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Sunday, July 15, 2012

Squares Are Triangular

Filed under: General,Geometry — m759 @ 2:00 PM

"A figurate number… is a number
that can be represented by
a regular geometrical arrangement
of equally spaced points."

Eric W. Weisstein at Wolfram MathWorld

For example—

IMAGE- 16 points in a square array and in a triangular array

Call a convex polytope P  an n-replica  if  P  consists of
mutually congruent polytopes similar to P  packed together.

The square-triangle theorem (or lemma) says that

"Every triangle is an n-replica"
is true if and only if n  is a square.

Equivalently,

The positive integer n  is a square
if and only if every triangle is an n-replica.

(I.e., squares are triangular.)

This supplies the converse to the saying that

Triangles Are Square.

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