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Keywords: triangle tiling, Cullinane, Soifer, Beeson

Triangles are Square

by Steven H. Cullinane

A triangle-decomposition result from 1984:

IMAGE- Triangle and square, each with 16 parts

IMAGE- Letter to editor by Steven H. Cullinane, American Mathematical Monthly, June-July 1985-- 'Every triangle is an n-replica' is true if and only if n is a square.'

The above 1985 note immediately suggests a problem—

What mappings of a square  with c 2 congruent square parts to an
equilateral triangle  with c 2 congruent equilateral-triangle parts are "natural"?

IMAGE- Square and triangle, each with 16 parts

(In the figure above, whether the 322,560 natural permutations of subsquare-centers
in the 16-part square map in any natural way to permutations of subtriangle-centers
in the 16-part triangle is not immediately apparent.)

Here is a trial solution of the inverse problem—

IMAGE- Trial mapping of triangle to square

Exercise— Devise a test for "naturality" of  such mappings and apply it to the above.


Update of March 26, 2012:

For related material, see

Beeson,  Michael, Triangle Tiling , series of preprints, 2012

Soifer, Alexander, How Does One Cut a Triangle?, second edition, Springer, 2009

As of March 26, 2012, neither Soifer nor Beeson have referenced the 1985 Monthly letter.

Page created January 16, 2012.