PROFESSOR: So for

today’s lecture we have Jason Ku

guest lecturing. And he’s the president of

OrigaMIT, which you should all check out Sunday afternoons,

origami club at MIT. He’s an origami designer

and a grad student in mechanical

engineering, and he’s going to talk about the more

artistic perspective on how origami design

works, in particular in the representational and tree

method of origami design world. So take it away, Jason. JASON KU: Hi. I’m Jason. Eric gave a bit of

an introduction. I’ve been folding

origami instance maybe I was five years old,

and I’ve been designing origami for maybe

the past 10 years. I’m a PhD student in

mechanical engineering, working in folding things

on the micro and nano scale. So that’s how this is

applying to my research. I’m here to talk a little

bit about origami art and how the concepts we’ve

been talking about in class apply to origami in actually

designing and folding artwork out of paper. These are all the

websites that I’m going to be pulling

pictures from. So if we can’t

use these pictures in future versions

of this lecture, then you can still

see some of the media. I want first make the analogy

of origami art to music. Many, many people

make this analogy, and it’s actually

very apt analogy. In music you have

composers, you have people who produce a work of

music, design the structure, design what the main

aspects of the piece are, in terms of a

structural sense, but aren’t necessarily

performers themselves. Now in origami, the

performer and the composer are usually one and the same. But hopefully in the future,

that won’t always be the case. In music the composer

usually makes a piece for multiple instruments

or multiple voices or things like that, so most the time

can’t do all that performance. And some people are more

gifted in the performance side. Some people are more gifted

in the composition side. And I think it’s a

fairly apt analogy. There’s tons of

mathematics in music. There’s tons of

mathematics in origami. But there’s also this level

of artistic complexity, which we’ll see later in this lecture. I’m going to concentrate mostly

on representational origami. Representational

origami is traditionally representing living

things in our world, but it’s pretty much, you

see something not necessarily living, but you see

something and you want to make that form

for the form’s sake, to represent that form. And this is different

than, say, patterning to create artistic patterns

on a sheet of paper, tessellating, making

geometric polyhedra, or making more abstract art

that doesn’t necessarily have a relation to

a real world object. So I’m going to start with a

little bit about origami art. We’ve heard this. Eric mentioned this particular

individual, Akira Yoshizawa. He is widely understood to be

the father of modern origami. He was born in 1911. He was around for

a very long time. Unfortunately,

passed away in 2005. I was lucky enough to get

to meet Akira Yoshizawa when he attended a convention

in North Carolina when I was maybe around 10. He was very powerful

and influential in the world of origami, because

he was one of the first people to start creating new models,

be able to look at an object and create that object

just from folding. He was one of the first

people to actually try to make a large

number of new models, as opposed to the

past many centuries when only a few traditional

models were known or pursued. This is a picture of

Yoshizawa right here. He’s fairly happy

in this picture. But he’s holding the logo

of the US organization in origami, OrigamiUSA,

which is this sailboat. But as you can see, different

than the traditional origami crane or the frog or

things like that, you see a lot of curves in his

work, a lot of shaping. He uses a technique called

wet-folding, in which he weakens the paper

to some degree, weakens the paper fibers

by applying water, shaping the paper, and letting it dry

so that it holds that form. And you can see in this

sparrow– this is particularly one of my favorite

works by Yoshizawa– it really has the essence

of this little bird, but is actually very

simple and elegant. Origami design isn’t

all about making the most complex

thing in the world. It’s really trying to

represent a subject elegantly. And I think this model does

a very good job with that. But you can see here, very

clean surfaces, not a lot of extra creases

that you can see. Traditionally, wet-folding

uses thicker paper and is slightly

more substantial. So here are some

of his other works. And I want to start

out with Yoshizawa because he was represented

as the father and the master, and many, many of the

origami designers, if not all of the origami

designers that I’m going to continue to

talk about were heavily influenced by Yoshizawa. So I’m going to first talk

about the traditional style. And I’m going to compare it

to, say, the crane or the frog. These types of models are

characterized by straight, well-defined, polygons

in the final form, typically folded flat. Little shaping is

traditionally needed to go from the base of the

model to the final form. It’s very geometric,

these models, characterized by very

straight, precise creases. So here is, I think,

a very good example of this traditional style. While there are

some curves here, everything is very

well-defined, maybe just a slight shaping here. But even that is

fairly well-defined. But you can see Komatsu, Hideo

Komatsu, a Japanese folder, uses really clean, large

polygons of open paper without creases on them

to represent polygons on the model. The folded form. His design process isn’t

really using tree theory. I mean, all origami design

is subjected to the condition that no two points on

the unfolded square can increase in distance

in the folded form. That’s a property called

developability of the paper. The paper’s not going

to stretch, basically. So all origami design is

subject to that condition, but you don’t have to deal with

necessarily these things called uniaxial bases. Pretty much all of these

models are non-uniaxial. His design process is

kind of a trial and error process of folding along

different 22.5 degree grids. 22.5 degrees is, I

guess, 1/8 of pi. 1/16 of 360 degrees. And it’s a particularly nice

and useful discretization of angles in origami design. All the traditional bases are

based on this 22.5 degree grid system. And there’s a certain

elegance of that. Actually I think, the mouse

is based on a 30 degree grid system, but is kind

of an exception, but follows the same principles. He keeps folding

a piece of paper and tries to get these

geometric shapes that really are able to by themselves

capture the model. And I’m going to use some

of that design technique later in a design example. He has a small but

very distinguished repertoire because his process

is less algorithmic– I mean, he has algorithms, I’m sure,

that are difficult to describe, but his process is

actually very artistic. And while it’s

very exact, I think it’s one of the most elegant

examples of origami design. Here’s another example

of the traditional style. As you can see, there’s slightly

more curves and things in it, but it’s fairly

well characterized by these straight creases. Heavier paper for wet-folding. This model on the left

