Paper Folding: Extracting the cube root of a number

In the previous post on paper folding, we have learned how to use paper folding to extract the square root of a number on the number line. In this post, we are going to learn how to extract the cube root a number by paper folding. The steps in constructing the fold is very similar to extracting the square root.

Paper Folding Instructions 

  1. Get a rectangular piece of paper and fold it in the middle, horizontally and vertically, and let the creases (see green segments in the applet) represent the coordinate axes.
  2. Let M denote (0,1) and let R denote (-r,0).
  3. Make a single fold that places M on y = -1 and R on x=r.
  4. The x-intercept of the fold is \sqrt[3]{r}.
The GeoGebra applet below visualizes the fold. Drag points P and Q to satisfy the conditions above. Note that you can also move point R

[iframe http://mathandmultimedia.com/wp-content/uploads/2011/07/paperfoldcuberoot.html 565 432]

Exercise: Prove that if M on y = -1 and R is on x = r, the intersection of the fold and the x-axis is at \sqrt[3]{r}.

The continuation and proof of this theorem can be found here.

For GeoGebra enthusiasts, you can download the GeoGebra file here.

Reference: My old notebook. Sorry, I don’t know to whom I should attribute the paper fold.

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