TITLE: Uniqueness of an equilateral triangle decomposition into three similar polygons, exactly two congruent
QUESTION [7 upvotes]: A favorite fun problem I give to students and (even non-mathematical) friends is the following:
Find a decomposition of an equilateral triangle into three similar polygons, exactly two of which are congruent.
There is a cute solution which the persistent reader will find.  (By the way, the problem is not mine.  I believe I saw it in an old Hungarian problem book, but I don't recall for sure.)
The funny thing is that everyone seems to find the same solution as my own.  I sort of expected other solutions to crop up.  So I am led to ask:

Is there a unique such decomposition (up to isometry)?

Apologies if this pushes the MO boundaries a little.  I am probably not alone in wondering about this, though.  And it seems to compare well with a few other "elementary geometry" questions on MO.
For a hint leading to (what I think is) the standard (unique?) solution, see this:
http://en.wikipedia.org/wiki/File:Tile_3,6.svg

REPLY [3 votes]: From the Akiyama et al. paper (arXiv abstract) which Gerry cites,
Figs. 10b and 12b,
both factal limit cases with ratio $r = 1/\phi$: