It is easy to find the midpoint of a side of a square piece of paper by simply folding one of the corners onto an adjacent corner. This done, we can also find 1/3 of the side, as shown below.

We do not need anything more sophisticated to get 1/4 of the side than finding another midpoint, but, as the second diagram shows, starting with 1/3 of the side we still can get 1/4, and then also 1/5. The procedure is general enough and leads to a successive construction of a 1/k-th of the side for any whole k. To see why this is so, consider the diagram below:

,

where the corner B has been folded onto the point Q, such that AQ = AD/k, for a given k, not necessarily an integer. We shall assume the side of the square to be 1. Let AP = x. Then PQ = 1 - x. The Pythagorean proposition gives

from which x = (k² - 1)/(2k²).

Further, angle PQR = 90^o, which implies similarity of right triangles APQ and DQR:

or

For the midpoint T of DR, we get

which proves the construction.

Remark

1. After completing this page I have discovered R. J. Lang's online paper where he describes this method as Haga's Construction after Kazuo Haga.

2. Another procedure known as the crossing diagonal method has a longer history and an exiting story that evolved around 1995.

1. C. Abbott, What Good Is Proof by Induction Anyway?, Math. Horizons, Sept. 2005, pp. 8-9
2. R. J. Lang, Four Problems III, British Origami, no 132, October 1988, pp. 7-11

Paper Folding Geometry

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• Angles in Triangle Add to 180^o
• Broken Chord Theorem by Paper Folding
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• Egyptian Triangle By Paper Folding
• My Logo
• Paper Folding And Cutting Sangaku
• Parabola by Paper Folding
• Radius of a Circle by Paper Folding
• An Interesting Example of Angle Trisection by Paperfolding