How to use origami to achieve any property (Part 1)

           Origami is a pretty powerful tool, if you know how to apply it to a given situation. Here is an algorithm you can use to creatively solve nearly any geometric problem, given some amount of creativity.

The first step is to reduce your situation to lines. For example, I recently observed someone struggling to cut circles out of parchment paper for lining cake tins. This can easily be solved with origami, once you’ve reduced your problem to lines. In this case, what you have to do is approximate the circle as a many-sided polygon.

The next step is to notice any symmetries present in the system. In the circle paper cutting example, we can see that there are many lines of symmetry along the diameters - wherever you see a line of symmetry, you’ll most likely make a fold. 

The second line of the diagram shows the process. When you start, you fold along the dashed lines to form a right triangle. You’d probably want to fold the triangle a few more times than I’ve shown here until it’s a really pointy right triangle unless you’re OK with octagons. You cut along the dotted lines and the unfold into your finished (rather low polygon count) circle!

In this example, the algorithm ends here, but for some problems you might need a few more steps. In particular, in situations where different parts are moving, consider the type of motion in question. Are you dealing with a rotational hinging motion, a linear motion, or a combination (like a screw). In hinging motions, you want one or a set of mountain folds to dominate your structure. In linear motions, you want parallel and alternating sets of mountain and valley folds, sort of like an accordion. In the screw motion, you do the same but with the folds at angles.

At this point, you might be thinking, “Hang on a moment, this is total BS! You’ve just chosen a very simple example that happens to conform to your model; not everything’s going to turn out like that!” I’d say that’s a pretty reasonable response. The thing is, though, this algorithm really does apply to basically everything. In fact, if you apply the parchment paper circle algorithm in reverse, and go a little bit further, you basically get Robert Lang’s famous Treemaker algorithm. Check out langorigami.com/article/treemaker/! If you read the article, you’ll notice that the first step is very similar to what I advocate - making a mathematical tree model is a form of reducing the problem to lines. The Treemaker then unfolds these lines into (again, very low polygon) circles, which it then packs on a page, and then voila - you have your folding pattern.

Actually, there’s another cool mathematical theorem that shows that you can make any shape (or at least any well-behaved shape) by folding a piece of paper, making a single cut, and unfolding. It’s called the, uh, Fold-and-Cut Theorem. You can watch a nice Numberphile video on it here.

This article was getting really long, so I decided to split it up into parts. Stay tuned for part 2!

Origami Letters

 https://www.letterstrellis.com/activities/origami-letters 

Folding a letter size sheet (8-1/2 x 11 inch) into its own envelope for mailing.  The above page has a video instruction, and a template to follow for the folds.

There is a QR code that can be stuck to the back of the envelope to assist the recipient how to open the letter without damaging it.

Finally, some information on "letterlocking" - a number of methods, including some fancy folding and tucking (and cutting) that were used to assure that a letter had not been tampered with in transit.  Mary Queen of Scots locked her last letter using some of these techniques before she was beheaded. 

MIT foldable surgery tool

             As biotechnology improves, we are gaining new surgical capabilities, allowing us to move towards less invasive procedures which have obvious advantages. “However, sealing and repairing tissues is still a major challenge for minimally invasive surgery,” says Sarah Wu, a graduate student at MIT mechanical engineering. This is because sealing and repairing tissues often involves many sensors, haptic feedback, and tools. To combat this Sarah Wu’s team developed a sort of fabric that is engineered to function as a tissue-sealing patch. One side contains sticky microparticles infused with a liquid that prevents contamination with bodily fluids and the other contains a non-sticky layer that limits contamination with bacteria and the like. This creates a nice, foldable sheet which you can do origami with to your heart’s content. The key is to imagine this fabric as a sheet of paper. For instance, one fold pattern allows it to integrate nicely with a laparoscopic stapler to form a sort of surgical tape, or a balloon catheter for sealing things with cylindrical symmetry. Appealingly, this patch is biodegradable even in the body.

The Zen of Folding and Forming a Crease

Before

Before you put your first crease in a piece of paper, pick it up, and familiarize yourself with its feel. Is it stiff or flexible?  Is it smooth or rough? Are the edges cut cleanly or does it have a ragged edge? Does the paper bend nicely or does it feel like it is brittle like a dry cracker as if it were going to break as you bend it?  Good folding paper should be thin and lightweight so it does not get too thick with many folds. It should be strong and withstand at least a few folds back and forth along the same line.

During

Feel how the paper handles under your fingers. Crinkle it gently and feel its resistance. Use pads of your fingertips: they are very sensitive and can detect tiny differences in surface and edge features.  In fact, when folding a square in half, if you adjust the paper so the edges touch your fingertips you can tell when the edges are lined up exactly even without looking: your fingertips will tell you.  Then grasp both layers, don't let them slip, and make the crease without letting go.  Same for a diagonal: as you position one corner over the other, let your fingers feel when both sides of the corner are aligned, then grasp the layers together and make the crease.

When forming a crease, start with a wide curve and loosely holding the paper. Gradually (don't rush!) make the curve smaller and tighter, feeling the curve form itself into the crease. When the curve is small and tight, go ahead and press the crease into the paper.

After

Forming the crease in a high quality origami paper can be gentle (using the pads of your fingertips) or firm to very firm (using a fingernail or the slick side of a sharpie pen) to press the crease into a permanent mark. In most standard pieces, a good firm crease can be obtained with a fingernail or hard object making the two layers look and feel like one when you are done.

Look at the finished crease after you are done: are the edges lined up and straight?  Did anything shift between when you knew the edges were lined up and after the crease was completed?  Is the diagonal fold on a square coming to a perfectly symmetrical point (if it's off center one side will show more than the other)? Did a part of the paper get stretched from too much pulling or pushing? Imperfections are just opportunities to practice and make it better, but you have to look carefully at your work to verify that your crease is as good as it can be. Don't just fold and look away.

Practice

Becoming familiar with your paper and how to handle it, how to make it obey your objective,  An exercise that can be done anywhere, is to shape an edge of a piece of paper into an "S".  Start with wide soft curves, but tighten them bit by bit, and as the curves get tighter and smaller, line up an edge with the crease.  Adjust the lengths of each of the 1/3 so they balance out and converge on an exact 1/3 of the edge.  When the curve gets small enough right when the lengths are visually equal, set the crease firmly.  

Creasing Many Layers

When taking a piece and making a crease when there are already many layers in place, it is important to fold all of the layers together, and very important to prevent the inner layers from slipping. One way is to grasp all the layers firmly on both sides of the crease while the fold is being formed. Don't let the inner layers slip away from the crease: this can easily happen if there are many layers already in the piece being folded.

Fold on!

Auxetic Materials and Face Masks

When ordinary materials are pulled from side to side, the top and bottom tend to shrink towards each other.  It's the behavior of squeezing a balloon: some part comes out somewhere else.

Auxetic materials behave in ways that are surprisingly opposite of ordinary.  https://en.wikipedia.org/wiki/Auxetics  When the sides are pulled away from left to right, the top and bottom actually expand higher and lower.

From a shopping bag I folded a pattern similar to the offset zig zag of the auxetics diagram, but interestingly there was little movement in one direction, while there was great flexibility in expanding in the other.  The expandability looked like it might be useful when used in a face mask to cover different sizes of faces, but sealing the edges is a challenge.

Here is a brief video to observe the behavior of this piece.

TED-Ed origami video

    Just a quick thing I wanted to share here, I recently came across this beautifully animated video by TED-Ed about the mathematical rules behind origami, specifically, two-dimensional folds. It's great, I hope you enjoy it!

Origami landing legs

    Say you are designing a reusable rocket (don't worry, it's just a thought experiment). A rather important part of a reusable rocket is the landing legs. However, random sticks won't do, you must also include shock absorbers. Enter, origami!

    A few years ago (sorry I'm a bit late on this), researchers investigated the use of an origami structure to build a shock absorber. Their design includes multiple compressible hexagonal prisms. When one end of the chain is compressed, it causes a compression wave to travel up the structure. Subsequently, the prism then expands, which causes a rarefaction wave to also travel up the structure. However, the rarefaction wave travels faster than the compression wave, causing the two to meet, and cancel each other out. Why does this occur? I have literally no idea. I've been ruminating on this post for several months and still haven't found out.

    Anyways, here's a photo of what the unit's fold pattern was: 

Here's one of a folded unit:

Here is a picture of the full mechanism:

I should note that on some of the other media regarding this article the setup involved adjacent prisms being mirror images of each other.

Here's the paper: https://advances.sciencemag.org/content/5/5/eaau2835

I got all my images from it.

creases

 Crease [noun]: the indentation caused by a fold

In our various discussions of various delightful pieces of origami, we here at Ourigami Origami Informational Content Recipes Inc. LLC Eastern Massachusetts division often tell you to do things like “fold so that the creases lie on top of each other” and “fold from the edge to the crease”. But what do we really mean? Well, as you might have heard, a crease is the line that’s the result of a fold. So, if you “fold so that the creases lie on top of each other” you are creating a new fold halfway between the two previous folds, and “folding from the edge to the crease” is creating a fold halfway between the previous fold and edge. However, this usually comes with a couple assumptions - specifically, that the fold is perfectly sharp and linear - and an important distinction when folding.

Depending on the force you apply when folding, the crease will turn out very differently. If you apply a large amount of force, your crease will be essentially flat. However, if you don’t, it’ll be a glorified bend in the paper, with the radius of the bend depending on exactly how much force you put. Sometimes, this is desirable, as when you want to bend over another, sharper fold.

Another important distinction is mountain vs. valley folds. These terms are really relative, they merely depend on the way you hold the paper. A “valley fold” means the fold is pointing away from you and a “mountain fold” means the fold is pointing towards you. When there are two of the same type being creased to each other, you usually have slightly less precision than when there are two of different types creased to each other as then you can fit one crease into the other.

The Importance of Equal Dividing

The Importance of Equal Dividing

Dividing things (line segments, circles, squares or rectangles) into equal parts is of course a geometric and math challenge, but more important than the academic / intellectual "hey I can solve this puzzle!" accomplishment, is the idea of fairness.

Fairness, as in equal parts, is key to the "right" way of doing things.  Even very young children and some monkeys, recognize unfairness before they have words for it, that's how deeply ingrained the idea has been: fairness is in our DNA.

My early experience with fair division was splitting one dessert between my brother and I. The governing principle is this: "whoever makes the cut, the other one has first pick". So it was always an objective to make the cut as even as possible to equal parts. When the number of parts increases, so does the complexity of the solution. How, for example, would you have a piece of cake divided for three cake eaters, in a fair and equitable way? 

Higher numbers only create much more complexity.  For example, if there are 3 people who have to share dessert, who makes the cut(s) and who chooses? At home, you might just appeal to a higher authority: Mom divides the portions and gives them out randomly (maybe each portion is covered, and the 3 dessert eaters pick their piece without seeing the contents. No complaining afterwards, just eat your dessert.

Dividing a pie or cake into equal portions for seven friends can use math and origami methods: https://ourigami.blogspot.com/2020/12/fujimoto-approximation-of-dividing-line.html where the aim is to be as "fair" as possible.

This is one area where use of origami can help in the field of ethics and right behavior.

Origami Masks 2021

Masks, masks, masks. 

(in a) 2008 TED talk by renowned origami artist and physicist Robert Lang. "As weird and surprising as it may sound,” Lang said, “origami may someday even save a life."

https://api.nationalgeographic.com/distribution/public/amp/science/2021/01/we-need-better-face-masks-and-origami-might-help

Origami Safety

Origami Safety - Before You Do Anything

What could possibly be dangerous about origami? It's just folding bits of paper!  

Usually I will suggest to people:  Just Do It - the more mistakes you make, the more situations you deeply know what to avoid, so by eliminating the failures in action, you are moving in a very efficient way to finding that narrow path of success, of "what works".  But I will ask you to read this while you are still thinking about doing origami.  It is for your own safety.

1. Cutting Paper

If you buy nicely cut paper, with pretty colors, that is fine. But going into production or making many pieces such as in Unit Origami,  it is possible you will need a lot of pieces, and cutting with sharp edged blades can do damage to fingers and other body parts.

1a. Rotary Cutters

Rotary cutters are circular blades inside a cover so no fingers can touch the sharp cutting edges. These are very good for precise cuts because the flat bed portion is marked with inch and centimeter rules so edges are straight and corners are perpendicular. The only limitation is thickness:  how many sheets of regular paper, and how thick a piece of paper can be before it's not good for the rotary cutter. Replacing blades on these should be done by someone who won't damage themselves.

1b. Exposed Blades

Avoid all exposed blades, whether it is a guillotine or straightedge and razor.  Scissors are not really easy to do a long straight cut unless you have good fine motor skills.  I worked with someone in a print shop that had a hydraulic powered guillotine that could cut a ream in one slice, but once the paper is clamped in place, both hands had to be below the working platform to push two buttons at the same time so no limbs were anywhere near the blade.  Even the nice clean cut requires attention to detail: when I asked someone else to cut some paper for me it was cut but not perfectly square, which wasted an entire ream of paper. Slightly out of square is very frustrating if you are hoping for actual square pieces. Oh well.

2. Paper Cuts

If you handle paper, no matter how carefully, one day you might suffer a paper cut - that is when the edge of the paper lands just right against a finger and with surprising speed makes an ouchie. The way to avoid paper cuts, is to avoid being rushed: take your time, handle the paper with care and respect. 

3. Starting With a Rectangle

If you have a piece of paper from the printer, or pick up a menu or brochure from somewhere, it will most likely not be a nice square. Even the waxed paper liner in your lunch sandwich might have a jagged edge, if it doesn't have some sauce all over it. Let's just start with a letter size piece of paper that came out of a printer, which dependably will have its edges at right angles.  Folding one corner to the opposite side, forms the diagonal of the square. Any paper that is a single layer can be trimmed off to leave a square.

3a. If the paper is folded over so there is a good sharp crease between the square portion and the excess, a knife (dull is not a bad choice here for safety) can be used to cut off the strip and make a nice square.

3b. There is a way to curl the excess paper and make a clean tear - (need a video to show...)

How Can I Memorize a Complex Series of Folds?

How Can I Memorize a Complex Series of Folds?

The answer is so obvious I should not have to write it down, but if you are reading this section I am going to confirm your most likely guess for a technique to memorize a long and perhaps complex series of folds. Fold your piece. Do it again. Repeat.

Begin a piece with the instructions.  After a few pieces are done, start without the instructions and see how far you get.  When you get to a point where you are stuck, take a peek at the instructions, just enough to get you back on track. Do it again, and the next time start from scratch and see how far you get without the instructions. 

Select a piece which you like, or want to make so it can be gifted to friends, family, or even strangers, someone you'd like to meet. Some simple pieces are kept by friends because they are a pleasing shape - and everybody likes a gift. If you have memorized it you can make one on demand anywhere you are. Your friends will be amazed at your ability to remember "so many steps".  All you do is take one step at a time.

One of my friends makes rings out of dollar bills, and gifts them to wait staff. Most of the time they are welcome. 

Innovation

While you are folding without the instructions and seeing how far you can get, is also a good time to explore alternate paths to your end product. As you make more of the same piece, try variations to "make it your own" - specific design features that are not on the original instructions, but can make it look nicer in the finished piece. Innovation in this sense can be a small detail or a radical new piece.

An example of easy variations is the "dollar bill ring." It is usually made with the "1" on top of the "jewel" in the finished ring. The "1" is expected to be at one end of the dollar bill folded into a narrow strip. If however you start with making a strip going down the cetner of the back of the dollar bill, the small square that forms the jewel can look like the face of Ganesha, or Ganesh, the Indian deity with the head of an elephant on a human body. There are two eyes and a trunk curling down the center.  Other features can be highlighted, but you have to look closely to find them.  The 10 dollar bill and highter denominations has the numerals printed in a color shifting ink https://www.uscurrency.gov/sites/default/files/downloadable-materials/files/en/dollars-in-detail-guide-en.pdf . The color shifting ink can be an attractive "top" or "jewel" on the ring. I'm usually too cheap to use anything other than a George Washington one dollar bill. If the design ever changes, or when the paper converts to plastic in some future form of currency as is the case with many international currencies, I'll have to stock up on the paper bills much as classic Coke drinkers hoarded their beverage when it was ending production of their favorite drink.

Serendipity

Without the directions to follow, sometimes the folding can lead you to a lucky surprise. Also in the dollar bill folding, there is a shirt-and-tie design. Purely by accident and without looking at the directions, I made some extra folds that added a nice pattern to the shoulders, similar to the stitched decorations on the shoulders of a western style shirt. These types of customizations are unique to my dollar bill shirt and tie pieces, and can be thought of as a signature feature. It's the shirt on the left in the poster https://ourigami.blogspot.com/2014/03/conference-poster-credit.html.  The standard shirt and tie has no triangular decoration below the tie up to the shoulders. 

The Perfect Origami Paper

The Perfect Folding Paper

There are some standard, nearly-perfect square cut paper that comes in packages of 400 or 500 multicolored sheets, which take a crease neatly and don't fall apart after a few folds back and forth. If you have been to an origami store, or craft store you have seen them.  In different sizes, colors, and wonderfully consistent.  The colors are pretty stable (they don't fade quickly).

Every folding project, actually, might demand something different for your objective.  An artistic item such as a figure with a flowing robe, might do well with a nice piece of paper made from long fibers and can withstand "wet folding" where the paper is wetted with a starch or "sizing" material, that stiffens the paper after it dries - and keeps the shape of the flowing robe after it dries.

Lighter, more transparent papers for their own projects.

"Traditional currency", or money, was historically made from linen fibers which are strong, survive handling, crumpling, washing, and much abuse as money changes hands and thus is able to take a good crease.  Newer currency is more plastic than paper, and can be very difficult to fold and maintain a crease of any kind.

I tried folding a light fabric to make some visual patterns, where double and triple layers could darken the color. It would have been put in a sandwich of two pieces of glass in a frame, and held up to a window or light to see the patterns. What a disappointment to find that some synthetic fabrics simply do not take a crease no matter how much spray starch or high iron temperature was used! The material just sprang back with a mind if its own. A sharp straight line crease was impossible, but a smoother construction might have been perfect for it.

For many decades now, short fibers and coatings were the combination that made paper inexpensive to produce and print, and looked good. But try to fold a crease, and you will find quickly that it has already started to crack and break into pieces.  So before deciding to use paper that looks good, try to crease it first. 

As it turns out, patterns on paper for folding origami can be their own disappointment because many origami pieces have enough folds that the pattern is buried inside the folded completed work, so most of the nice looking pattern that you can see when the paper is flat and unfolded, will not even be visible after the folding is done.

For practice, the extra paper that comes off the printer can be cut into squares and used as experimental platforms. 

The perfect origami paper is different for each piece for which you wish to use it.  Try as many as you can put your hands on:  magazine paper is flimsy but can be used for their color (in the advertisements), menus as a distraction only have to last until your meal arrives, and adding machine paper can be made into hexaflexagons.  Paper napkins can be a challenge because they are fragile and more ephemeral than something more solid. The large white or brown paper some restaurants use on their table tops are always fun to make a giant sized piece, and you can discover for yourself where the weaknesses are in the design and pattern.  Foil can be its own challenge because it can tear easily but has an important advantage in that it retains its shape when you are done, much better than paper, which tends to unravel over time.