Shulou(Shulou.com)11/24 Report--

The original title: "is your origami technique taught by your math teacher?" "

Did everyone play origami when they were kids?

Today, I would like to share with you the mathematical story contained in the origami we have played since childhood.

Let's fold a pentagram together! Although we Chinese have special feelings for the five-pointed star, there is no clear boundary between human love for the five-pointed star. As early as in ancient Greece BC, people were deeply attracted by the charm of the pentagram. It's no ordinary pentagram, it's the badge of the Pythagorean club!

The symbolic figures in the picture, and the three-dimensional lines like the modern overpass, make people seem to feel an infinite movement with a cycle of 5, repeated and never-ending!

Many readers learned to use origami to cut five-pointed stars when they were children. The following picture directly shows the process of this folding method. You can try it with a piece of paper by yourself.

The last cut seems to be arbitrary, so strictly speaking, the cut figure can only be said to be a "pentagram", not necessarily a positive pentagram.

The Roman numerals in the picture show the order of creases. Origami art seems simple, but it contains profound scientific reasons. The method of origami is not single.

For example, to correct the pentagram, people do not have to use the complicated folding starting line-up as above! In fact, a common knot is enough! The prop used is just a long piece of paper tape.

Not everyone knows that our daily knot-tying is actually creating one beautiful pentagram after another.

By "folding" the parabola may be thought that origami can only fold a straight figure, because the crease can only be straight in any case. In fact, this is a misunderstanding!

Enough straight creases can sometimes make a beautiful curve. Please cut out a rectangular piece of paper ABCD with paper. Fold as shown in (a) below so that point A falls on the edge of CD after each fold. Countless creases will form a curve as shown in figure (b). Such a curve is geometrically called the envelope of a crease.

The envelope curve of figure (b) is a parabolic arc. When you throw pebbles, you will see a beautiful arc of pebbles in the air. This arc is the result of the simultaneous action of gravity and inertia on the stones. Suppose you throw the stone at an alpha angle to the level, and the velocity of the stone is v0, then the position coordinate of the movement of the stone at the moment is:

After eliminating the time t, we will get a quadratic function of x. Therefore, the image of a quadratic function is also called a parabola.

Interestingly, when the initial velocity of the throw is constant, but only the throwing angle is changed, we will get a series of parabolas as shown below, and the envelopes of these countless parabolas also form a parabola, which is called "safe parabola" in physics.

Let's go back to the subject of origami and study why the crease envelope mentioned earlier is a parabola.

As shown in the figure, the Cartesian coordinate system is established with the midpoint O of AD as the origin and OD as the positive direction of the Y axis.

Let AD=p, then the coordinate of point An is (0jiao-p / 2), let A'be any point on CD, EF be the crease on paper when A folds to A', and T on EF satisfies TA' ⊥ CD. Below we prove that the trajectory of the T point is the envelope curve of the crease.

In other words, the trajectory of the T point is a parabolic arc. The remaining problem is that it must be proved to be tangent to the crease. For this reason, if the slope of the straight line AA' is k, then

It is noticed that the crease EF is the vertical bisector of the line segment AA', and it is easy to find out that the equation of the straight line EF is

Envelope is one of the topics in differential geometry. It was founded in 1827 by the German mathematician Gauss. The picture below is another interesting origami envelope.

Cut a round piece of paper, take any point An in the disc, and then fold the paper as shown in figure (a) so that the folded arc passes through point A, so as to get numerous creases in figure (b). The envelope of these creases is an ellipse with point An and the center of the circle as the focus and the long axis as the radius. Readers might as well try to fold one for themselves.

The most magical origami is probably the "Miura folding method". It was invented by Professor Gongliang Miura of the Institute of Cosmic Sciences in Japan. This kind of origami method can make inanimate paper have the function of "memory"!

As you know, when we want to fold a large piece of paper small, we often fold it perpendicular to each other. Whether the creases of this folding method are "mountains" or "valleys" are independent of each other. Thus a variety of possible folding combinations, the total number is very large! When a large piece of folded paper is fully unfolded, it is difficult to fold it back to its original position.

In addition, this kind of mutually perpendicular folding method, the crease is often very thick, so under the action of tension, it is inevitable to cause damage! The "Miura folding method" is also called "double-layer wave type expandable surface". It is different from the "mutually perpendicular folding method" in that the longitudinal fold is slightly serrated.

In this way, when you open a piece of paper folded by the Triple folding method, you will find that as long as you grasp the diagonal part and stretch it in any direction, the paper will automatically open both vertically and horizontally.

Similarly, if you want to fold such paper, just squeeze one side at will, and the paper will return to its original state, which is equivalent to remembering the original state! The paper is folded by the three-pool folding method, and the whole piece of paper forms an organic link. There are only two kinds of crease combinations: full expansion and full return. Therefore, it will not be damaged because the crease is not aligned when folding.

Figure (b) above shows the situation when folding with the "Miura folding method". It is easy to see that the creases here are staggered with each other. Figure (a) is a common folding method, and it is not difficult to find that the creases here have dangerous bumps at the overlap!

Today, the magical Miura folding method has been widely used. In the grand plan of human conquest of space, the application prospect is particularly outstanding for the construction of large areas of solar sails, artificial moon and so on.

Source: "Mathematics Story Book for Children" by Zhang Yuannan, Zhang Chang, Editor: Zhang Runxin, this article comes from the official account of Wechat: Origin Reading (ID:tupydread), author: Zhang Yuannan Zhang Chang

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