here is box pleated. So as opposed to the

22.5 degrees structure, box pleating is characterized

by only multiples of 45 degrees. So pi over 4. And so you see the

grid here, this model is based on a fairly

large grid, so you can get the detail

that it needs. These are not

uniaxial bases, again, but they’re still limited

by this stretch ability constraint. And those were

styles that stemmed from the traditional

crisp folding of say the crane and the crab

and the frog and all these traditional designs. The non-traditional

style is more an extension of Yoshizawa’s work

and shaping and curved folding and things like wet-folding. There is much shape

that needs to be done to create the

essence of the model. The model is encapsulated by

not necessarily the structure as much, but of the final

shaping, the undefined shaping that you kind of

put into the model. Here’s an example

of an English folder named David Brill, who

is an investment banker, if my memory serves me. He now lives on a golf course. And I think he’s retired now,

but he likes to fold paper. But you can see

here, a good example of this style, thick paper. The character of

the model is really defined by these curved

tension folds, which is slightly different than at

least the traditional style. And oftentimes, it’s

very, very difficult to replicate to any of

these types of models, because it has so much to do

with subjectivity as opposed to objectivity, as in

the traditional style. Here’s another good

example, Michael LaFosse. He’s a paper folder who actually

resides here in Massachusetts. He’s in Haverhill,

“Have-er-ill,” something like

that, Massachusetts. He is unique in origami

designers in the fact that he is also an

avid paper maker. So he actually makes a lot

of the media which he folds. And that gives this

intimate relationship between the life

cycle of the paper. He’s able to make

specialty paper that’s really necessary to make some

of the most complex works out there. He’s gone to culinary school. He was a chef for a while. And he was also a marine

biologist for a while. So these origami

artists have come from many different

walks of life. This next folder, Eric

Joisel, he’s a Frenchman, lives in Paris. He was a former clay sculptor. And actually, I think you can

really see that in his work, the kind of solidness

and really cohesiveness of his composition. All the detail and

texturing are very well thought out in terms of the

subject as a complete piece. Heavily influenced by Yoshizawa. A lot of this use of texture,

incorporating texture into his models, he was a

big pioneer in that area. This texturing is fairly

obviously non-uniaxial. He doesn’t go

through a tree method and represent each

one of these points as a stick in a stick figure. These flaps don’t

lie along an axis. They don’t hinge

perpendicular to that axis. Yet he’s able to create

these amazing forms in paper. He has stopped

doing clay sculpture and does origami full time now. He’s very well known for his

depiction of the human form. This is taken from a

collection of masks. He’s done numerous,

numerous masks that are really very expressive. He was one of the first people

to really, for me at least, evoke emotion and convey

emotion in his work. But you can see here, the

structural crease pattern for this face is actually

very, very simple. It’s kind of represented

by a few pleats. But the amount of work used to

transform that very simple form into this very expressive,

curved work of art is kind of astonishing. Here’s a more recent work

of the entire human form. You see how this is starting

to come as some sort of blend between the traditional

and nontraditional forms. It lies somewhere

along the spectrum. But it’s a very

complex model, so it had needs to have this

structural complexity. But at the same time, he

shapes it to an extent that very few people can do. I’m paraphrasing a

quote of his, but he’s of the opinion that

if you can reproduce exactly a piece of origami

then it’s not really art, because you’re not putting

anything more into the model, if it doesn’t have something

unique and original and something that can’t

be reproduced in the model. Here’s two fantastic subjects

in terms of art, to me. Very Escher-like and

it’s self-referencing. This is called

the Self-made Man. And I forget the

title of this work, but he’s basically

emerging from the paper. You see that his arm and leg

are not actually finished. I think this is called

Birth, actually. But really, using paper to

express an artistic idea, very few people get to

that stage of competency with the technical and

being able to infuse that emotion into the subject. So Eric Joisel is a pioneer

in that realm of origami art. Here are three

very, very complex– These are very recent

works, probably within the last year or two. Lots of use of texturing

to make the armor here. Lots of planning,

tree theory included. These are mostly

box pleated models, but you can’t really

tell from here because of his impeccable

ability to shape a model. I’m going to remind you

guys that everything I’m showing to you is a

representational work. Each one of these is made

from a single uncut square. Pretty much, I

believe everything I’m going to show you

today has that property. AUDIENCE: When people fold

these, do they fold them by hand or do they

need special tools? To me, this looks like

it would be completely impossible to just

fold it by hand. JASON KU: These are

actually fairly large works. Each one stands maybe

about that tall. So the paper’s very

large to begin. But yeah, I believe

he just uses his hands and this wet-folding

technique to allow things to be held in place. I mean, many people, including

Eric Joisel, use clips and braces and things like that

to hold certain things in place while he’s working on

other areas of the model, but it’s pretty much by hand. Some people use tweezers

or things like that. But most of it is by hand. AUDIENCE: Does he add color

or shade or those things? JASON KU: Sometimes. For example, that mask I

think was speckled with paint after, before or after. There are different opinions

on this idea of origami purity. I like Robert Lang’s

definition of origami, that it’s any work

whose primary structure is defined by folding. And that’s a very broad

definition of origami. But I think it

works really well. So if the subject matter is

still heavily characterized by the folding and

not some other thing that you do to

the model, I think most people are OK with

that, as long as you’re not trying to pass it off

as something it’s not. There are many origami designers

to do multi-sheet things and do very complex works and

very beautiful pieces of art. I think Joseph Wu is a

great example of this, who I don’t have

pictures of his work. But he doesn’t try to pass them

off a single sheet origami. He is a very skilled designer. He could do it with

a single sheet, but he finds that the

solution is more elegant using multiple sheets. Any other questions? Just one more picture of

