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 paper-folders – especially modular paper-folders – 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 longest edge of the right-angle triangle that is half the equilateral triangle – its hypotenuse in the language of geometry – is one edge of the original triangle and therefore has length 2 units. The shortest edge of the triangle – at the bottom in the diagram – is half the length of one of the original edges, and therefore has length 1 unit. This ratio of 2:1 of these edges, with a right-angle at the end

of the shortest edge, is the key to constructing angles of 60o and 30o by folding paper. (For those of you familiar with Euclidean geometry, this is a triangle described by the condition ‘right-angle, hypotenuse and side’ which is a condition for congruence.) The diagrams below show the classic way to fold an angle of 60o in the middle of the top edge of a rectangle. To do this, first crease the paper in half to mark the centre-line, then fold the right-hand edge to the centre-line to crease the right-hand half of the paper into two quarters. (Look at the first diagram – it’s easier!) To form an angle of 60o swing the top left-hand corner across to the quarter-way fold, making sure the crease starts at the centre line.

The shaded triangle in the right-hand 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 right-angle at the end of the shortest edge. This creates the ratio of 2:1 (with the right-angle at the end of the shortest edge) for which we are looking.

The fact that the angle of the shaded triangle at the centre of the top edge is 60o means that the angle next to it along the edge (its supplement) is 120o, and this is bisected by the fold that we have made. By folding the right-hand top corner over to lie along the first folded edge we can create three equal angles, each of 60o, at the centre of the top edge of the rectangle.

Folding the other (right-hand) corner across, so that two raw edges lie together, bisects the angle of 60o on the right into two angles of 30o each. Of course, we don’t have to make the angle of 60o first: we can simply swing the right-hand top corner to the quarter-way fold (the third diagram below). Again the shaded triangle has the crucial 2:1 ratio between its longest and shortest edges, with a right-angle at the end of the shortest edge. This time the angle in the shaded triangle at the centre of the top edge is 30o, so the angle next to it, along the edge to the right (its complement) is 60o, which is bisected by the fold.

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:

**Ian Harrison**