Proof of rational fraction division using crossing diagonals

A model can be correctly proportioned by dividing the initial square into rational fractions e.g. by reserving 3/13 of a paper for a head, the head will be the correct size after folding.

 

This creates the problem, how can one divide a paper into any given rational fraction? This can be accomplished with approximations and a ruler, but a more pure method is to use folds to construct the fraction. A technique that is often used to do this is the Crossing Diagonal technique.

 

Example for folding 3/13

1) Fold the easy fraction 5/8 at EG (a). This can be found by simply folding sections of the paper in half.

2) Fold the main diagonal AG.

3) Fold a line connecting CE.

4) The height at the intersection point D will create the fraction 5/(5+8)=5/13 at GF (b).

5) From here it is easy to work out how to find 3/13.

 

For a full explanation of the mathematics and more techniques, see the article Origami Constructions by Robert Lang www.langorigami.com/article/huzita-justin-axioms

 

Also see Robert Lang's fiddler crab for an example of a model which does require division by 13: www.langorigami.com/crease-pattern/fiddler-crab-opus-446

 

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