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Triangles Are Square
by Steven H. Cullinane
| American Mathematical Monthly,
June-July 1984, p. 382 MISCELLANEA, 129 Triangles are square "Every triangle consists of n congruent copies of itself" is true if and only if n is a square. (The proof is trivial.) -- Steven H. Cullinane |
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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. Steven H. Cullinane
501 Follett Run Road Warren, PA 16365 |
The above 1985 note
immediately suggests a problem—
What mappings
of a square array of
n
2 points to
a triangular array of n
2 points are "natural"?
(In
the figure above, whether the 322,560
natural permutations
of the 16 points in the square map in any natural way
to permutations of the 16 points in the triangle
is not immediately apparent.)
Here is a trial solution of the inverse problem—
Another trial mapping, from
square to triangle,
based on gnomons of the square:
Another
approach to the
square-to-triangle mapping problem —
For the square model referred to in the above picture, see (for instance)
Picturing
the Smallest Projective 3-Space,
The
Relativity Problem in Finite Geometry, and
Symmetry
of Walsh Functions.
Coordinates for the 16 points in the triangular arrays
of the corresponding affine space may
be deduced from
the patterns in the projective-hyperplanes array above.
This solves the inverse problem of mapping,
in a natural way, the triangular array of 16 points
to the square array of 16 points.
Note
that the square model's 15 hyperplanes S
and the triangular model's 15 hyperplanes T —
— share the following vector-space structure —
| 0 | c | d | c + d |
| a | a + c | a + d | a + c + d |
| b | b + c | b + d | b + c + d |
| a + b | a + b + c | a + b + d | a + b + c + d |
(This vector-space a
b c d diagram is from
Chapter 11 of Sphere
Packings, Lattices
and Groups ,
by John Horton Conway and
N. J. A. Sloane, first published by
Springer
in 1988.)
Page created January 16, 2012.
