Text written in Basque and translated automatically by Elia without any subsequent editing. SEE ORIGINAL

Art + mathematics = papiroflexia

The Spanish Association of Papiroflexia has held its annual convention in Bilbao, where we have seen that this discipline between art and science does not only serve to create decorative objects.

In view of the following figures, it seems incredible that they have only been obtained by folding paper. The magic of it? No, the art. And, to a large extent, mathematics.

The history of paper bending is as old as that of paper itself. It originated in China in the 1st or 2nd century, but it was spread by the Japanese, so the word we know is Japanese: origami.

According to the laws of orthodox paper flexure, a single square paper can be used to form a figure, and neither scissors nor glue can be used.

Assisted by JOSÉ IGNACIO ROYO. From the UPV-EHU: It seems that through papiroflexia you can only make boats or birds of paper or typical airplanes, but not. Look at the figures that can be made. See what kind of dinosaur can be made with these restrictive laws.

Comet, fish, bird and frog. From these four basic models, an infinite number of figures can be generated. This was done until the beginning of the 20th century. But today, each figure is designed on its own before the paper begins to fold.

Assisted by JOSÉ IGNACIO ROYO. From the UPV-EHU: For example, if I want to design a horse, I have to split six points: head, tail, and four legs. And they must be equidistant. If I undo the folds of that horse, I'll see those six points spread out on square paper. This is a geometric problem: how to divide six equally spaced tips with the greatest possible spacing.

Artists who practice papiroflexy don’t use brushes or chisels, but their fingers are not their only tools. Mathematics is a very useful tool for converting a flat sheet of paper into a three-dimensional object, since this limited space offered by paper must be optimally distributed.

Assisted by JOSÉ IGNACIO ROYO. From the UPV-EHU: Mathematics is used to solve problems. And if these problems are geometric, we can do it easily, because ancient mathematicians thought a lot about geometric problems. And we can take advantage of their work.

Assisted by ROBERT LANG. The Origami artist: Mathematics and mathematics are very close. Origami is a limited art. We have to use a square paper, we can’t add paper, we can’t remove anything, cut it, or break it. What we can do is limited to the mathematical properties of paper.

Robert Lang’s story is an excellent example of the close relationship between papiroflexy and mathematics. This Californian physicist and engineer spent 15 years researching laser in a NASA lab and decided to leave everything to write a book about origami.

Assisted by ROBERT LANG. The Origami artist: Origami came earlier. I started practicing origami when I was six years old, so I’ve spent my whole life folding paper. I've always had this hobby, but I never thought it would be my job.

The books so far included guidelines for building models, but what Lang wanted was for people to be able to make new designs. He discovered that all origami designs fulfill four mathematical axioms, and based on this idea, he developed a computer program to create paper models from any scheme. Lang applied mathematics to origami and is now on the opposite path, applying the principles of papiroflexy to engineering projects.

Assisted by ROBERT LANG. The Origami artist: The clearest example of origami’s contribution to engineering is found in the structures that need to be expanded. In fact, in space structures: telescopes, antennas, solar panels... they have to be very small when they go inside the rocket, but once they are in space, they have to take a flat or almost flat shape. If you go backwards, you have something that has a flat shape and you want to reduce it in a controlled way. One logical way to achieve this shape change is to fold.

Robert Lang designed an umbrella-shaped pattern so that the Eyeglass space telescope, the size of a football field, could be reduced to three meters. Even on the ground, papiroflexia has been used for simulations with car airbags, among others.

But papiroflexia is not only mathematics, it is also art; and this is evident in this advertisement with pieces by Robert Lang, where everything but the car is folded paper.

Assisted by JOSÉ IGNACIO ROYO. From the UPV-EHU: Mathematics is not the only thing, but it helps. Papiroflexia is not a brain thing we do to cultivate the brain; it’s a game and a beautiful art.

Assisted by ROBERT LANG. The Origami artist: Origami also serves to create beautiful art that has nothing to do with engineering. The same technique and design is equally valid for engineering and art. And that's wonderful.