some of Joisel’s work. He actually made an entire

orchestra of these little guys. This is two sheets. The saxophone is

a different sheet. But again, he’s not

trying to pass them off as being the same sheet here,

whereas in here, the weapons actually are from the

same sheet of paper. And these multi-subject

pieces, each one of these, it’s not all three of them

together as one sheet. Just to clarify. But these multi-subject,

trying to represents clothes, and weapons,

and the human, and all these types of things,

is becoming more and more a way to a push the limits

of origami design. So again, trying to breathe life

into the paper is really what Yoshizawa’s mantra was,

and so is Eric Joisel’s. All right. So I’m going to move on to this

independent concept of really the ability that

we have right now to pretty much– We have the

algorithms to make anything we want and really

trying to capture that is this idea that I

call this modern realism. The style, like

Eric Joisel’s work, kind of follow

along the spectrum of this rigid structure

and this free-form shaping, but really try to capture

this realism of the subject. So I think Robert Lang is

one of the foremost origami designers in this kind of area. He’s a guy from

California who is a pioneer of algorithmic

origami design. You’ve heard his name

a number of times. He has kind of

codified tree theory, if not one of the

pioneers of establishing that research himself. He wrote the program TreeMaker

that you guys are all probably using to do your homework. He was at Caltech Ph.D., and

was a laser physicist for NASA. And he decided maybe

less than 10 years ago to quit and do

origami full time. So that’s what he does now. So here’s a number of his works. Very complex, very exact. For example, he

was a huge pioneer of what we call the Bug Wars. When we had these

tools at our disposal to make very complex

trees, we can represent very, very

complex subjects. And that led to this Bug

Wars of trying to one-up each other on how many legs you

could make or things like that. So here’s a centipede,

for example, with lots and lots of legs. And the exactness to which

we can specify the tree is phenomenal. For example, the

scorpion here is a design that Robert Lang

has approached– a subject he’s approached–

many, many, many times. This is a design that

I particularly like. It’s very clean in its folded

and its structural forms. But he actually used

TreeMaker and designed each of these pairs

of legs to actually be increasing in

length as they go back. So really being very

exact with the proportions of the model, the

proportions in the tree. And tree theory

really allows you to do that, to capture that. Here’s a slide for the

mathematicians in here. This is not one

square sheet of paper. This is probably the only

model here that isn’t. But it’s what we call modular

origami, making a single unit and sticking them all together

in a very complex and elegant way. Here’s a representation of

some of the tessellation work that Robert Lang

has been working on. This is a vase form. And all these, or at

least these three, were very much characterized

by using mathematics to find these forms. And while they’re very

heavily rooted in mathematics. Mathematics, as I’m sure

all of us can appreciate, is an elegant subject

in and of itself. There are elegant

solutions to problems. And in origami it’s

particularly nice, because these elegant

solutions often are very elegant and

pleasing to the eye, as well. So this is also a Klein bottle. It’s kind of a joke. But it topologically does

intersect and things like that. So in an interesting work. I want to move on to a

guy named Brian Chan, who is an alumni of MIT. He got his bachelor’s, his

master’s, and his Ph.D. at MIT. He defended his Ph.D.

in 2009, but he’s still around Cambridge. He is a big pioneer of pushing

the limits of complex folding. He’s picked up origami

design very quickly. And so it is possible to do. So I encourage all

of you to try it. Here is an example of a very,

very complex centipede that he designed kind of in

response to Robert Lang’s. There’s a huge history

of really trying to one-up each other in origami. And it really helps

spur the creativity. And playful competition is

very useful to any subject. These multi-subject

things, like this rose, the stem and

the petals itself, all from one square

sheet of paper. He uses color change. One side of the paper is red;

one side of the paper is green. There have been tons of

people that design just the rose part of the rose, and then

they make an additional stem and stick it on. This is the first

one-piece model of that. And was somewhat

influential in that respect. Here’s a very complex,

textured character from an anime TV show. I forget which one it’s called. AUDIENCE: Rozen Maiden. JASON KU: Rozen

Maiden Thank you. That is correct. But you really can see his

use of color change here. Again, being able to make

this cross in the fabric here. The zigzags of lace, and

this texturing of the dress, very, very complex, in its form. But these are all

actually uniaxial bases, all come from this

idea of tree theory, being able to map

things on your subject to the sheet of paper

in an algorithmic way. Here’s a very complex, another

anime work, a Neko Bus. Neko is, I believe,

Japanese for cat. And it’s very, very complex. Again, similar to the centipede. Lots and lots of points. But this tree is actually

kind of represented by– There’s a head

region and there’s many points sticking

out on both sides. And then this flap kind of

comes over and attaches up here, and you’ve got the tail. Here’s another example

of a multi-subject model. Every year there’s

a design challenge in New York for origami. And this was the

sailing ship category. And he kind of went

another direction with it. He did make a sailing

ship, but this is a kraken attacking the ship. He’s got a little person

in one of his tentacles. Part of the ship

and the ship itself, and it’s all one square sheet

of paper without cutting. And if that wasn’t enough,

then the MIT seal, as well. One square sheet of

paper without cutting. The mens and manus,

so the mind and hand. And I believe this isn’t

traditionally a crane, but yeah. The last person I

want to touch on is a guy named

Satoshi Kamiya, who’s represented as probably

the foremost pioneer of super-complex origami. He is a little further

on the spectrum on the traditional

style than many of these other

super-complex folders, characterized by kind of

very exact, straight creases, this texturing for example, a

unique balance between making a very cleanly folded–

The Japanese traditionally make very clean subjects in

terms of exactness and form. Here’s a little more

shaping in the wet-folding. But again, this is one

of my favorite works of his, another Lord

of the Rings character. What’s neat about this

sea turtle actually, the diagrams for it

were just published. I first all this work in

2001 or something like that. But it has these plates

on the back, this texture, but it also has plates on

the front of the model. So you can actually

pick it up, and it looks very, very convincing. Here are some more

models by him. Again, you can see a lot

of this texturing here in this wasp, very

clean folding. A dog, multi-headed

dog, a caribou with very complicated antler

patterns, and this dragon. And again, you see the

crisp, clean folding, but at the same time very

well-planned and well-designed 3D structure to be

shaped afterwards. Here’s another work that

I particularly enjoy. Really lending this texturing

he applies throughout the model, and it’s a very

cohesive subject, from an artistic sense. It’s very complete. There’s the same level

of detail everywhere on the model, which

is very useful. And I’m going to kind of

end this artistic side with a model which

is widely regarded as the most complex

single work in origami. This took Kamiya over the

course of a year to fold. There’s thousands of

scales on this guy. And again, it’s one square

sheet of paper without cutting. You see that it’s

a very long model. You’d think that

this subject would be much better represented

by a long rectangle or something like that. But actually it’s

very symmetric. This crease pattern,

which we’ll look at later, actually has an

asymmetric crease pattern and is quite ingenious

in how he decides to accomplish this

form and structure. If you’re interested in learning

more about the origami art side of things, there’s this

phenomenal document documentary which you can and

purchase online. Or I’ve believe OrigaMIT

has a copy of this, and we’ll probably be

screening it some time this semester or next. It’s called Between the Folds. And it features, among others,

both Erik and Marty Demaine, Robert Lang, and many more. And there’s a picture

of Stata from the film. Now we’re going to move on a

little bit to origami design. We’ve learned what

the algorithms are behind a lot of origami

design, but now we’re going to see how that applies

more directly to creating a representational work of art. If you’re really

serious about wanting to get into origami design, this

book, Origami Design Secrets by Robert Lang, is really

the first major book on the methods of

origami design. Most origami books are

traditionally about diagrams, trying to fold specific models. This is the first

book really to lay out some of the ground rules

of how you create models. And it goes through a number of

the things we’ve talked about. So just to review a little bit

about tree theory, the idea, the process is you start with

a subject, like this picture I took at a Japanese

museum of a little crab. You kind of draw a

little stick figure of what that crab might look

like in a one-dimensional form, characterized just

by the lengths of these flaps and

the connectedness. You go from here to here

to an origami base, which has all of those flaps

of the right length and connected in the right way. And then you shape it

into an origami model. Now this method, this step here

might seem hard to you guys. With a little experience,

it’s actually very reasonable to assume that someone

fairly well versed in the vocabulary

of origami will be able to accomplish that step. This step, again, this kind

of child’s play, somewhat. It’s actually not,

to do it really well, to represent this model

as a stick figure, and we’ll see that when we

try to go through an example. This step is the one where

algorithms and mathematics really help to do a

lot of the work for us, and essentially is kind of the

easy part from our perspective, because it’s kind of

methodical and there’s algorithms involved

to help us out. The most artistic

and free things we can do with

origami design are kind of this step in the

shaping and this step in defining the proportions. In this step really you define

what the abstraction you choose to characterize

in your model. Like here, we are

choosing to represent all four legs on either side. You don’t have too. But we also decided to model

the eyes and the claws as is. But an underbelly to a crab. We could have modeled

that with the texture. We could have modeled the

little mouth parts of the crab. There are many things we

could choose to model on here that we don’t choose to. So this is one level

of abstraction. And this comes with

a lot of choice. Here, there’s lots of algorithms

and math to help us out. But as we’ll see,

there is actually a lot of choice going

from here to here as well, artistic choice,

and from here to here, again, probably the most blatant

way that an artist can put his style in essence

into an origami work. AUDIENCE: What is

the extra fringe? JASON KU: Which one? This? AUDIENCE: Diagonal from the top. AUDIENCE: On the left. JASON KU: On the left. Oh, this? OK. So I modeled here

the body of this crab as a flap coming from here. I kind of wanted a flap

to cover the rest of this. And so that’s why I’ve added

this leg of the tree there. While branch edges– this

is a branch edge, it doesn’t terminate– will provide

paper in that region, as we’ll see later,

branch edges of the tree, rivers in the space

allocation, really don’t lend themselves to

being shaped very easily. And so if I isolate

that body segment as a leaf edge

for itself, then I can actually do

control a little more about how I’m able to shape it. Good question. So we’re going to

review a little bit about uniaxial bases. This these are the definitions

that Erik Demaine posed in the algorithm, I

think in lecture four. Again, you have

this uniaxial base. It has these

characteristics that it’s in the positive space above

the z equals zero plane. And that’s kind of

represented here. The intersection with that

plane is the projection. So if you shine

the light above it, it would cast a shadow of

a stick figure out, which is exactly kind of what we want. We want to make an origami

