"That n 2 points fall naturally into a triangular array
is a not-quite-obvious fact which may have applications…
and seems worth stating more formally."

— Steven H. Cullinane, letter in the
American Mathematical Monthly  1985 June-July issue

If the ancient Greeks had not been distracted by
investigations of triangular  (as opposed to square )
numbers, they might have done something with this fact.

A search for occurrences of the phrase

"n2 [i.e., n 2 ] congruent triangles" 

indicates only fairly recent (i.e., later than 1984) results.*

Some related material, updated this morning—

This suggests a problem—
 

What mappings of a square  array of n 2 points to
a triangular  array of n 2 points are "natural"?

http://www.log24.com/log/pix12B/120708-SquareAndTriangle.jpg

In the figure above, whether
the 322,560 natural permutations
of the square's 16 points
map in any natural way to
  permutations of the triangle's 16 points
is not immediately apparent.

 

* Update of July 15, 2012 (11:07 PM ET)—

Theorem on " rep-n 2 " (Golomb's terminology)
triangles from a 1982 book—

IMAGE- Theorem (12.3) on Golomb and 'rep-k^2' triangles in book published in 1982-- 'Transformation Geometry,' by George Edward Martin