## Origami part I

A paper lion by Joseph Wu, photographed by Sigurd Kranendonk

In the randform post about tesselations it was already mentioned that there are not so many mathematical things which can be visualized easily or visualized at all. However next to the tesselations there is a very popular area of mathematical visualization which is Origami. Origami is an active field of mathematical research as well as of arts.

Origami can be linked to such diverse regions like e.g. partial differential equations (describing e.g. hornbeam leaves) , folding of solar paddles and cell simulations (to which I will come later).

The Wikipedia-origami site gives some good overview. For people who are more interested in mathematical aspects Tom Hull´s site gives an introduction (although it would need an update..:)) and he has a nice collection of literature. Strangely I could not find a link on his site to the recent major origami conference OSME4, although he was in the technical committee.

Joseph Wu’s page containes the largest Origami database and lots of other stuff. It is from his site I borrowed the above “news lion”. The lion is not a “true” origami, since it is made of 10 pieces of paper and since not all the folds look straight if I look at the ears…:).

An always beautiful overview of new artistic deviations of Origami can be found on dataisnature, like this new entry about J. McCabe or Pauls entries for the paperfolding week. Paul liked our tesselation entry which gave me actually a little spur to write this Origami post .

However for everyone who wants to dig deeper into Origami math Erik Demaine‘s site is a must. Erik Demaine is famous for his contributions to solving the fold-and-cut-problem, which states that any polygonal shape (plane graph) — which you can imagine to be drawn on a sheet of paper — can be cut out, by folding the paper in an appropriate way and then by cutting along ONE line.

A polygonal shape includes single (possibly nonconvex) polygons, multiple disjoint polygons, nested polygons, adjoining polygons, and even floating line segments and points, so basically everything polygonal you can imagine to draw on a paper, including for example a logo like e.g. this flat lion.

Artist/physicist Robert Lang has on his page a program called treemaker, which sets parts of the above involved mathematics (Straight-Skeleton Solution) into a computer algorithm for determining a crease pattern. His Straight-Skeleton Solution for convex polyhedra was derived independently from Demaine et al. as I understood from Demaines page.

The mathematical Origami research is in the field of
Discrete and Computational Geometry
(the corresponding wikipedia link is not very instructive, thats why I linked to this workshop of our colleagues Günther Ziegler et al.)

For understanding more about how the fold-and-cut problem works see the instructions by Erik Demaine, Martin Demaine and Anna Lubiw of how one can cut out a logo for the CCCG.

### 5 Responses to “Origami part I”

1. Walter Says:

Great post! So much references, I didn’t find the time yet to take a look at them.

Do you know the following animation: http://youtube.com/watch?v=fQ0SSt8VAGg

I didn’t find out yet who did it and how it was done.

2. Walter Says:

Thanks also for the link.I didn’t know the animation. It is suberb. Despite the animators are quite busy to list all their sponsors…..

However I knew this animation:

quite a case for “Kottan ermittelt” I would say.

you can use LaTeX in your math comments, by using the $shortcode: [latex] E = m c^2$