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.
