path can also be traced out (on the plane or in space) by giving a series of instructions for the pencil (or “turtle”) to follow, Move x units far in direction y, then Turn through angle z, then continue in a sequence of Moves and Turns for perhaps different x, y, and z values.  In space, a Turn may also include a roll, like an acrobatic airplane move.  To complete a circuit, the last Move connects the path to the starting point.  In the plane, instructions that repeat exactly the same Moves and Turns trace out regular polygons:  equilateral triangle, square, hexagon, etc.  In space, polyline paths can dip and twist in many directions before completing a circuit.