Infinite Cranes 1

A lot has been done with the traditional origami crane (orizuru), including folding many cranes from a single sheet of paper. One method of doing this is Renzuru, where a piece of paper is cut multiple times (but never separated from itself) and then folded into many cranes. Another approach was taken by Robert Lang, who produced a crane tessellation from one square, no cuts.

I’ve followed Lang’s approach (one square, no cuts) in my design. It yields a grid of cranes, each touching the rest of the paper (which forms a flat square) at a point on the end of each crane’s wingtip. The cranes alternate between positive and negative 45 degree angles along the rows and columns of the grid. The spacing between rows and columns of cranes is adjustable.

The unit tile of this tessellation is a diamond touching the base plane of the paper at a point at the tip of the diamond’s corner. The diamond formed at the first iteration has a large connection (the same width as the diamond) with the rest of the paper.

By applying a fractal pattern, the connection at the base can be reduced to a single point (at the limit). Once complete, the diamond can be folded into the preliminary base, and a crane folded from that. There is no leftover paper after accounting for the plane and the cranes.

In practice this design has significant problems with too many layers building up very quickly. The number of layers doubles each time the connection between the diamond and plane is thinned, and it starts with more that a few layers. The diamond can be formed cleanly to two or three iterations, but at that point there are so many layers towards the bottom that folding the crane becomes awkward at best. The tessellation part, however, is relatively straight forward.

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