The limit shapes of midpoint polygons in R³

For a polygon in the d-dimensional Euclidean space, we give two kinds of normalizations of its mth midpoint polygon by a homothetic transformation and an affine transformation, respectively. As m goes to infinity, the normalizations will approach "regular" polygons inscribed in an ellipse and a generalized Lissajous curve, respectively, where the curves may be degenerate. The most interesting case is when d = 3, where polygons with all its mth midpoint polygons knotted are discovered and discussed. Such polygonal knots can be seen as a discrete version of the Lissajous knots.