Month: December 2022

Miura fold

You can see some examples of this fold in the Elliot Building origami display (here is a link to my post about that). Since our first assembly of the display, I have made several more!

Astrophysicist Koryo Miura developed this fold as a way of packing things such as solar panels flat, so that they could be very easily opened (and closed) in space. Because of this space connection I asked a colleague from Physics & Astronomy to donate something more space-related for me to fold, to replace the math research poster I had originally used for our display. In this video, you can see me demonstrating the two-points-of-contact folding and unfolding that the Miura fold provides:

Click on the picture to watch the video.

I will update the Elliot Building displays soon, to include this space-themed example. I will also add some examples made from geological survey maps, donated by a colleague from Earth & Ocean Sciences, because the Miura fold is also excellent for storing maps.

Thank you to Arif Babul  (Physics & Astronomy) and Duncan Johannessen (Earth & Ocean Sciences) for the donations!

Menger sponge

The Menger Sponge is a three-dimensional fractal curve; if you have seen the Cantor Set (in one dimension) or the Sierpinski Carpet (in two dimensions), the Menger Sponge will look familiar to you. If you haven’t see those other things then this is a good opportunity to spend some quality time on Wikipedia.

In his book Project Origami, Thomas Hull gives instructions for folding models of portions of the Menger Sponge out of business cards. He also includes this dire warning:

Instructors in any class using this activity should be forewarned that it is normal for students to become addicted to making business card cube structures.

For our final mathematical paper folding project of December, we assembled a Level 1 Menger Sponge model:

Photograph of our Menger Sponge business card model.
Our level 1 Menger Sponge, built out of 120 business cards.

This model used up my entire supply of old University of Minnesota business cards; to make the Level 1 sponge we will need another 2,280 business cards. If you have any defective business cards (former institution, old job title) you would like to donate to our cause, please let me know!

The octahedron is dual to the cube

Earlier I shared a video of me folding a cube made from Fuse’s wireframe modules. While I was working at the University of Minnesota’s Math Centre for Educational Programs, I learned a different way of folding a wireframe cube. I don’t know the original designer, because the video I used to reference is gone. Here is a video of me slowly folding the module (in case you want to learn yourself) and then a time-lapse of me assembling it into a cube:

Two hands holding a partially-assembled paper cube.
Click on the image to watch the video.

https://echo360.ca/media/f2f16367-8010-4097-b635-2ab4debbd49d/public

The cube is one of the five platonic solids (named for the ancient Greek philosopher Plato). A platonic solid is a solid whose faces are all the same, whose faces are all regular polygons, and whose vertices all have the same number of faces meeting together. Faces of the cube are regular 4-gons (squares), and at each vertex three faces meet. Now, if you put a vertex in the middle of each face and then connect two vertices together if and only if their corresponding faces share a mutual edge you get another solid. We call that the dual of the first one. It turns out that the dual to a platonic solid is always also a platonic solid. Grab a cube and see if you can work out what its dual is!

Go ahead; I’ll wait.

You could always fold one yourself; you just need 12 pieces of paper!

The shape you have (perhaps) just drawn or visualized is called an octahedron because it has eight faces. Each face is an equilateral triangle. In this next video I’ll show you how to fold a model of an octahedron, which I also learned at the Math Centre for Educational Programs. It’s made out of six “water bomb” modules. While at UMN, I regularly helped teach local teachers how to fold these and help their grades 3 and 4 students assemble the model, as part of the Girls Excel in Math (GEM) program there. That means that this model is not too hard to put together; it is certainly easier than the wireframe cubes.

Watch to the end to see a demonstration of their duality!

Two hands holding a partially-assembled octahedron.
Click on the image to watch the video.

https://echo360.ca/media/e7b2d811-dbd2-4bf6-be4d-9f3082d9a705/public

Folding a cube with wireframe modules

Tomoko Fuse’s wireframe unit (from The Complete Book of Origami Polyhedra) is great for making cubes. This video might help you follow his instructions, and work out how to lock the units together. Watch to the end to see my final product!

