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American Mathematical Monthly, Vol. 92, No. 6
(June-July 1985), p. 443
 

LETTERS TO THE EDITOR

Material  for this department should be prepared exactly the same way as submitted manuscripts (see the inside front cover) and sent to Professor P. R. Halmos, Department of Mathematics, University of Santa Clara, Santa Clara, CA 95053

Editor:

    Miscellaneum 129 ("Triangles are square," June-July 1984 Monthly ) may have misled many readers. Here is some background on the item.

    That n² points fall naturally into a triangular array is a not-quite-obvious fact which may have applications (e.g., to symmetries of Latin-square "k-nets") and seems worth stating more formally. To this end, call a convex polytope P  an n-replica  if  P  consists of n mutually congruent polytopes similar to P  packed together. Thus, for n ∈ ℕ,

    (A) An equilateral triangle is an n-replica if and only if n is a square.

    Does this generalize to tetrahedra, or to other triangles? A regular tetrahedron is not a (2³)-replica, but a tetrahedron ABCD  with edges AB, BC, and CD  equal and mutually orthogonal is an n-replica if and only if n is a cube. Every triangle satisfies the "if" in (A), so, letting T  be the set of triangles, one might surmise that

    (B) ∀ t ∈ T (t is an n-replica if and only if n is a square).

     This, however, is false. A. J. Schwenk has pointed out that for any m ∈ ℕ, the 30°-60°-90° triangle is a (3m²)-replica, and that a right triangle with legs of integer lengths a and b is an ((a² + b²)m²)-replica. As Schwenk notes, it does not seem obvious which other values of n can occur in counterexamples to (B). Shifting parentheses to fix (B), we get a "square-triangle" lemma:

    (C) (∀ t ∈ T, t  is an n-replica) if and only if n is a square.
   
    Miscellaneum 129 was a less formal statement of (C), with quotation marks instead of parentheses; this may have led many readers to think (B) was intended. To these readers, my apology.