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Euclidean GeometryOrigami and Paper Folding

Vrijeme čitanja: ~25 min

Using straight-edge and compass is not the only way to construct geometric shapes. Another technique uses no tools at all: Origami.

The word Origami (折り紙) comes from the Japanese oru (to fold) and kami (paper). The goal is to make objects out of one or more sheets of paper, without any additional tools like glue or scissors. You can create incredibly beautiful and impressive designs – all of these figures were built using nothing but rectangular sheets of paper:

Building shapes like this can take a lot of time, and it is important to be extremely accurate. But with a bit of practice, you can do it yourself!

You just need a square sheet of paper. To start, fold the sheet along its two diagonals.

Next, fold it along its horizontal and vertical centers – but in the opposite direction.

Now take two opposite corners of the sheet and bring them together as shown. This forms a smaller square which is open at the bottom.

Fold the left and right corners of the square towards its center line. Then turn it over and repeat the same.

Now fold the top triangle down, along the horizontal line, and then open up the folds from the last two steps.

This one is difficult: take the bottom corner of the paper and fold it all the way up, along the horizontal line you just created. Some of the folds you made before will be inverted. Then turn over and repeat.

Make sure the two “legs” are pointing down. Then take the left and right corner and fold them towards the center line. Turn over and repeat.

You’re almost done! Slightly open the right side, and fold the head up towards the top. You will have to turn it inside out. Then repeat the same with the tail on the left.

Reverse the fold as shown to create a beak. You can decide how long you want it to be by picking the location of the fold.

Finally, fold down the two wings, and pull them apart.

This crane is one of the oldest and most famous Origami models. We have many more instructions for Origami models you can try!

Origami Axioms

Just like drawing with straight-edge and compass, there are a few axioms of different folds that are possible with origami. They were first listed in 1992, by the Italian-Japanese mathematician Humiaki Huzita.

You can fold a line connecting any two points.

You can fold any point P onto any other point Q. This creates the of the line PQ.

We can fold any two lines onto each other. If the lines intersect, this creates the of the angle between the two lines.

Given a point P and a line L, we can make a fold perpendicular to L passing through P.

Given two points P and Q and a line L, we can make a fold that passes through P and places Q onto L.

Given any two points P and Q and any two lines K and L, we can make a fold that places point P onto line K and at the same time places point Q onto line L.

Given a point P and two lines K and L, we can fold a line perpendicular to K that places P onto L.

It turns out that these axioms are even more powerful than the Euclidean ones. It is possible to trisect angles and double cubes using just paper folding! Of course, it is impossible to fold any curved lines, and you still can’t square the circle with origami.

Applications of Origami

Origami is an ancient art, and for the longest time, it was mostly a recreational pursuit, without real-life applications. However, it turns out that techniques developed for Origami can be incredibly useful in technology and engineering:

Origami in Space

Satellites need large solar panels to generate power. Unfortunately, the rockets that carry satellites into space only have very limited space for cargo, and any additional weight costs a lot of fuel.

Origami techniques allow solar panels to “unfold” when they reach space. Some particularly clever folds are incredibly compact and require very few motors and other mechanical components.

Origami in Medicine

In medicine, similar ideas from Origami are used on a much smaller scale. In 2003, researchers developed Origami Stents: tiny tubes that can be inserted into blood vessels. They are initially folded up but can expand inside patients’ blood in order to enlarge clogged arteries or veins.

Foldable Bridges

The British and American military used Origami to develop foldable, mobile bridges. These were important for quickly crossing rivers or anti-tank ditches, and could be deployed much faster than previous designs.

They can also be used for disaster relief, to quickly give emergency vehicles access after earthquakes or tsunamis. This image is of a prototype designed at Hiroshima University in Japan.

Origami under the Sea

The depths of the oceans are some of the least explored areas on Earth. Animals that live there are often squishy and delicate, which makes them very hard to examine.

Here you can see a “trap” in the shape of a dodecahedron that can fold around marine organisms, allowing them to be studied. It is remotely controlled and only needs a single motor to drive the complex folding motion of its five arms.

And there are many more applications of Origami in everyday life: houses that will compress rather than crumble during an earthquake, unfolding airbags in cars, self-assembling robots, more efficient packaging, and lightweight aircraft.

Origami in Nature

It turns out that we humans are not the only ones harnessing the power of Origami: nature has been doing so for millions of years.

Here you can see the wing of an earwig that can be folded up using an ingenious pattern. When opened, the size of the wing expands by a factor of 10 – the highest “folding ratio” in the animal kingdom:

When expanded, the large wings snap into a stable position that allows the insects to fly. But it only takes the lightest touch for the wings to retract. When folded up, they are compact enough to allow earwigs to tunnel underground. Many other insects, bats, leaves and flowers use similar folding patterns to fit large surfaces into small spaces.

Scientists are studying these plants and animals, hoping to mimic their abilities in engineering and technology. Potential applications could include foldable electronics in smartphones, expanding solar panels for satellites, or even self-folding camping tents.

Origami even appears in your own body: every human cell contains around 2 meters of DNA, the molecule that carries all your genetic information. If you could combine the DNA from all cells in your body, their length would be more than 140 times the distance from Earth to the sun!

To fit all that DNA in your body, without it getting twisted or torn, every strand is curled, folded, and held in place by special molecules.

A similar process is also used by other complex molecules that appear in living organisms. For example, protein folding is one of the most complex problems in biology. Understanding it better can help scientists develop new drugs in the future.