base that associates itself with a stick figure. And then we partition

the faces into flaps. So there’s all

these definitions. I think to put these in

kind of layman’s terms from an origami designer’s

point of view, what do these really mean? Really, the important

characteristics that we want are that the flaps lie along

or straddle a single line. Because if they do that, then

we could just fold it in half and it will have that

property of everything being above an axis and

everything lying along an axis, and that the flaps hinge

perpendicular to that axis. The reason why we need the

flaps to hinge perpendicular to the axis is if they don’t

hinge perpendicular to the axis then you will not be able

to create a projection to the plane that is a

one-dimensional stick figure. If these hinges are

tilted then that line will project to a line instead

of a point, like we’d want. We’d want it to project to

a single node on the tree. In any of these

uniaxial bases, think about the base being thinned

in the limiting case, that we can create folds parallel to

this axis and thin this model until it’s right along the axis. And then in that

limiting case it is a stick figure, essentially. And once it is a stick figure,

layering and orientation of the flaps really

don’t matter, because it is the stick figure. So this is kind of an

informal definition, but we’ll use these

later in the lecture. So what is a flap? We kind of made this argument

a couple lectures ago. So we want to model a flap, so

that we can kind of stick it together. And this is kind of

an intuitive sense of we take a sheet of paper,

we thin it a little bit, we hinge it perpendicular

to some axis. And when we do and

we unfold the paper, we see that it

takes up this kind of quarter octagon of paper. Now, if we continue

to thin this, if we make it

really, really thin, you see how deep the boundary,

this fold that we make, will little closer and

closer approximate a circle. Everyone see that? It’s kind of like an umbrella. I like this analogy

with an umbrella. That you have a

single point that’s the center the

umbrella, and when you close the umbrella,

all of the umbrella kind of maps to a single line. And so it’s neat to

see on the paper. It’s kind of what

you could think of as a projection to this tree. Lines on the unfolded

square, these lines at the edges of the circle,

map to a single point on this flap, or infinitely thin

flap, or essentially the tree. This is kind of a

leaf edge of our tree. And so everything along this

line maps to a single point, is compressed onto

a single point. You can do that with any

point going up this flap. We can actually pick

off a point here, and we see a line of constant

elevation with respect to this flap. And so now we’ve created

a very, very simple tree. Instead of one leaf

edge extending off of the rest of the model, we

have a branch edge, and then a leaf edge. And this branch edge is

corresponding to this strip of paper here of constant width. That’s what we call a river. And we see that a circle is

just a limiting case of a river. Rivers separates two parts of

the model off from each other by a constant distance. That’s what that constant

thickness strip of paper means. And the circle is really

just a limiting case off that river that separates

only a single point away from the rest of the model. That’s all I want

to say about that. And we can actually

tile these rivers onto a plane to create

arbitrary trees. So here’s an example

of the correspondence. I call these

circle/river packings. That’s the common term

in origami design. This is a circle/river packing. It’s kind of a space allocation. It’s an idealization. The model we make

is actually not going to be infinitely thin. So each flap is going to take up

more space than these circles. But it’s a good idealization. This circle/river packing

or this space allocation actually maps

uniquely to as a tree. So if we go through

it, this point, this circle here might map

to this line on the tree. It can actually also map to this

line, this edge or this edge, as well. Because I don’t really care

how these flaps are oriented. The tree is just

supposed to preserve length and connectedness. It doesn’t really have to do

with where they’re mapped. And so we’ll see some

examples of that later. But we can kind of

go through this tree and see all the

different aspects of it, how the edges correspond

to circles and rivers on the packing. And we’re going to do a

little bit of practice for that, because I think

that was one of the day hardest parts for me starting

out in origami design, was being able to be comfortable

with going from a tree to a space allocation, from

a space allocation to a tree. Getting that concept in my

head was kind of difficult. So how about we

practice a little bit. We have this space allocation

of maybe two circles, a river, and three more circles. I’ll just give you a second to

see which one of these trees is represented by

this space allocation. Or should I say

how many of these. Because some of these

trees might be equivalent. So we’re going to start with

the upper right one here. Does it correspond to

this space allocation? Yes or no? No. Why? AUDIENCE: [INAUDIBLE] JASON KU: Yeah. So the topology’s kind of wrong. You’ve got three equal

length flaps up here, which is what we want. We want three equal

length flaps separated off from the rest of the model by

a river of the same length. That makes sense. But instead of separating

off two flaps, two leaf edges of maybe

twice the length, it separates three of the same

length, which doesn’t quite work out. So the distances and

the connectivity’s kind of off here, just

terms of the numbers. So this one’s wrong. How about this one? Yes. Right? It has the right topology. This one? No, again. The wrong typology. There is, again, three separated

from two by a branch edge. But it doesn’t have

the right lengths associated with this

space allocation. And how about this one? Yes. Right? I’ve transformed this

tree from here to here. I just moved them around

with respect to each other. They’re equivalent, in terms

of how we choose our tree. And this will be important

when we actually use TreeMaker. Because it doesn’t matter how

we orient things in our tree, we can manipulate where we

put our circles on the paper to get the same tree. The mapping from here

to a tree is unique. Mapping from a tree

to a space allocation is not, which leads to

interesting design choices that you can make in

designing an origami model. Yeah, those two. One more time. We’ll go through this

one a little quicker. I’ll give you maybe

five seconds or so. So we’re going to start with

this one, the first one. Does that map to this

space allocation? AUDIENCE: [INAUDIBLE] JASON KU: Yes, it does. We have two equal length set off

from the same length to equal, and then one twice as big. And see how I’ve actually

added a redundant node here. I’ve split this leaf edge into

a branch edge and a leaf edge. That’s kind of a redundant

node that I don’t really need. It doesn’t really change

the topology of this at all. It would just map to

a line right here. Everyone see that? How about this one? No. Right? For a number of reasons

that I won’t go into. How about this one? No. Again, the distances and

the topology are wrong. This one? No. This is actually one of the

trees from the slide before. And this one? Yes. This is actually just a

manipulation of that tree. So yay! We’re awesome. Now, going the other way

is not necessarily unique. So there would be

multiple answers here. Is this a correct

representation of this tree? Yes or no? AUDIENCE: No. JASON KU: It has the

correct topology, right? It has three equal length

flaps separated off by a river from three equal length flaps. That’s what we have here. But this river is actually twice

