Quarter the Cross

Yesterday, I noticed that some of the math teachers I follow on Twitter were challenging their students, and themselves, with the #quarterthecross problem.  The problem is simply to find interesting regions of a five-square cross that take up exactly one-quarter of its area.  This looks like a great way to get kids to explore fractions and geometry.

nested_cross

A solution that occurred to me would be to use an infinity of nested crosses.  If the area of the cross shrinks at each step by one-third, then the combined area of the colored regions converges to one-quarter of the large cross.  This works because the alternating geometric series

\dfrac{1}{ 3} - \dfrac{1}{ 3^{2}}+\dfrac{1}{ 3^{3}}- \dfrac{1}{ 3^{4}}+\dfrac{1}{ 3^{5}}...

converges to \dfrac{1}{4}.  Note that if the area shrinks by one-third, the width of the cross must scale down by the square root of 3.  So every two steps the width shrinks by one-third, placing a little cross precisely inside the middle square of a larger cross.

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3 Responses to Quarter the Cross

  1. Bob Rosinsky says:

    It’d be neat to use your algorithm to generate CGI animations. Here is a short list of programmable variables: 1) Display resolution; 2) Rotation and other transformations; 2) Frame duration; 3) Flicker rate/frames per second; 4) RGB values–hue, saturation, and luminance.

    Using a quarter cross algorithm for rendering motion graphics could be an interesting tool for testing the limits of visual perception and for creating art.

    Animating wacky tessellations is fun. However the results sometimes play awful tricks on the mind and body.

    Like

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