Introduction: Origami - Made With Math

Origami 折り紙, Japanese pronunciation: [oɾiɡami] or [oɾiꜜɡami], from ori meaning "folding", and kami meaning "paper") is the art of paper folding, which is often associated with Japanese culture. "

It 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, robotics and engineering

Scientists use math as some kind of pipeline to truth.Modern scientists use origami by applying mathematical rules and procedures to improve rockets, science ,robotics and nearly every aspect of our every day to day life.

The bending, curvature and design of origami need a lot of math and the only way to make all the elements play together is by following mathematical methods and procedures.

To understand this process we will create a few basic Origami Designs and afterwards we will see how mathematicians and artists create Origami designs and build the future.


Although there are more than few brands of traditional/special Origami paper out there; in my experience every medium that can create a fold is suitable and can be used in Origami design. I used plain A4 printer paper.

Step 1: Basics: Creating an Origami Fortune Teller for Kids

A paper fortune teller can be constructed with these steps using a plain A4 paper:

  1. The corners of a sheet of paper are folded up to meet the opposite sides and (if the paper is not already square) the top is cut off, making a square sheet with diagonal creases. The four corners of the square are folded into the center.
  2. The resulting smaller square is turned over. The four corners are folded in a second time
  3. All four corners are now folded up so that the points meet in the middle.The players now work their fingers into the pockets of paper in each of the four corners.

You can always download and print the instructions. I have also included a printable design and a full color printable design you can download and play!

Completing these tasks, you may have noticed that you are using many properties of geometry and math converting a huge square to a smaller one by subtracting (folding away) triangles?

Math can be fun after all.

Step 2: Intermediate: Creating an Origami Flasher

Modern technology is trying to make everything smaller.

Origami helped create from miniature robots that administer drugs to satellite solar panels that fold and can be stored safely in a rocket

We will now create an Origami Flasher.

Satellite solar panels are delicate and not forgive mistake while handling them.

Imagine being a scientist trying to figure a way to safely store solar panels inside a rocket, trying to find a way to withstand the immense forces during the launch and then deploying them in space.

Origami to the rescue.

Scientists figured out a way that is based on an Origami Flasher and use Origami to fold solar panels in a similar design. They used math to convert a large area of solar panels to fold and fit inside a small cubic like space

I have attached a pdf file for you to download if you want to follow along.

In order to make the Origami flasher you need to follow these steps:

  1. You fold you paper horizontally. Then you fold your paper vertically. Follow the lines.
  2. Fold and unfold your paper in half. Fold the top edge and then the bottom edge to the line in the center.
  3. Now fold top and bottom then flip and fold top and bottom to the center line.
  4. Now flip and fold in half.
  5. Unfold carefully. Try not to rip the paper.
  6. Flip and cut or fold(to the back side) the gray area.
  7. Find the middle square and follow the mountains (red lines) and caves (red dotted lines). Do it slowly and try a quarter of paper at a time.

That' s it ! You made it!!!

Step 3: Introduction to Origami Mathematical Design

In order to understand the math behind Origami we have to take a trip to Ancient Greece.

There Euclid an ancient Greek mathematician practiced math and discovered the five axioms that are the base of modern geometry.

Euclid's Axioms are:

  1. You can join any two points using exactly one straight line segment
  2. Given a point P and a distance r, you can draw a circle with centre P and radius r.
  3. You can extend any line segment to an infinite line.
  4. Any two right angles are congruent.
  5. Given a line L and a point P not on L, there is exactly one line through P that is Two or more lines are parallel if they never intersect. They have the same slope and the distance between them is always constant.

Euclid’ axioms basically tell us what’s possible geometry.

It turns out that we just need two very simple tools to be able to sketch Origami on paper:

  • A straight-edge is like a ruler but without any markings. You can use it to connect two points (as in Axiom 1), or to extend a line segment (as in Axiom 2).
  • A compass allows you to draw a circle of a given size around a point (as in Axiom 3).

Axioms 4 and 5 are about comparing properties of shapes, rather than designing.

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 angle bisector midpoint perpendicular bisector of the line PQ‾.
  • We can fold any two lines onto each other. If the lines intersect, this creates the perpendicular bisector midpoint angle bisector angle bisector 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.

Using these mathematical principles, designers create Origami that has multiple functions and forms, because form follows function.

Step 4: Advanced:Creating the Crease of an Origami Scorpion (How Artists and Scientists Do It)

I think it is time to delve inside the mathematical thinking process of Origami.

I will try to demonstrate how artists and scientists use math and geometry to create an advanced Origami Design like a this Scorpion(Pic 1).

  1. We first need to represent every feature (legs,body,claw etc) by a crease (an object that will later be folded). We do that by using abstract circular objects.
  2. These circles have an arrangement on the paper. The arrangement has to be represented by similarly bigger or smaller circles on a square paper. The bigger the feature (legs,body,claw etc) the bigger the circle. So the arrangement of these objects will create the central crease pattern of the scorpion. In our case the scorpion has a big body, a big tail and smaller claws and even smaller legs. (Pic 2)
  3. We then use geometry rules to create the rest patterns of the crease. (Pic 3)
  4. Afterwards we put a line from the center of each circle. When 2 lines intersect in a V-like pattern we have a fold. (Pic 4)
  5. There are a lot of steps in order to add more lines(and definition to our object) but unfortunately, i cannot cover them in a single instructable. The thing is though that every step in the way is based on math and geometry rules.
Pictures were taken from Robert J. Lang 's amazing work. You can also download a pdf version the Scorpion. Also from this excellent youtube video of Veritasium about Origami.

Step 5: What to Do Next:

Origami Design is a Quest.

You 'll have to keep learning, keep designing and your designs will become better, have more definition and you 'll find way more purposes for them. The sky is the limit!

There are a few Resources I can offer to help you in your journey:

1. Software for creating origami

  • Origamizer by Tomohiro Tachi (free for non-commercial purposes)

2. Origami Designs and Creases

3. Books

  • Origami Design Secrets: Mathematical Methods for an Ancient Art by Robert J. Lang
  • Unit Origami: Multidimensional Transformations by Tomoko fuse
  • Origami Tessellations: Awe-Inspiring Geometric Designs by Eric Gjerde
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