as long as any of these flaps. So this is actually

a little bit shorter than the length of

any of these flaps, not really working for us there. How about this one? No, for pretty much

the same reason. It actually has a very

similar, if not identical, tree to this one. How about this one? No, topological problems there. This one? Yes. So it’s got three equal

length flaps separated off by a large 2x river, I guess. And this one? No, for the same reason here. And here we can actually

see three different packings of a very similar,

if not same, tree. And this goes to

show you that there could be many

different ways we could put these disks on a sheet of

paper that could either improve efficiency or be more useful. For example, this packing

has a central flap. We may or may not want

that central flap. You can see that a central

flap will use more paper than a flap at the corner

or the edge of the paper, because it has 360

degrees of paper that you have to fold

as opposed to 180 or even just 90

degrees of paper. So typically in

origami design, if you have a flap you need a little

bit of bulk in, a little more paper, you might

want to consider making that a central flap. If you want to make a very thin,

maybe an antenna or something, a corner flap might

be a better choice. So correct, in that sense. Now again, I want

to stress the fact that this is an idealization. These are circles. They don’t really account for

all the paper in the square. This paper between the

circles and the rivers is not really used. Pretty much everything in

this no man’s land here actually maps to a single

point in the tree right. This is a bad example, so

I’ll use this correspondence. This kind of looks like a

bikini or something like that. That’s neither here nor there. But this space all maps to

one of these branch nodes. Everyone see that? Because in the situation where

we thin this model infinitely, this is kind of extra

space that we kind of just don’t even deal with. In reality, that

extra space will have to go into either

the rivers or the circles in the packing. So the reason why uniaxial

bases are nice in this model is because since all they hinge

creases of the model, basically the boundary of the

flap with the model, hinge 90 degrees to some

axis, then its projection maps to a single point. So if we cut off all the

flaps along the hinge creases, we should actually get

a very similar mapping to what we have here. And here’s an example of

a fairly complex model. But you can see, I’ve

just highlighted the locus of possible hinge

creases on this model. There’s a unique way to do this. I won’t go into it. But there is a unique way

to add these hinge creases. But as you can see, the

idea is very similar. But instead of having these

curves of constant width, you have these discrete angular

curves of constant width. So for example here, you have

a river of constant width that changes directions

at a discrete corner, but it’s still a strip

of constant width. Everyone see that? So maybe we could

go ahead and see– If we had this crease

pattern and we didn’t know what the model

was, we could actually pick off the tree and figure

out what this model is. So maybe we start with

these two points down here. They’re all points separated

off the rest of the model by a certain distance. And that distance here, all

these lines that are connected must be at the same location,

the same node on the tree, because all those hinge creases

must map to a single point. So these two flaps

connect with each other because they share this

set of hinge creases. And so that’s that

point right there. I’m going to ignore these two

flaps at the bottom for now. We have this big long river. I’m just going to deal

with the big points first. And that connects to

two more big points. Everyone see that? I don’t want to go too fast. And actually, you can do that. You can just keep doing that,

and methodically picking off distances on this hinge

crease representation, this is discrete

space allocation and fill in the whole tree. Anyone can think of

what this might be? Maybe a four-legged animal with

antlers, like maybe a moose. So this is a model

I designed, I think, my freshman year

as an undergrad. AUDIENCE: What

happens when you have the squares inside the squares? JASON KU: That’s an

excellent question. First, I want to answer

one other question before I get to that one. Here, these polygons,

these squares, you could think of maybe

putting a circle in them. And the square is taking up

more paper than the circle, and so the flap

that this represents would be the largest circle

that would be fully contained in that square. And that would be the

length of that flap. What does it mean to have

instead of this point separated off from the rest of the model

have a line separated off from the rest of the model? Can anyone guess

why would you want that as an origami designer? Well, you might

want that property if you want not just

a point separated off from the rest of

the model, but a line. You might want thickness. It’s a qualification

of thickness of that flap at the extreme

distance away from the model. And I don’t have

this example with me. I talked about it yesterday

at the OrigaMIT lecture. But let’s say you wanted

to model a butterfly wing. It’s not well

characterized by a stick. Its thickness is

kind of important. So how I designed

a butterfly wing is I separated a line

off from the rest of the model, something

similar to this, so that I would have enough

paper to kind of spread out that idealized single point. I could spread the end of that

point to have some thickness and to make a full

butterfly wing. And what you’re

saying is what does it mean to have these points,

these single leaf edges, separated off kind of

surrounded by river? You see what that means? Yep, this is just a river. Rivers, again, don’t have to

go all the way across a model. They can also connect. You’re separating

these two points off from the rest of the model by

a certain constant distance. So excellent question. And as I promised before, I

want to take look a little bit about the structure

of this model. It looks very symmetric, right? And you’d think that maybe

it would be well represented by a rectangle of paper

instead of a square. How do you fit

this into a square by still having this detail? How do you think this

texture was made? Anybody? It’s kind of just pleating

the paper back and forth. If you’ve ever taken

a sheet of paper and pleated it to

form a texture, kind of a

one-dimensional problem, but you’re pleating it. But after you pleat

it, it’s smaller. If we take a look at the

crease pattern here– This is actually

a crease pattern to an earlier version

of this model. This is slightly

less detailed, if you can imagine, than

this model right here. What do you think this is? Maybe the scales, right? This is the head region. We can actually do

a rough version, perform a rough version of

this kind of hinge crease representation, and get

an idea for the structure of this model. So here, we see the tail. I’ll talk about this later. We have the two back

feet separated off from the rest of the

model by a distance. That’s this distance here. Two more feet. This is kind of the neck region. And here’s the head. OK. So this looks kind of weird. I haven’t really been specific

about the details here. But what does that pleating do? Well, it shrinks the

useful area of the paper, because I pleated it. So that’s why here the

length of this flap is this distance here. That’s the length of the tail. But when I make pleats,

this thing shrinks. And it actually shrinks

to this distance. This whole thing is cut in half. So we make these

pleats, it shrinks, and then it can lie

along this segment. Then this area here

also shrinks by half. So the length is here. And it is able to cover

this aspect, this part of this middle

river with texture. Please ask questions,

because this is complex. AUDIENCE: [? What’s the ?]

