Origami + Math + Mobiles

image001So exactly how do origami and math relate to each other? The connection with geometry is   clear and yet multifaceted; a folded model is both a piece of art and a geometric figure. Just unfold it and take a look! You will see a complex geometric pattern, even if the model you folded was a simple one. As a beginning geometry student, you might want to figure out the types of triangles on the paper. What angles can be seen? What shapes? How did those angles and shapes get there? Did you know that you were folding those angles or shapes during the folding itself?  Can you come up with any relationships between a fold and something you know in geometry?

purple3Any basic fold has an associated geometric pattern. Take a squash fold – when you do this fold and look at the crease pattern, you will see that you have bisected an angle, twice! Can you come up with similar relationships between a fold and something you know in geometry?

On the other hand, if you are a person who likes puzzles, there are a number of great origami CubeOrigamichallenges that you might enjoy trying to solve. These puzzles involve folding a piece of paper so that certain color patterns arise, or so that a shape of a certain area results. But let’s continue on with crease patterns… For instance, the traditional crane unfolded provides a crease pattern from which we can learn a lot. Pick a point (vertex) on the crease pattern. How many creases originate at this vertex? Is it possible for a flat origami model to have an odd number of creases coming out of a vertex on it’s crease pattern? How about the relationship between mountain and valley folds? Can you have a vertex with only valley folds or only mountain folds?  

image06How about the angles around this point? You can really impress your teacher (or your students) with this…of course, you will need to understand it first! There is a theorem called Kawasaki’s Theorem, which says that if the angles surrounding a single vertex in a flat origami crease pattern are a1, a2, a3, …, a2n, then:

a1 + a3 + a5 + … + a2n-1 = 180      and        a2 + a4 + a6 + … + a2n = 180

In other words, if you add up the angle measurements of every other angle around a point, the sum will be 180. Try it and see!   Can you see that this is true, or, even better, can you prove it?

The study of origami and mathematics can be classified as topology, although some feel that it is more closely aligned with combinatorics, or, more specifically, graph theory.  You can investigate these connections further on your own.

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Origami, a Japanese word, combines the word oru (to fold) and the noun maki (paper).

A Bit Of HistoryOrigami origin is unclear. Some historians claim the origin of origami began as a Japanese tradition of folding important documents/certificates. “Origami Tsuki” means “certified or “guaranteed”. The phrase stems from their ancient custom of folding certain special documents – such as diplomas for Tea Ceremony masters, or masters of swordsmanship – in such a way as to prevent unauthorized copies from being made.  Others claim origami came about after the invention of paper and made by Cai Lun in AD 105. This was the first usage of the word “origami” traced in Japan. The word “origami” came to be used occasionally for another kind of ceremonial folding, namely for “tsutsumi”, or formal wrappers, by the beginning of the 18th century. However, its use for recreational origami of the kind with which we are familiar did not come until the end of the nineteenth century or the beginning of the twentieth. Before that, paperfolding for play was known by a variety of names, including “orikata”, “orisue”, “orimono”, “tatamgami” and others. Exactly why the switch came to “origami” is not clear, but it has been suggested that the word was adopted in the kindergartens because the written characters were easier for young children to write. Whatever the origin, Japan has fully practiced the art. It is so valued in Japan that it has become part of religious ceremonies.

What is origami? Origami is the art of folding paper and creating three-dimensional figures of: people, animals, object, and abstract shapes. The material for creating origami is a piece of thin paper; although any paper may be used. The paper is normally cut into a 15 cm. square that is plain white on one side, and decorated on the other (color or decoration). Some creative origami artists try to experiment with cardboard, cloth, wire mesh, sheet metal, and even pasta. I bet there a million other possible ways to be creative in origami.

The four most common bases of origami are the kite base, fish base, bird base, and the frog base. Bases are the starting shapes for different figures. Adding additional folds you can create figures of virtually any shape. Some of the folds specialize in modular origami, or making multiple copies of a simple single shape and forming the pieces to make an elaborate structure.  See the cube to the right.

In 1999, Joseph Wu provided this simple yet encompassing definition.

Origami is a form of visual / sculptural representation that is defined primarily by the folding of the medium (usually paper). 

Here’s another interesting theorem:  
Theorem:
 Every flat-foldable crease pattern is 2-colorable.

In other words, suppose you have folded an origami model which lies flat. If you completely unfold the model, the crease pattern that you will see has a special property. If you want to color in the regions of your crease pattern with various colors so that no two bordering regions have the same color, you only need two colors. This may remind you of the famous map-maker’s problem: what is the fewest number of colors you need to color countries on a map (again, so that two neighboring countries aren’t the same color)? This is known as the Four Color Theorem, since the answer is four colors. As an interesting aside, this theorem was proven in 1976 by American mathematicians Appel and Haken using a computer to check the thousands of different cases involved. You can learn more about this proof, if you like.

But back to our theorem since I know that it is buggin’ you (look to the right!!).  Can you see that you need only two colors to color a crease pattern? Try it yourself! You will see that anything you fold (as long as it lies flat) will need only two colors to color in the regions on its crease pattern.   Here’s an easy way to see it: fold something that lies flat. Now color all of the regions facing towards you red and the ones facing the table blue (remember to only color one side of the paper). When you unfold, you will see that you have a proper 2-coloring!

MovingOnToMobiles

image021   Mobiles are not just to hang on a baby’s crib. They’re modern art.  Mobiles were first made by Alexander Calder, an American artist who started as a boy making wire and wooden toys.  Calder created a whole circus of animals of and performers with wire and cloth.     image020    Mobiles are sculptures with parts that move. Mobiles mean movement. You’re mobile when you move and jump and play. Mobiles move when they are suspended freely in space.

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Mobiles are shapes – circles, triangles, rectangles and squares – floating in space, suspended by a string or wire. If the shapes are balanced, they will spin and float, turn and twirl.  Touch them or blow on them, and watch them move.

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Generally mobiles hang from a ceiling, but some are mounted on pedestals.  All parts of a mobile should swing freely so that the movement of the mobile is maximized!

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Links – More about Origami

purple3Links – More about Mobiles

Download the Make your own custom made origami mobile.

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About napmath

I recently retired from full-time teaching. I taught students in High School math classes from Basic Math through AP Calculus. I have been into integrating technology into my classes since 1981. I am Nationally Board Certified, am the proud recipient of a Presidential Award of Excellence in Math Teaching, text book author, Golden Apple Scholar, Certified SMART Board Trainer, Regional Technology Teacher of the Year, and give workshops across the country - geometry, Common Core, mathematics, Interactive Whiteboards, and technology. I am currently working as an adjunct faculty member at Illinois Wesleyan University and authoring books for SMART Boards and Promethean Boards with Vision Technology. http://visionstechnology.com/
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