The Impossibility of a Tesselation of the Plane
into regular triangles of mutually different sizes, one
of them being minimal.
Theorem by Karl Scherer
Published in : Elemente der
Mathematik, Band 38, Nr 1, p 1-4. January 1983
Download it here.
Related problem :
Squaring the plane with squares of different integral sizes. The problem
was solved by
Jim M Henle and F.V. Henlein 2005; see http://www.math.smith.edu/~jhenle/stp/
or download the pdf file of the theorem here.