Thursday's post Triangles Are Square posed the problem of
finding "natural" maps from the 16 subsquares of a 4×4 square
to the 16 equilateral subtriangles of an edge-4 equilateral triangle.

http://www.log24.com/log/pix12/120116-SquareAndTriangle.jpg

Here is a trial solution of the inverse problem—

http://www.log24.com/log/pix12/120116-trisquare-map-500w.jpg

(Click for larger version.)

Exercise— Devise a test for "naturality" of
such mappings and apply it to the above.