How does origami involve math

So, when you fold your paper and it looks off, get out a protractor and measure those angles. Remember, origami is also an art of geometry and precision. How do we read origami crease patterns?

The blueprint is right in front of you. Take an origami crane. Unfold it. The creases are the blueprint. You can follow the creases, refolding the paper back into the crane. The step-by-step pattern uses a dash-dot-dot line to signify a mountain fold. And a dashed line to illustrate a valley fold. Crease patterns may follow these types of lines.

Some may include two different colours to differentiate mountain from valley creases. And could overwhelm the reader. Instead, the base of the design is created. Origami crease patterns use mountain and valley folds. The vertex of a mountain fold is pointing up. Modular origami, and origami in general, do not involve glue, tape, or any other type of adhesion. This requires origami designs to be well-engineered to be structurally sound and not fall apart on their own. This requirement for strength translates well into architecture, and indeed, several buildings in recent years have been based on origami designs.

Geometrical figures, proportions, tessellations, symmetry, angles, and other math concepts can all be made with origami. This allows hands-on demonstration of ideas and allows for greater learning opportunity. Because origami has structural qualities beneficial for architecture, it provides endless learning opportunities and acts as a physical example of mathematical concepts.

It should not be considered merely as a hobby. How is origami used to solve problems? Why is origami so hard? What was the original purpose of origami?

Why we do not use pencil in space? What are some ways origami has been used by designers and engineers? Is Origami still used today? Is Origami an engineer? Can I sell origami? Can you make money from Origami? Is Origami copyrighted? How does origami relate to science? How many times can you fold something? Why is origami important to Japanese culture? Who invented origami? What is the hardest origami to make?

What is the biggest origami? Why is origami so popular? Why do Japanese do origami? What does origami symbolize? Did origami originate in Japan? How did origami get its name? What does the origami crane represent? Here's a short and idiosyncratic answer , but you should really read Neil Strickland's thorough answer with pictures.

It is important that you understand that geometry and topology are very different. Topology is sometimes called "rubber sheet geometry", meaning that in topology, stretching an object or changing it's shape will not affect it as long as you do not create any holes or patch up any holes.

To a topologist, a coffee cup and a doughnut are the same, while a geometer sees them as completely different. If you read Neil's answer, you noticed that he mentioned a subway map, which is just a network of points connected by lines, just like an origami crease pattern! Studying origami crease patterns can help us learn about networks such as subways and phone networks, and how to make them faster and more efficient.

But don't take my word for it. Thomas Hull, an assistant professor of mathematics at Merrimack College in North Andover, Massachusetts, is the expert in the field of origami and topology. Tom is currently teaching a course in combinatorial geometry, and you can view the course's syllabus and assignments.

If you are looking to do more in depth research in this field, your first step should be to contact Tom. His Web site was even mentioned in a story on ABCnews.

Now back to the origami theorem that I mentioned earlier, which can be seen from two points of view. 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 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. 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! A more rigorous proof goes as follows: first show that each vertex in your crease pattern has even degree the degree is the number of creases coming out of each vertex - we discussed this earlier! Then you know the crease pattern is an Eulerian graph, that is, a graph containing a path which starts and ends at the same point and travels along every edge such a path is called an Eulerian cycle.

Don't try to prove this unless you are an experienced mathematician!