Origami – The Math Behind the Paper Folding

I am about to start teaching an online geometry course, and it has me missing some of the things I use to do with my students to help them discover relationships, and work with angles and symmetry, which was origami. Origami is the art of paper-folding – and using it in geometry is a great hands-on and visual tool to help students discover angle relationships, symmetry, linear relationships.

Origami is something I am sure most of you are familiar with and maybe have even attempted to create some origami art yourself. I have two friends who are origami wizards and often post their creations on FB – and it’s pretty amazing the shapes they create. When I recently went to the Museum of Math in NYC there was a whole exhibit devoted to Origami.

In my class, obviously, we did relatively simple constructs – folding one piece of paper into things like cubes, birds, shapes. The focus being on the folding and shapes created from each fold and looking at the angles and relationships that developed after each fold. But – as I have discovered, there is some really complex math behind origami, and really complex shapes that are created all from one sheet of paper that are simply astounding. I just found this Ted Talk from 2008 by Robert Langdon that discusses the mathematics behind Origami and how because of mathematics, folds that before were impossible are now possible, allowing for origami constructions that are astounding. Those of you who teach geometry, I think this will be very interesting to you, though I think other math subjects as well will find some applications. At the end of the video there is also a link to some templates for folding some more intricate origami constructs.



A Math Nerd’s Dream Museum

img_3760I went to the National Museum of Mathematics (MoMath) today – what else would I do while in NYC?!!  If you were unaware, this is yet another img_3766great attraction to add to your to-do list next time you are in New York City. I was lucky enough to have a few hours today to myself and thoroughly enjoyed my hands-on experiences – me and several hundred school-age children.

The museum is focused on providing hands-on, interactive img_3782mathematical experiences so students can see, create, and play with mathematics.  There are games, art exhibits, bikes with square wheels to ride, cars to control around a mobius strip, img_3780angles, tessellations, fighting robots, logic puzzles….it was really fun, and there was a lot of ‘learning’ embeddedimg_3775 in all of the exhibits, though I did find I was the only one reading – the kids wanted to just ‘do’. But can you really blame them?


One of my favorite exhibits when you walk into the museum is the wall of etchings done on metal plates. There are parabolic lights above them that move and due to the angles the metal etchings are at, it appears the whole display is moving and that the etches are 3D when in fact they are flat. The etchings themselves are beautiful – lots of mobius strips in there!!  I tried to capture it on video but it doesn’t do it justice.

Another favorite was the art exhibit showing the amazing geometric img_3770sculptures of Miguel Berrocal – famous for creating sculpture puzzles – i.e. sculptures built by pieces fitting together. There are numerous sculptures on display along with puzzle books showing the steps to build some of the sculptures. There are also two hands-on opportunities to try to build some of the sculptures. I tried my hand at the above sculpture, “portrait de Michele”, which they recreated the pieces using a 3D printer and then provide ‘directions’ to build.  My results are below….I was very proud of myself!

There was a little bit of everything – I made myself into a human fractal tree (that’s me as the trunk if you look really close). And then I made two 3D shapes (sphere and star) by putting together flat plates with 2D shapes (circles and triangles) in a layered order so that they end up looking 3D.  That was a challenge trying to piece the different sized shapes in the right order.

There was a lot more fun to be had – from the square tire bike to the shape challenges to building polyhedra. All in all, a fun-filled few hours doing some math and experiencing students enjoying doing math as well. If you ever get the chance to get to NYC, be sure to include the MoMath in your itinerary!