Necker Tiler
The Necker Tiler App lets you create patterns (tilings) like the one shown in the background and the three shown below. These patterns each are projections of a polyhedral surfaces built from rectangles in ℝ3; see below for more detail. For more pictures like this, see the gallery.
Constructing the surface
The polyhedral surfaces described here are embedded in ℝ3; and built out of rectangles, each parallel to one of the three coordinate planes. The surface is determined by three functions f,g,h: ℤ ⟶ ℝ>0. How this works is explained by the following figure, which explains how several edge lengths are given by values of these functions. The remainder of the lengths of edges in this figure follow from the fact that all these polygons are rectangles, so opposite sides of each quadrilateral have the same length.
