Trisection of angles in origami and paper-cutting [paper art theory]

Trisection of angles in origami and paper-cutting [paper art theory]

In origami and paper-cutting, the problem we often encounter is to divide a corner into three equal parts, then fold and cut it, but there must be a very serious problem before folding and cutting. Set before us, cut this angle into three equal parts!

How to divide an angle into three equal parts? According to Galois' theory, you simply can't do it using just a ruler and compass, so you can just give up on the attempt. However, you may be able to find some approximations. But in this origami or paper-cutting, what is needed is absolutely precise angle division!

You may have seen the angle-dividing method introduced below in many places, but if you haven’t met it yet, now is a good opportunity to catch up on it!

Since we are folding paper and not deriving geometric propositions, we directly choose a corner on a square piece of paper.

What we need to find now is how to draw the two dotted lines in the picture, and those two dotted lines, plus the top realization and the bottom edge, exactly trisect the angle we set.

Since origami itself is a practical work, we will study the paper itself.

Suppose we can draw 3 equal right triangles as shown in the picture.

These triangles are equivalent to three equal parts of the angle we focused on before (there is a right triangle inside, with equal sides and equal angles), so we only need to know how to "place" them there.That’s it!

Choose any height h close to the bottom edge. Any height is fine, just like in the picture. Use the horizontal line of this height as the crease, fold the bottom edge upward, and then restore it, leaving this fold. mark.

What we need to use at this time is the blue line drawn in the picture, and the length of this line should be twice the height h of the previous step, which is twice the h.

We can make some "rulers" in the paper. The method is very simple, which is to fold the bottom edge upward twice in a row according to the previous height h, so that another crease is obtained.

Having such a "ruler" makes it very convenient for us to find the imaginary line.

Our next operation is to fold point b to point B, and fold point d to point D. Points b and d are on the left side of the figure, and point B is on the earliest crease made, and point D is Its on the crease closest to the bottom edge. After folding it, we can simply use a pencil to lightly draw the red line.

In this way, we find the side that finally completes the trisection of the angle! If you still have doubts, you can simply verify it.

< br />

Please indicate the link to this article when reprinting: http:// zhidiy.com/zhezhirumen/452/