Rijuta Dighe

# Mathematics of Origami

When we think of Origami, the first thing that comes to our minds is the traditional paper folding art. As a child, most of us have made that paper boat or the paper crane to play with. Such paper toys made from 'rough' pages of notebook were sure to bring joy.

In reality, Origami goes way beyond creating paper toys. It is not just a form of art, but an elaborate mathematical science, which has helped humans send satellites in space! Mathematical laws and theorems are the most important part of governing the fundamentals of Origami. It is a mathematical art, since it is a pattern of creases.

In 1893, Indian mathematician T. Sundara Rao published a book titled '*Geometric Exercises in paper folding*' which used paper folds to demonstrate geometric theorems. According to historian of mathematics Michael Friedman, it became "one of the main engines of the popularization of folding as a mathematical activity". The field hugely evolved in 20th century, with mathematicians like Margharita P. Beloch showing Beloch fold (A fold equivalent to solving cubic equation), Humiaki Huzita rediscovering the Huzita-Hatori axioms (Basic geometry using paper folds), and Robert Lang furthering the axioms

It is at the same time, that Japanese Origami grand master, Akira Yoshizawa raised Origami from an art to a science of math and sculptures. According to his own estimation made in 1989, he created more than 50,000 models, of which only a few hundred designs were presented as diagrams in his 18 books

If you build an Origami model, and then unfold it back, it creates a pattern of creases on paper. This pattern is essentially a blue print for the model. Origami creators build such blue prints to develop new models. Now, these patterns are not made randomly, but are defined by set of mathematical laws. After all, isn't Geometry basically a definition of patterns!

Robert Lang, one of the biggest Origami theorists of modern times, addresses 4 specific laws of Origami in his ted talk:

1. **Two color-ability**: You can color any crease pattern with just two colors without ever having the same color meeting.

2. **Maekawa's theorem**: At every Vertex (corner), the numbers of valley folds and mountain folds always differ by two in either direction.

Taking Red as the 'main' side here,

Mountain folds (Where main pointed side comes up like a mountain) - Valley fold (Where main side goes down like valley) = 2

3. **Kawasaki's theorem**: At any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even. This can also be stated as, all alternate angles around a vertex will form a straight line.

4. **A sheet can never penetrate a fold. **

All of these images have been taken from Robert Lang's website: https://langorigami.com/

Visit https://langorigami.com/artworks/ for more of his gorgeous, amazing work!

I also suggest watching his TED Talk:

__https://www.youtube.com/watch?v=NYKcOFQCeno&t=79s __

It is one of the most interesting TED talks, even if you know nothing about Origami. The talk is purely mind blowing!

As a result of studying Origami mathematics, researchers have been able to apply paper folding to solve mathematical problems such as the

**Delian problem** (Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first),

**Angle trisection** (construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass),

**Haga's theorems** (a set of constructions to divide the side of a square at an arbitrary rational fraction), and many more!

Techniques such as,

Wet folding (dampening the paper with water to give better curved folds), Flat folding (using flat pieces of paper joined at hinges), modular origami, etc have even more more complex patterns governed by more mathematics!

Exploring Origami through mathematical eyes has also helped solve real world problems and have assisted in the development of many real world applications such as these:

**Miura Map fold**- An Origami tessellation of valley and mountain folds which was used to make maps.

**Space Exploration**- Sending satellites to space requires large satellites to fit into a small area and Origami works best there! Watch this video for more information on Space Origami: https://www.youtube.com/watch?v=Ly3hMBD4h5E

*An origami-inspired deployable solar array developed by Brigham Young University, NASA Jet Propulsion Laboratory, and Lang Origami. Credits: BYU*

**Heart Stents**- Origami and Medical technology got related with these heart stents developed by a British - Japanese team at Oxford university. It uses the water-bomb base origami technique, a rounded cube form that folds flat and is inflated via a hole in a corner of the construction (The same one used in kid's blow up box!)

*Credits: (2017, August 7). From Origami to a Prototype Stent. Retrieved from https://www.mddionline.com/origami-prototype-stent*

**Air Bags**- Air bag is one of the best examples of precision design. It requires a great deal of engineering aptitude, through experimentation and computer simulations. These computer simulations came from Robert Lang's Origami software which he had used to make Origami simulations for insects! The air bag design has to be placed at the same position in all cars, must inflate quickly, and should work for all human body types. This is where Origami comes into picture. An Origami fold is used to fit air bags into cylinder, which gets unfolded into a cushion during impact.

**Origami Robots**- Pioneered by Dr. Devin Balkcom, who first made Robot with Origami skeleton at CMU's Robotics institute in 2004, a number of research labs are involved in developing Origami Robots. Origami Robots are those Robots, whose exoskeleton design and morphology are made using folds. With the recent advancements in shape changing materials, Origami Robots are becoming better than before!

*What essentially look like simple projects are actually completely autonomous, shape transforming Robots developed at MIT's CSAIL lab!*

*Credits: Rus, D., & Tolley, M. T. (2018). Design, fabrication and control of origami robots. Nature Reviews Materials,3(6), 101–112. doi: 10.1038/s41578-018-0009-8*

I will be writing more specifically on Origami Robots in the upcoming articles

Other Applications: More applications of Origami can be found in architecture and civil engineering, different packaging products, and building Solar panels, among others!

Now that you know, how Origami impacts so many aspects of our lives, I hope you can look at it beyond a simple art form. Do encourage children to pursue Origami and do let me know what are your thoughts on the science of Origami in comments!

After all, an art amalgamated with mathematics and science, further enhanced with technology forms the heart of marvelous engineering!

*Keep Folding :)*