Two hands holding a partially-assembled origami cube.
Click on the image to start the video.

https://echo360.ca/media/617d6ee8-d35e-4676-a783-3a1bad7c22f9/public

Elliot Building Mathematical Origami display

Visit the classroom wing of the Elliot Building to see a display of mathematical origami starting in early December 2022. We hope our mathematical art brightens your December exam period!

Most of the models were constructed by members of the Association for Women in Mathematics student chapter at UVic and their friends. A few of the models you will see are listed below, and I will create some more detailed blog posts here for a few of them too, if you would like to learn more.

The Miura Map Fold

Koryo Miura, professor emeritus at the University of Tokyo, developed this fold in the 1970s, as a way of folding things like solar panel arrays so that they could be easily opened and closed for use in spacecraft. When used on somewhat rigid paper, the Miura Map Fold lets you unfold (and refold) your paper by just pulling (and pushing) on a pair of opposite corners.

For more details, visit https://en.wikipedia.org/wiki/Kōryō_Miura.

Self-similar Wave

From Thomas Hull’s book Project Origami, this is a self-similar origami model and can go on as long as your paper, and skill, permits.

Hyperbolic Paraboloid

Another model from Thomas Hull’s book Project Origami, this model might remind you of a surface from Multivariable Calculus. This model can also become as detailed as your skill allows.

Hexahedron

This model is due to Molly Kahn, and is an example of modular origami. The final shape is built from three identical units, which lock together.

Five Intersecting Tetrahedra

A tetrahedron is a platonic solid having four faces, each of which is an equilateral triangle. This model consists of five wireframe models of tetrahedra, interlaced together. Each tetrahedron has six edges, so it took one student 30 units (Francis Ow’s 60-degree unit) and about nine hours to build this model, following instructions from Thomas Hull’s book Project Origami.

Pentagon Hexagon Zig-Zag modules

This module is due to Thomas Hull, and is also called the PHiZZ module. It is useful for constructing wireframe models whose faces are pentagons or hexagons, because those result in sphere-like models. Graph theorists will recognize them as three-regular planar graphs, and many of our models are also properly three-edge-coloured because our graphs are Hamiltonian. The PHiZZ module can also be used to create a wireframe model of the torus, by including faces with more edges so as to introduce negative curvature. The models we built are listed below.

  • Dodecahedron: this is a platonic solid having 12 faces, each of which is a regular pentagon. It requires 30 PHiZZ modules.
  • Buckyball: familiar to both chemists and football fans, this wireframe model has 20 faces that are regular hexagons and 12 faces that are regular pentagons.  It requires 90 PHiZZ modules.
  • Torus: there are many models out there; we followed one due to the topologist dr. sarah-marie belcastro. It requires 84 PHiZZ modules.

Sonobe modules

Either due to Toshie Takahama or to Mitsunobu Sonobe, the Sonobe module  is great for building all sorts of models. The models we built are listed below, and we followed instructions for making the module from Maths Craft New Zealand.

  • Cube: this is a platonic solid having 6 faces, each of which is a square. It requires 6 Sonobe modules.
  • Triakis octahedron: this is related to a regular octahedron, but each face of the octahedron has a pyramid built over it. It requires 12 Sonobe modules.

More platonic solids

You will find a wireframe model of a cube (following instructions from Tomoko Fuse’s book The Complete Book of Origami Polyhedra), which required 12 units to build.

You might wonder what is inside that cube! It is an octahedron, yet another platonic solid, which has 8 faces that are each equilateral triangles. This model demonstrates that the octahedron is dual to the cube. You will also find some octahedra on their own, not inside a cube. These octahedra are built from “water bomb” modules, and although I don’t know who to credit for the original design you can find instructions here. Each one requires 6 units to construct.

Recommended resources

Tomato Fuse, The Complete Book of Origami Polyhedra.

Mark Bolitho, The Art And Craft of Geometric Origami.

Thomas Hull, Project Origami.