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# Appreciating Math through Origami

**Introduction**

Origami as an art, and a craft, has engaged humans for 1500+ years. The general perception of origami is that it is a creative and artistic skill to craft beautiful shapes and objects from paper.

**Origami and Math**

The mathematical underpinnings of origami are a recent development in the long history of origami. Interestingly, we have an Indian connection here - one of the first books that systematically presented the mathematical ideas in origami is the 1893 book “Geometric Exercises in Paper Folding” by Tandalam Sundara Row from then Madras, in India.

**Applications of Origami**

The application of computational methods to origami in the last three decades gave rise to modern origami, which finds real world applications in very diverse and previously unexpected areas - in space exploration, consumer electronics, health, architecture, safety, to name just a few. Modern origami or more broadly, folded structures, is an active field of mathematical and computational research.

**Origami in the Classroom**

The benefits of origami in improving students’ skills is also well-researched and documented.

Origami is also a very powerful mathematical tool in classrooms. Many geometrical and mathematical constructions are possible with paper folding, which are not possible with the ruler and compass. For example we can trisect an angle, solve a cubic equation, fold a sheet of paper into polygons/ polyhedral shapes, etc.

I have observed senior origamists teaching origami methods to school children. I observed that few things excite and capture the imagination of children as origami does. However, the mathematical ideas in origami are not usually discussed in these interactions. To address this gap, I design my courses with two primary objectives:

To demonstrate that paper folding is a useful mathematical tool

To teach some essential skills required to pursue a mathematical analysis of origami.

I recently concluded a session for GenWise Online on Origamics. This course was targeted at students in the Grades 5-7 age band. The rest of this blog captures my experiences from the course.

I tried to introduce diverse mathematical ideas in paper folding. Some of the ideas we explored in the course included:

The seven axioms of origami

Origami bases

Haga’s origamics activities

Fold and cut theorem

Optical illusions with paper folding

2D Nets of polyhedral structures

Modular origami with sonobe units

The above content was delivered over five sessions, running on alternate days, with each session running for an hour.

**Day 1 - 7 Axioms of Origami - An Introduction**

These “axioms” are basically a list of all mathematical operations that are possible when folding paper. These were formulated in 1992 by the Italian-Japanese mathematician Humiaki Huzita. Students did all the folding operations listed in these axioms. Some students started to observe the interesting consequences of some of these folds. For example, Axiom 1 states that ‘we can fold a line connecting any two points P and Q’; this creates a crease, which is the straight line between points P and Q. Axiom 2 states that ‘we can fold any two points onto each other’; this creates a crease which is the perpendicular bisector to the line segment PQ. Likewise every axiom creates a fold which corresponds to a mathematical construction.

By introducing these axioms I wanted to establish the fact that paper folding is a powerful mathematical tool.

**Day 2 - Kazuo Haga’s “Origamics”, Origami Bases**

This was an open-ended session. The activities invite the students to experiment with different folds, observe a pattern, make conjectures, and finally prove their conjectures.

There are many interesting activities to choose from Haga’s origamics. I chose to do the “folding turned-up parts” activity. Students found it challenging at the start, but made some attempts at coming up with conjectures. We also discussed the concept of various ‘origami bases’. ‘Origami bases’ fold patterns which can be manipulated, and further folded into more complex shapes. Computational approaches to origami analyses the target shape and identifies the base fold that is needed to create complex structures.

During this activity some interesting learnings happened in the class. One student while folding the windmill base found the crease patterns ‘beautiful’. We discussed that identifying and appreciating beauty in shapes is very much part of mathematical thinking.

**Day 3 - ‘Fold and Cut’ Theorem**

In the third session we did an activity which is often used by magicians: the ‘fold and cut’ theorem. Students are invited to fold a sheet of paper and with one single straight cut, create a hole which could be a square, rectangle, triangle, irregular quadrilateral, etc.

Students played around freely with different folds and cuts. When they opened the folded paper they saw different shapes. For some shapes (e.g. square), they understood how to fold and cut, while it was challenging for others (e.g. triangle). After allowing some time for free exploration, we gathered for a discussion on some possible strategies. One strategy that works is based on Axiom 3 which states that ‘we can fold any two lines onto each other’. This axiom implies the statement of the ‘fold and cut’ theorem, which states that any shape that is made of straight lines can be folded and cut with one single straight cut. This connection between Axiom 3 the ‘fold and cut’ theorem helped in strengthening the mathematical thinking towards paper folding.

**Day 4 - Optical Illusions through Paper Folding, Sonobe Units**

Students were introduced to the concept of 2D nets, of 3D objects. We then moved on to modular origami projects using the ‘sonobe unit’. Sonobe unit is a popular unit in modular origami where multiple units can be assembled to create complex 3D shapes. The Sonobe unit can be used to build many polyhedral structures. While folding the sonobe unit, one student observed that the crease patterns were similar to those of the windmill base. This was a moment of making connections between different fold patterns.

**Day 5 - Building 3D shapes with Sonobe Units**

On this final day, we invited children to assemble sonobe units into a cube. Some students also tried making other 3D shapes. Students absolutely enjoyed creating a mathematical object finally, after all the productive struggle during which they learned different mathematical ideas related to paper folding.

While there was a lot of “doing” and working with hands, I had designed my course with the express objective of introducing mathematical ideas in origami to students. I was happy when students mentioned they had found more math in the course than they expected, and were all too happy with the experience!

I invite educators to actively consider this fun and playful tool in their classrooms to inspire joy and confidence in Math! I would love to hear your experiences...

**About the Blog Author**

Hari Krishna loves developing educational activities to encourage curiosity in children. Hari is a mentor and course designer at GenWise. He taught at the School of Energy and Environment, City University of Hong Kong, an Atal Innovation Mission Mentor of change, and was a teacher at RSC Salters’ Chemistry Camps. He is also an accomplished athlete and loves playing Volleyball.

Hari has a Masters and PhD degrees in Chemistry from the Indian Institute of Science, Bengaluru.

Hari was part of the team of mentors who facilitated the GenWise Summer School Early Explorer (2019).

We are proud to announce that Hari will run the Learning Math through Origami once again, thanks to the excellent response to the first edition! Click here for more details.