Quarter the Cross: An Elegant Construction

Last month, I presented an infinite pattern of nested crosses as a response to the  #QuarterTheCross challenge on Twitter.  The idea is to find interesting ways to shade one-quarter of the area of the cross built from five squares.nested_crossNow David Butler, a math lecturer at the University of Adelaide, has found an elegant way to construct this pattern with the classic tools of compass and straightedge (I used some simple MATLAB code).

As the crosses get smaller, each new one must shrink by \sqrt{3}.

I asked David to explain his construction.

The segment marked “1” is \dfrac{1}{3} of the “radius” from a corner of the outer cross to the center, so the construction generates an inner cross with the required radius, \dfrac{\sqrt{3}}{3}=\dfrac{1}{\sqrt{3}} as long as the original radius.

A nice feature of this construction is that the compass is only used near the beginning.  After that, the construction always provides a scaffold for the next inner cross.

 

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