**Origami Numbers**

**3 Quarks Daily**by Jonathan Kujawa

*Invicta by Robert Lang*

Last time at 3QD we talked about constructible numbers. We
followed in the footsteps of the ancient Greeks and asked what sorts of numbers
we can get as the lengths of line segments made using only a compass and
straightedge. In the end we found that starting with a line segment of length
one, a compass and straightedge, and enough patience, we can make any rational
number, the square root of any rational number, and, indeed, any number which
can be made by a finite sequences of additions, subtractions, multiplications,
divisions, and square roots. Even crazy numbers like √(13/2 +14.3(√7)) are
constructible. In short, compass and straightedge constructions are equivalent
to being able to solve arbitrary quadratic equations.

On the other hand, that's it. Logarithms, exponentials, π,
and even the roots of higher powers are all impossible. No matter how hard we
try, the cubed root of 2 is not constructible. The 2000-year-old challenge to
double the cube is forever out of reach.

*Drawing an ellipse via two tacks a loop and a pen. (from Wikipedia).*

But, of course, this all depends on our initial decision to
only allow a compass and straightedge. If you also gave me two pins and a
string I could use them to make an ellipse. "So what?", you say.
After all, what does the ability to draw an ellipse buy you? Well, it has been
known for centuries that you can double the cube and trisect the angle if you
are allowed to use parabolas and hyperbolas. In 1997 Carlos Videla determined
exactly which numbers are constructible using a straightedge and the conics
(circles, ellipses, hyperbolas, and parabolas). In short, the addition of the
conics allows you to take cube roots. No more, no less. Remarkably, in 2003
Patrick Hummel proved that the hyperbolas and parabolas are redundant. Every
number you can construct using a straightedge and the conics can be constructed
with just the compass, straightedge, and ellipses. Give me an ellipse and I'll solve your cubic
equation!

Of course, this is all very Euro-centric. We use the
compass, straightedge, and conics because we are in the Western tradition and
that's what the Greeks used.

But what about those of us who grew up in Japan? There the
ancient geometric tradition is origami. We could just as well ask which numbers
are constructible using paper folding. We can make a straight line in the form
of the crease made when we fold the paper. If two creases intersect, this makes
a point. If we've made two points, we can make the straight line which connects
them by folding a crease through the points. Since we can make lines and
points, we can say a number is origami-constructible if we can start with a one
by one sheet of paper and use paper folding to make a line segment of that
length; much like we did for the compass and straightedge.

Which numbers are origami-constructible?

Surprisingly, despite the centuries of effort in
understanding the mathematics of Greek geometry, the mathematics of origami has
only really taken off in the past few decades. Nevertheless, lots of progress
has been made!

It seems pretty obvious paper folding is more limiting than
the compass and straightedge. After all, with paper folding all you can ever
make is a straight line. Not a circle to be seen. Indeed, in the mid-1990's
Dave Auckly and John Cleveland wrote a paper in which they identified four
geometric operations you can do with paper folding (folding a crease to connect
two dots, folding the paper so that one crease lies exactly on top of another
crease, etc.) and they described exactly which numbers you can get using their
operations. In particular, while you can get square roots of the form √(1+a2),
you can't get arbitrary square roots like you can with a compass and
straightedge. That is, there are definitely numbers which are compass/straightedge-constructible
but not origami-constructible.

Or so it seemed! Auckly and Cleveland overlooked some of the
possible paper folding operations [1]. Unknown to them, in 1989 in the
Proceedings of the First International Meeting of Origami Science and Technology,
Huzita and Justin separately introduced an expanded list of paper folding
operations. Most crucially, Auckly and Cleveland missed the now famous sixth
operation:

*The magical sixth move!*

If you already have two lines and two points, in origami you
can fold the paper so as to put the two points on top of the two lines. It
seems like no big deal, but we will see there is magic in this move!

The good news is that about ten years ago Robert Lang and
Alperin and Lang proved Justin's list of seven paper folding operations truly
do exhaust the possible moves you can make with origami [2]. There is no worry
about another oversight.

The even better news is that the magical sixth operation is
all you need! With the magical sixth operation alone, you can make any number
constructible by the full set of seven operations [3]. It amazes me you only need this one operation.

But what of the question of which numbers are
origami-constructible? Auckly-Cleveland showed that their four operations
definitely do less than the compass and straightedge, but what happens when we
add the remaining paper folding operations? It turns out that including the
magical sixth move is equivalent to being able to solve arbitrary cubic
equations! With origami we can double the cube, trisect any angle, and solve
any cubic equation! Despite the fact that origami seems more limited than a
compass and straightedge, it's actually far more powerful! For example, forming
the cube root of 2 is impossible with compass and straightedge but only takes a
couple of folds and is downright trivial with origami.

The history of the sixth operation has a surprising twist to
it. In 1936, Margharita Beloch, an Italian mathematician, showed how to use the
magical sixth move (usually called the Beloch fold in her honor) to solve an
arbitrary cubic equation. In the intervening decades her work was nearly
forgotten. For decades nobody paid much attention to the mathematics of paper
folding. It is only recently that it and her work has been properly
appreciated. She was half a century ahead of her time! I recommend the survey
article by Thomas Hull if you'd like to learn about Beloch's method.

The recent development of origami mathematics has caused a
revolution in paper folding. Robert Lang, Meguro Toshiyuki, and others use
mathematics and computers to build designs undreamt of by past origami masters.
Here is a single sheet of paper, properly folded:

*Chrysina Beetle by Robert Lang*

Recent work by Erik Demaine and Tomohiro Tachi shows that
any three-dimensional shape made from polygons can be folded from a
sufficiently large sheet of paper. They even gave an algorithm and software for
doing this.

Why all the interest in the mathematics of origami? Well, of
course, it's worth studying just for the fun of it. But in recent years origami
has also become super useful. Being able to predictably and efficiently fold
and unfold is a valuable skill in the modern world. It is used in everything
from the design of airbags and solar panels on satellites to advances in
medicine. Knowing what you can and can't do by folding is a key first step.
This spring Nova showed a documentary on the mathematics of origami and all the
cool things which we now know how to do with folding. I highly recommend it.
You can watch the Nova episode here.

[1] You can find the full list of operations here.

[2] Huzita's list had only six operations. But since both
lists give you the same collection of origami-constructible numbers people
usually don't make much of a distinction between the two. Also, Hatori
independently discovered the same list of seven operations.

[3] This is discussed by Hatori here.

~~~~~~~~~~~~~~~~~~~~~*People who know me will be astonished to find this article here. The mere mention of algebra is enough to send me into anaphylaxis.*

*But hey, math nerds are people too, and I can see how this would be serious catnip to the numerically inclined.*

For more Robert J Lang Origami, go

**HERE**

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