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Folding
angles of 30 and 60 degrees For those of us who fold paper regularly, creating angles of 90o and 45o is almost second nature. We create a right angle by folding an edge, or a crease, back on itself, and we can bisect a right angle (or any other angle) by bringing the folds/edges that form the two arms of the angle together, and creasing between them. The mathematics of folding angles of 60o or 30o is only slightly more difficult to understand. (The other angles that paperfolders  especially modular paperfolders  often need to construct are the angles of 108o and 72o, useful in making polyhedra with pentagonal faces, and other related forms.) The method of constructing angles of 60o and 30o by folding is based on the symmetry of an equilateral triangle  one that has three edges of equal length. To make the explanation clear we shall use as an example an equilateral triangle that has edges of length 2 units. We need to look at just one half of it.
The shaded triangle in the righthand diagram above is an example of the half equilateral triangle described above. It has a longest edge  hypotenuse – half the length of the top edge of the paper, and a shortest edge one quarter the length of the top edge of the paper, and a rightangle at the end of the shortest edge. This creates the ratio of 2:1 (with the rightangle at the end of the shortest edge) for which we are looking. With small modifications to these ideas we can fold an angle of 60o or 30o at a corner, or at the centre of a square, like this:


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