Origami and Geometry: Squareness

Squareness
Many origami pieces from the very simple to the very complex, depends on having a nice square piece of paper with which to start.   With a few folds it is possible to verify the squareness of a piece.  First fold one corner to the diagonally opposite corner, and see if the other two corners (at the ends of the crease) meet up with the folded line.  If they do, that's a good sign.  Now do the the same with the other two corners.   If everything lines up, the diagonals are fine, but you still need to check two more folds.  Take two adjacent corners and match them up with the other two (I've heard this called the hot-dog bun fold - use your imagination).   If all the edges line up, turn the paper 90degrees and check it the other way.


If all four folds line up the corners and edges, you have a square.  And generally your "test" folds are not wasted because they are often the basis for starting folds in various pieces.


If they do not...  at least you know you will probably have to compensate, or start with another piece that is square.

Most "origami paper" (or "kami") is cut nicely square.  If you want to cut your own, be very careful with blades -- also, if you cut more than 3 or 4 sheets at the same time, watch for some of the layers (below) slipping around underneath.

Not all origami constructions use a square, but many do, and they all benefit from starting with a square square.
You can always return to Table of Contents.

-----
2012-01-09
Copyright 2012

Origami and Geometry: How to Fold Media

How to Fold Media

If you're thinking "I already know how to fold paper," you're right of course.  But there are some general techniques for folding precisely that help arrive at a consistent end result.

It's also no longer "paper" in the conventional sense, but many materials can be "folded" - with care, aluminum foil (foil backed paper has both the benefit of great "memory" - the fold stays where you fold it) can be folded and "stays" in place, where conventional paper tends to "relax" and lose its shape.

Speaking of losing shape, one technique for keeping a paper folded piece in place is to use "sizing" (also called "starch" in ironing).   Wet folding can produce very natural looking shapes, and once the sizing / starch dries it can be as stiff as a starched, pressed shirt.

Whatever you fold, here are some general suggestions:

* Do Not Hurry

* Find the points that should match, and watch that they do not slip out of alignment (say match up corners), and starting with a large circular shape gently reduce the size of the curve (reduce the radius of curvature) until you're ready to "press home" the crease.   

* A sharp crease makes a precise mark from which later folds are made - if the first one is not exact, later folds will be farther off the desired mark.

* At all times make sure your alignment points have not shifted while everything else is being handled, particularly if there are many layers under your fingers.

* When handling multiple layers, grasp all the layers together and do not let the layers on the inside of the bend / fold escape:  make them stay under the desired crease.

* As the paper or media allows, make a crisp sharp crease, either using your fingernail, or a flat, hard, slippery object to "set the crease".   A Sharpie pen is ideal, or see if you can find a "whale bone" (a smooth piece of plastic, many years ago polished bones were also used).

* Apply the crease evenly; don't start on one end and "push" the folding tool to the other end,   This will often result in an uneven fold with stretches and sags.

* Use square paper (as perfectly square as possible) if you can get it, if that is what the design calls for it.

The next section will talk about squareness.

Or, return to Table of Contents.

-----
2012-01-08
Copyright 2012

Origami and Geometry - Lessons

Finally I am going to just get started with writing out some observations on the relationship between origami and geometry.   The outline will be in the right-hand side menu on the "Origami and Geometry: Lessons" page, with links to individual pages as they are written out, and edited to include some video clips as appropriate and as they become available.

Or, go to that page using this link.

Origami and Geometry: Safety

Safety

Safety is important in all activities, and some of the dangers particularly related to origami should be part of the reminders of anyone who is going to handle paper on a regular basis.

1. The most obvious danger is the paper cut:  it can be very painful and inconvenient because the most common paper cuts are in parts of the hand that are used a lot.   A good approach to avoiding paper cuts is  to take one's time handling the paper.   I have found that many paper cuts result from rushing and hurrying.

2. As we will see, many modern and classic pieces are based on a traditional square of paper.  The sharp instruments used to make your own "perfect square" must only be used with supervision, whether it is a knife blade or paper cutter.   Rotary cutters when used correctly are generally safer than the traditional large blade paper cutter.

3. When young folders are first learning to fold, they should be reminded to ask what paper is allowed to be used for folding.

4. Most paper sold as "origami paper" for folding are manufactured avoiding toxic materials, but it is good practice to avoid any transfer of colorful dyes or paper coatings to one's mouth (don't lick the paper, or wrap food in paper unless you know it's safe; for example, wax paper made for handling food).   Certainly wash hands regularly, before and after folding, as a healthful practice.


At any time, return to Table of Contents.

-----
2012-01-08
Copyright 2012

Origami and Geometry: Examples

Examples

Some examples of origami being more efficient than tools of geometry in some activities include:

1. The perpendicular bisector of a line is readily found by creating the crease formed by placing one endpoint on the other, and folding.   Note that all points on this line are the same distance from the selected end points.

2. A 30-degree and 60-degree angle can be constructed by two folds

3.  1/8th of the side of a square can be found in two folds.

4.  3/7th of the side of a square can be found in five folds.

Later we shall see how the use of folds in paper achieve three dimensional geometric shapes as well as constructions that could be very difficult to accomplish with traditional tools.

Before we start folding, here are some reminders on Safety.


Or, return to Table of Contents.

-----
2012-01-08
Copyright 2012

Origami and Geometry: Introduction

Introduction

This begins a series of lessons on the close relationship between origami and geometry.  The subtitle could be "Anything that can be done with the traditional tools of geometry, can be done with folding."    

Traditional tools of geometry include three items:  a straightedge, compass, and protractor.  Many, if not all constructions that are made with these tools, can be directly performed by folding.   Folding a flat sheet of paper immediately creates a straight line through any two points you select.  The compass and protractor are not difficult to demonstrate an equivalence as well.

Using the manipulation and folding of paper can in some instances actually be more efficient than using the traditional tools.   Some examples are given in the next page:  Examples.

Or, return to Table of Contents.

-----
2012-01-08
Copyright 2012

folding ukelele

http://www.ponoko.com/design-your-own/products/-supermini-foldable-ukulele-kit-6556

Brian Chan's design on bamboo, extending folding to an entirely new realm.  Parts are laser-cut on ponoko's 2d/3d personal factory, you design, upload, have it made, and in a few days the part appears on your doorstep.  There is a bigger (more complex) design for another uke, but this is perfect.

See it in action here: