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.

Dividing a Line Segment into Odd Number of Parts of Equal Size and a Fujimoto Approximation

Dividing a segment into even numbered parts of equal size is easy, just fold the ends together and get half, repeat to get a quarter, again to get an eighth, another time to get a sixteenth, and so on.  This does not rely on the squareness of the paper, just a straight segment of a line.

There is a kind of "cheat":  to make seven exactly equal segments, fold a segment three times over, and you have eight equal parts. Fold one of them over (or cut it, although cutting is frowned upon among origami purists) and you have seven equal parts! When I showed this to second graders at Lynch School for Japan Day one year, the chorus of boos and "noooo!"s clearly revealed their position that they were not impressed. But it is easy, direct, and rather precisely achieved without guessing.

Making odd numbers of parts of equal size requires a more analytical approach.  If you don't happen to memorize the exact methods in Lang's summary http://www.langorigami.com/wp-content/uploads/2017/09/origami_constructions.pdf, a quick way to get there is by a very efficient approximation method.  Fujimoto's approximation is explained in http://www.teachersofindia.org/sites/default/files/5_how_do_you_divide_a_strip_into_equal_fifths.pdf, and every fold divides the "error" in half, so by the 4th fold (5th approximation), the "error" is 1/16th of the deviation from your original guess. If your guesstimate were anywhere close to where it should be, the error between your guess and the exact 1/5 is very small. 

Extra credit: use Fujimoto's approximation to find sevenths of a segment.  Remember, this has no dependency on a square of paper, all you need is a line segment.

Extra extra credit: would this work on dividing an angle into odd numbers of equal angles?

One Eighth Along the Edge of a Square, and Three Sevenths

One eighth.  1/8th

The easy way to find 1/8th of the length of a line segment is to divide 3 times.  On a square, fold two adjacent corners together to get half, fold the corner to the center that you just found and get a quarter, and finally one more time, fold paper so the corner lands on the one-quarter mark.  That's three steps.

An interesting shortcut requires only two steps if you have a square.  Make a pinch or mark halfway between two adjacent corners, let's say on the right side:  upper right corner and lower right corner are lined up and the spot halfway between them is marked with a pinch. Take a corner on the opposite side of the square, say the upper left corner, and put it on top of the midpoint of the right side that you just marked.   You don't have to make a full fold, but just find the place (this will be on the bottom of the square if we are using the above landmarks) where the fold intersects the bottom edge.  Pinch this spot and you will have marked 1/8th of the distance along the bottom of the square.  That's two steps.

Challenge: prove this using the equation of a line going through two points where the slope of the line is expressed by the equation:
m = ( y0 - y )/( x0 - x )
where (x0, y0) is a specific point and x and y are variables on the x-y coordinate system. For a straight line, m a constant.

Hints:  Using the above references, and starting with a unit square (length 1 on each side), the midpoint between the top right and bottom right is (1, .5)(let's call that point B) and the upper left corner of the square is (0,1)(let's call this point A). The midpoint between those is -- (.5, .75)(let's call this C).    The line that forms the fold is the perpendicular bisector of the line between the upper left and midpoint of the right side.

The slope of the line through A and B is m = -1/2.  Its perpendicular is -1/m which would be m1 = 2.  The perpendicular bisector of segment AB would have a slope of 2, going through point C (.5, .75).  The equation for that line would be m = (y - .75)/(x - .5) = 2.

Sooo,  2x -1 = y - 3/4

y = 2x -1/4 = 0 where the square is on the x axis.

2x = 1/4

x = 1/8

Voila, the fold is exactly 1/8th of the length of the side of the square, in two folds, if you like, or two pinches, since there is no need to actually fold the crease, just pinch and mark the spot.

More:  if you mark two of these on the same square, and put one of the 1/8th marks on top of the other (that's five pinch marks), the part where the last fold intersects a side, is 3/7th of the length of the side of the square. This five-pinch procedure yields a particularly interesting artifact, that the side of a square can directly and exactly arrive at an odd numbered subdivision of a square. Other exact odd-numbered divisions from folding are demonstrated in http://www.langorigami.com/wp-content/uploads/2017/09/origami_constructions.pdf. 

Extra credit:  prove with algebraic equivalents that the last fold is actually 3/7 of the distance along one of the edges of the square.

Triangle

Triangle

In origami it is natural that the triangle is equilateral, just as the natural base is a square, where the sides are the same length.  Triangles are the first planar object that can be made with the fewest number of line segments (two line segments just are not up to the task).

There is an easy way to make a lot of equilateral triangles, starting with a strip of paper of uniform width.  Even if the first fold is only approximate, the following folds can be made "perfect" as long as the corner is tight and the edge aligns with the existing edge.    

https://www.cutoutfoldup.com/125-fold-a-series-of-equilateral-triangles.php.

Making an equilateral triangle (or equivalently, 60 degree and 30 degree angles) is as easy as making two folds on a rectangular piece of paper. Say we start with a letter size piece of paper, long side up and down (sometimes this is called "portrait mode") and fold it left to right in half:  the resulting fold is vertical. If you take say the upper left corner and put it on the centerline, while adjusting its location so the crease ends up including the upper right corner, and set the crease, you have a 30 degree angle from the top. Folding the paper over where the top edge has landed, makes a 60 degree angle.  Using the point on the centerline as a reference point, complete the equilateral triangle.  Voila!   Constructing 30 and 60 degree angles with two folds can be done almost anywhere with a bit of planning, and it's good to know this shortcut to make an exact 30 or 60 degree angle.

Strips of paper folded into connected equilateral triangles in this manner are very nearly ready to make a basic hexaflexagon.  https://en.wikipedia.org/wiki/Flexagon shows how to make a hexahexaflexagon.

I've used a hexaflexagon to write brief messages that are hard to decipher until the flexagon is returned to its state where the message was originally written.  

See also:

Triangle unit origami

Flexagon, Hexaflexagon, Hexahexaflexagon

2000 doves

 https://cathedral.org/visit-us/doves/

Maybe it should have been 2021 doves. I wonder if there is a video of them moving with the air currents.

Origami parabolas!?

             Last time I mentioned a fold that was equivalent to the shared tangent of two parabolas. In this article, I’d like to talk more about parabolas and origami - finding a parabola’s tangent is actually quite common in origami.

You might know that a parabola is the shape of a graph of a quadratic function; that is, the graph of something in the form of y=ax^2+bx+c.

You might also have heard of parabolas in the context of parabolic reflectors, where rays of light or sound emanating from one point are reflected parallel to each other or vice versa.

Parabolas in origami occur because of another definition of a parabola: the set of all points equidistant from a point (called the parabola’s “focus”) and a line (called the parabola’s “directrix”).

Okay, okay, maybe this is too obtuse and it seems like it’s not going anywhere, but let me explain. When you fold two points so they lie on top of each other, you get the perpendicular bisector of the segment containing the two points, that is, the set of all points equidistant from the two original points. Now, if you keep one of the original points fixed and move the other point around in a line, it stands to reason that the lines will all have a point that’s on a parabola - the parabola whose focus is the fixed point and directrix is the line on which the other point is moving.

Perhaps it’s time for the algorithm. Draw a point close-ish to one of the sides of a piece of paper. Then, take that side and fold it so that when it lands, it goes through the point. Make many folds of this type. Eventually, you should see a parabola emerge - there’ll be a heavily creased region, and a region without any creases, and the boundary between the two is a parabola.