Relationship between geometry and origami

The power of origami |

relationship between geometry and origami

The relationship between mathematics and origami has yet to be fully A set of Postulates, similar to those of Euclidean geometry, have been established. In the case of origami, we need to look at the geometry of the crease in a flat origami crease pattern, the difference between the number of. When I tried to teach some basic geometry and mathematics asked me what kind of relationship there was between geometry and origami.

relationship between geometry and origami

I especially recommend watching Robert J. Lang born May 4, is an American physicist who is also one of the foremost origami artists and theorists in the world. Nano origami is a new method that allows objects in the nano scale to be folded into simple three-dimensional objects Erik D. Demaine born February 28, is a professor of Computer Science at the Massachusetts Institute of Technology and a former child prodigy. Nano Origami Robots and Devices: George Barbastathis and a group of researchers are developing nano origami, a new method that allows objects in the nano scale to be folded into simple three-dimensional objects.

The folded tiny elements can be used as motor and capacitor, allowing for better computer storage memory units, faster microprocessors and the use of new nanophotonic devices.

Nano-size foldable robots can easily move though diseased tissues and organs in the human body. Scientists are trying to reveal protein folding and DNA secrets with molecular origami. You would probably prefer a folded air bag with the best folding model thought by scientists.

Parachutes are examples of the fact that smooth and systematic folding is vital. This was widely known at the time, and Euclid being able to do this was by no means unusual. 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.

relationship between geometry and origami

But their beauty lies in the fact that they can be used to construct geometric proofs of theorems that are immensely more complex than the axioms themselves. But there are also limitations to Euclidean geometry. Two of the most famous problems of antiquity were the trisection of the angle dividing a given angle into three equal parts and the doubling of the cube constructing a cube which has exactly twice the volume of a given cube.

Origami: mathematics in creasing

According to legend, the citizens of ancient Delos were faced with the latter problem when the oracle at Delphi advised them to double the volume of their altar, in order to avert the plague. This, however, proved impossible using only Euclid's straight-edge and compass methods, and the same goes for angle trisection. But it turns out that both problems can be solved using origami!

So we are left with the astounding possibility that origami geometry is more powerful than Euclidean geometry. The genius of origami Just as Euclid devised axioms for planar geometry, the modern mathematicians Humiaki Huzita and Koshiro Hatori devised a complete set of axioms to describe origami geometry — the Huzita—Hatori axioms click here to skip the axioms and see the rest of the article: Given two points andthere is a unique fold that passes through both of them.

Given two points andthere is a unique fold that places onto Axiom 3: 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. A beginning geometry student might want to figure out the types of triangles on the paper. What angles can be seen?

How did those angles and shapes get there?

Mathematics of paper folding

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.

Math Encounters - The Geometry of Origami -- Erik Demaine (Presentation Part 1 of 4)

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!

  • The power of origami

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.

relationship between geometry and origami