Log24

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.

Here is a trial solution of the inverse problem—

(Click for larger version.)

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

Coming across John H. Conway's 1991*
pinwheel  triangle decomposition this morning—

— suggested a review of a triangle decomposition result from 1984:

Figure A

(Click the below image to enlarge.)

The above 1985 note immediately suggests a problem—

What mappings of a square  with c ² congruent parts
to a triangle  with c ² congruent parts are "natural"?**

(In Figure A above, whether the 322,560 natural transformations
of the 16-part square map in any natural way to transformations
of the 16-part triangle is not immediately apparent.)

* Communicated to Charles Radin in January 1991. The Conway
  decomposition may, of course, have been discovered much earlier.

** Update of Jan. 18, 2012— For a trial solution to the inverse
    problem, see the "Triangles are Square" page at finitegeometry.org.