distance between the front and back legs? JASON KU: Yes. So this is the distance between

the front and back legs. But we have to cover

it with texture. There’s no texture here. So what we do is create

this extra flap here where the back legs are with

a length of half of this, and cover it with texture. So that’s what he’s done here. And so the same goes for here. It’s not quite half down

here, but this covers up the rest of that section. And there’s actually

some overlap so that they can mesh correctly. Then here, we have enough paper

to provide texture to the neck region, and then

there’s the head. It’s kind of an ingenious way

of actually the top and bottom, this top texture and

this bottom texture, folding up onto this line

segment which represents the length of the

dragon, and still having space for

these toes and feet. It’s an ingenious way

to distribute the paper, in this case. Here we can understand

another reason why we might want to

separate a line off from the rest of the model,

because then that line has some thickness, you have

a certain amount of space out there, and you

can actually then create more points

from that line being out at a certain distance. We can create a number of little

points, which are then toes. So I thought that

was pretty cool. One of my favorite

examples of structure. AUDIENCE: [INAUDIBLE] JASON KU: These were

all drawn by hand using a program very similar

to Adobe Illustrator. So yeah, it’s very tedious, and

lots of copying and pasting. But you should see the

more complicated version of this pattern. Because as you can see

on this model here, there are actually scales

on the feet part itself. These claws actually are longer

in proportion to everything else in the model. So we actually add

some more things. There’s also a

strip of paper here that has spines on the back. This crease pattern doesn’t

represent those things. So this is a simplified

version, if you will. AUDIENCE: What was the

starting size of the paper? AUDIENCE: Yeah, how big is it? JASON KU: It’s actually

an amazingly efficient use of paper. The length of the

dragon is pretty much this length right

here, which is actually quite impressive for the

amount of detail there is. The shrinkage

factor is something like to the length

of the squared to the length of the

dragon is not even a half. The overall structure of this

model is actually quite simple. The model itself is

maybe about this big. So I’m guessing the

size of the square was something like a meter,

if not a little larger. It’s a long time to work

with a single sheet of paper. All right. So we’re going to very quickly,

maybe for the next 10, 15 minutes, go through a

design example of a crab. And so we’re going to kind

of go through it quickly. To help you do your

homework, I just want to let you know about

some details of TreeMaker that might be useful to

you to be able to make a cleaner or nicer

crease pattern. So to go to a TreeMaker example,

I’m going to open up TreeMaker. I need to bring

TreeMaker over here. So we have TreeMaker. And let’s say we

want to make a crab. So how do you want

to draw this tree? Maybe I’ll just draw the

tree that we had before. First we have four legs

all of equal length. We could have them all

coming from the same spot. But traditionally, if we

take a look at a crab– That’s a cartoony

version of a crab, but we see that these maybe

our axis of our model is here. These legs actually don’t

need to split at the axis. We could actually model

this as in the tree, maybe we have our body

segment, and maybe we separate these four

flaps off from the axis by a certain distance so that

we actually can save paper. We don’t have to make each

one of these flaps this long. You see? So I’m going to add a

little line segment there. Repeat on the other side. You get the idea. Then maybe you

have some modeling of the thickness of the model. Then we have claws. One nice thing about this is

we could view just the tree. That might make things

a little easier. There’s lots of these

view characteristics that we’re going to

take advantage of. And maybe we want to

represent the eyes. Now, the lengths of these

edges in the program don’t really mean anything. So take that into note first. You actually I’ve to

click on each edge and specify its length

relative to all the others. So maybe we want

to make the claws half as long as the

branch connecting them. Bear with me. There’s no good way of

automating this process at this point. And maybe we make

the eyes– they’re pretty short– so

we maybe make them a quarter of the

length of those. The body segment, I don’t know. Also a quarter. This is really

kind of arbitrary, but you can play around

with these dimensions. And these guys, also a quarter. And the back legs

can also be one. Something like that. All right. When we’ve got that, we see that

we actually have circles there. Now, these circles

are kind of crossing. We don’t want

that, because paper can’t go to two points at once. What we can do now

is scale everything. So it tries to pack

all the circles such that none of the conditions

are being violated. So this is a valid packing,

except these points in the middle here, this

whole polygon is constrained. The green line segments

here are active paths. Basically, the distance between

these points on the tree and these points on the

paper are minimized, or they’re equal. So there must be a crease there. That is a key statement

of uniaxial bases, is that there must be a

crease along active paths. Now, these two points can

actually stand to get larger. That’s evident to the fact

that we can move these around and it’s not violating

any conditions. Well, if I move it over here,

it’s violating a condition. Whenever a condition

is violated then you have these red

lines that yell at you. But we can move this

around in this area without violating

any conditions. So it’s not happy. It’s not completely crystallized

or well-constrained, so it’s going to

yell at us when we try to build the crease pattern. TreeMaker was not able

to construct all polygons because a polygon was either

non-convex or contained one or more nodes

in its interiors. So these have nodes

in its interior, so was not able to

fill in this polygon. What we can do about

that is we don’t mind if these points

get a little bigger. Or we could add an extra point. So we never modeled

a body segment here. So maybe we just add

in a body segment. So scale everything here. We still have this problem. This guy is unconstrained. So what I’m going to do is

