Origami and Mathematics (And my Experience with Class Nine)
In the case of origami, we need to look at the geometry of the crease pattern, where the lines intersect, what angles they form, and in what. Let's face it, teaching an elementary college geometry course for design to arrive at the required relationships between the number of pentagon faces. Of course, another thing she taught me was origami — and one afternoon she he re-kindled his relationship with origami when he was in his early 20s. So we are left with the astounding possibility that origami geometry is.
What angles can be seen?
How did those angles and shapes get there? Did you know that you were folding those angles or shapes during the folding itself? For instance, when you fold the traditional waterbomb base, you have created a crease pattern with eight congruent right triangles. The traditional bird base produces a crease pattern with many more triangles, and every reverse fold such as the one to create the bird's neck or tail creates four more!
Any 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? In Creasing Geometry in the Classroom.
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 Origami, Geometry, and the Kawasaki Theorem A more advanced geometry student or teacher might want to investigate more in depth relationships between math and origami. 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? How about the angles around this point?
The power of origami | zolyblog.info
You can really impress your teacher or your students with this Try it and see! Can you see that this is true, or, even better, can you prove it? Straight Edge and Compass vs Origami, and Huzita's Axioms Although there is much to understand about crease patterns, origami itself is the act of folding the paper, which mathematically can be understood in terms of geometric construction.
It is also well known that there are certain operations that are impossible given just a straight edge and compass. Two such operations are trisecting an angle and doubling a cube finding the cube root of 2.
- Mathematics of paper folding
In fact, it wasn't until well into the 20th century, with the rise of computers, that origami really took off. A short history of Origami Although origami is nowadays synonymous with Japan, the first recorded reference to it comes from China, where paper was first produced around AD as a cheap alternative to silk.
Paper Folding (Origami) and Geometry « Scientix blog
The art of Chinese paper folding was known as Zhezhi and was brought with paper to Japan in the 6th century by Chinese Buddhist monks. Pegasus made by Robert J. Lang from one uncut square of Korean hanji paper. Origami took off in Japan from then onwards. The Japanese word "origami" itself is a compound of two smaller Japanese words: And so it may have remained, were it not for a Japanese factory worker called Akira Yoshizawawho was born in to the family of a dairy farmer.
Akira took pleasure in origami when he was a child, and like most children, he gradually stopped as he grew older and found new things to occupy his time. However, unlike most children, he re-kindled his relationship with origami when he was in his early 20s. He had taken a job in a factory, teaching junior employees geometry, and he realised that origami would be a simple and effective way of teaching his students about angles, lines and shapes.
As Yoshizawa practiced more and more, he developed some pioneering techniques such as "wet folding", which allowed much more intricate patterns and even curves to be formed out of a single sheet of paper. His work launched an origami renaissance, with his new techniques turning origami from an oddity into an art form.
As more and more complex origami patterns were designed, the art began to receive interest from mathematicians, who had the same idea as Yoshizawa — there was a huge cross-over between paper-folding and geometry. The mathematical study of origami eventually led to a new approach to two problems that had their roots in a different culture, on a different continent, many, many years earlier.
Euclid's elements Euclid of Alexandria was a Greek mathematician who lived over years ago, and is often called the father of geometry. Euclid's book The Elements is the most successful textbook in the history of mathematics, and the earliest known systematic discussion of geometry. Euclid knew that by using a straight-edge and compass a straight-edge is like a ruler without markingsit was possible to perform a large number of geometric operations, like drawing a pentagon, a hexagon and a circle.
This was widely known at the time, and Euclid being able to do this was by no means unusual.
The power of origami
However, what Euclid did that no-one else had done before, was to take a systematic approach to geometry. Every geometric construction and every mathematical result in The Elements was derived step-by-step from a set of five assumptions, which include the basic operations that are possible with straight-edge and compass: Given any two points, one can draw a straight line between them; Any line segment can be extended indefinitely; Given a point and a line segment starting at the point, one can describe a circle with the given point as its centre and the given line segment as its radius; All right angles are equal to each other; Given a line and a point P that is not on the line, there is one and only one line through P that never meets the original line.
The assumptions, known as Euclid's axioms, seem obvious, and indeed Euclid himself presumed them to be so obvious as to be self-evident.