make this guy a little bigger by selecting the

node and the edge. You have to do both. I can go here and scale

just this selection. And it’ll increase it by itself. Actually, nicely, this is

somewhat of a symmetric crease pattern, which

didn’t occur before. So you see these lighter

edges of the tree are fully constrained edges. These darker ones are not

fully constrained edges. So this guy can actually

also increase a little bit. So I’m going to scale selection. Now everything should be good. I can build the crease pattern. Guh! It built it, so whatever. So this is a foldable

crease pattern that will form

what we want it to. We can also go to

this creases view, and it will show the

creases of the model. It was not able to find

valid mountain-valleys. Anyway. So to make this

cleaner, you might want to deal with symmetry. So there’s an ability to

select diagonal symmetry and either add conditions

to make a node fixed to the symmetry line, so

add additional constraints to our system to

make it cleaner. We can fix them to

the symmetry line. We can fix it to the

corner or the paper edge, fix to any arbitrary position. Or we can select two

nodes and pair them about the symmetry line, which

is a very useful thing to do. I don’t know if it

will yell at me again. Yeah, it didn’t do anything. What I’m going to

do is go in here. There’s lots of things

you could do here. We could perturb all

the nodes, so they move if by a slight distance. And maybe if we try

scaling it again, it’ll find a valid solution. This was unfortunate. Scale everything. Kill the strain on this. Yes? AUDIENCE: I’m confused. Is it failing

because the problem is over-constrained

or under-constrained? JASON KU: It’s not

failing because of either. It’s failing because

certain creases get very close together. Now it’s failing

because it can’t find a correct

valley-mountain assignment for the crease pattern. So it’s able to build

the creases fine. So build crease pattern, fine. It just wasn’t able to construct

a mountain-valley assignment, which in the creases view

would usually give you mountain and valley assignments. AUDIENCE: That means

it’s not possible? JASON KU: It couldn’t find it. It’s not that it’s not possible. It just couldn’t find it. I want to say one other thing. Kill the crease pattern. We have a polygon bounded

by active paths that’s not triangular. But we can actually split it

any of these up into triangles. I’m just going to mention

that you can do this. You click on one

of these polygons, go here, stub, triangulate tree. It adds random points. And now all the

polygons are triangles, and that’s much easier to fold. Or not. That’s how you would do it. Anyway, I’m running

out of time, so I’m going to go back to

the presentation. But if you need any

help with these, or the tutorial with

this program, I’m around. You can contact me through

the OrigaMIT website, or you can come to it an

OrigaMIT workshop on Sunday and ask me questions then. So I’m going to quickly just go

right back to the presentation. Play slide show. Nope. Technical difficulties. Play slide show. So that was the

example of TreeMaker. Here’s an example of a

non-TreeMaker example that I designed this

weekend of a crab. I actually designed

this model it after I had drawn this picture. And I wanted to incorporate some

of the elements of this picture into my design process. So one of the first

I actually started was designing the

back so that it would have this

kind of structure with this polygon there, kind of

a Komatsu-like design process, and trying to make the

final form polygons and incorporate those

into my crease pattern. So that’s what this

area is right here. It has a very similar structure

to the tree we drew already. We have the four points for

the legs, the body segment. These are going to be the eyes. And so there are some

extra points just to make things easier to fold. The claws. Here’s the body segment. There’s these extra two

things on either side. And I made those

so that there could be an underbelly to

the crab, and that I could add some texture

in and things like that. But you can see some

of the constraints that I’ve put on it are that

I want this to be 22.5 degree folding, which is hard to

implement in TreeMaker. I’ve also shown

some of the thinning to make these points

thinner on the right. So if we actually

pick out the tree for this hinge

representation, we get something that kind

of looks like this. So we have our legs. Here’s our body segment. Here are the flaps that I make

into the underbelly, eyes, and the claws. And here’s the folded

proof of concept version that I folded last night. It’s actually a

really crappy picture. I apologize. But it’s up here. I fold it like 10:00 last

night, because I thought it would be useful for you guys

to see what a folded one might look like. But it actually turns out

to be somewhat non-uniaxial. And you can see some of the

texturing on the underbelly, if you take a look at

it and come up here. So that’s kind of describing

some of the design process of a real work that uses

the concepts of uniaxial bases, that I can make this hinge

crease representation, but then use some

shaping to modify it. If you’re interested in

learning about anything related to origami, there’s an

excellent online forum that you can ask questions

or show off work that you do or anything like that. And if you want to do

something slightly more local– this is shameless

self-promotion– but the origami club at MIT

welcomes you with open arms. We meet every Sunday

in the Student Center from 2:00 to 4:00 PM. You can find all sorts of

details on our website. So that’s about it. AUDIENCE: Are those

origami letters? JASON KU: Those are, yes. Each one of these

letters was a model. They’re all the same model I

designed, in which you have a 3 by 4 grid of flippable

squares of color change that you can flip to either

be in the all white state or all black state. And you could also do some

of these half pixeling. But you can basically make

any of these letters– I have a whole alphabet of

things– from a single model. I was lazy. I didn’t want to

design 26 models. I just want to design one model. So that that’s what those are. So that concludes the lecture. We’re just about at time.

Rozen maiden ð

TOTORO ð

SATOSHI KAMIYA, THIS LECTURE HAS FRICKIN EVERYTHING I NEED IN MY LIFE ð

read the book, much easier and more comprehensive!

The Tree Maker part dude, you never prepare your materials? I mean you can do the moose when you were just a Freshman however now as a Phd you can do shit!

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