Saturday, August 19, 2017

Crystals in cubes

My most popular model in recent years has to be my chemistry model kit (more updates for the model kit coming soon!). I have been thinking all along about how to use the model kit as an educational tool, and I now have raw footage for a video tutorial and plans to draw new diagrams with the hopes of publishing a chemical education journal article.

At CfC, I held an open discussion about origami education and was surprised to find how many teachers were in the room, but I didn't find hear much about origami being used in secondary or post-secondary education. With that stuck in my head, I started thinking about what other subjects could be more easily taught with a 3D model present. I came up with a few ideas that I still plan to try, but the one that stuck out was metallic crystal structures (which might have been influenced by finding out that Byriah Loper published a book last year).

Every introductory inorganic chemistry class includes the unit cells of the three basic metallic crystal structures: simple cubic (or primitive cubic), body-centred cubic and face-centred cubic (or cubic close-packed). Among other things, students are generally asked to identify the number of nearest neighbour atoms to each (equivalent) atom. Instead of the usual dot/sphere to represent an atom, I decided to make a wireframe structure where the nearest neighbours are all connected. By folding each of these connectors from the same size of paper, I could also show the difference in size of the unit cells.

The first model was the simple cubic structure. I decided to fold each connector from a 2:1 rectangle, similar to the chemistry model kit. In order to join together three units for a cubic frame, I needed a bit of extra paper as shown by the black outline labelled A

That extra 1/4 of the paper width was used to make the flap, as shown in the CP

Joining 12 units together gives you this:

Crystal structure unit cells

The simple cubic structure is naturally just a cube. For the other structures, I needed a larger cubic frame to represent the bounds of the unit cell in addition to the connector pieces. Following a bit of math, the frame of a body-centred cell is 2/√3 times the length of the connectors. The connector (A) and the frame piece (B) look like this:

Note that I added 1/4 of the paper width again for the cubic frame. The outer frame is folded just like the simple cubic structure, whereas the connectors have to be folded to join at an arccos(1/√3) angle. The following CP achieves this goal:

The unit only connects on one side, while the other side just hold the unit's shape. Ideally, the non-connecting end should be folded with an angle slightly greater than 45°, such as:

Using 12 frame pieces and 8 connectors (4 of which should be the mirror image of the others) gives this:

Crystal structure unit cells

Finally, the face-centred cubic structure. As the name "cubic close-packed" suggests, this structure is the unit cell from stacked triangular lattices so the angles between nearest neighbours are all 60°. In this case, my work is cut out for me by Francis Ow. The cubic frame is 2/√2 times the connector length (plus the extra 1/4 of the paper width), which looks like:

And, for completeness, the CP for Francis Ow's unit is:

12 frame pieces and 36 connectors (12 mirrored relative to the other 24) come together to make:

Crystal structure unit cells

Most importantly, the three structures together with the same original paper size look like this:

Crystal structure unit cells

Going back to the topic of education, many teachers that I met at CfC focused on the act of folding to teach students. A full class of students folding these crystal structures just to get a point would be unlikely to succeed, based on my experience teaching origami lessons. Instead, in a setting with few resources, I think this type of model would be great for a teacher to make for a class so that students can see the differences in three dimensions. With this and the chemistry model kit I don't expect to turn all students into origami artists; rather, I want to make education more accessible through origami.

Sunday, August 6, 2017

Cutting cubes

First off, I recently attended the Convention for Creators in Lyon, France and it was a bit of a surreal experience. I had looked at the guest list in advance and was surprised to only see a few names I didn't recognize. As Daniel Chang put it, "At most conventions, you want to meet with two or three particular folders. At this convention, you want to see everyone." Afterwards, I was fortunate enough to travel with Robert Lang, Dave Brill, Beth Johnson, Shuki Kato, Gen Hagiwara, Ondřej Cibulka and Alex Szarry-Gomez to Nicolas Terry's home and visit his shop, and I visited the Escuela Museo Origami Zaragoza in Zaragoza and the OAS (Origami Always Succeed) group in Barcelona.

Among the many ideas I had during the convention, one was to create an Instagram account: @rymacorigami.

In my last post I mentioned that I had been thinking about folding simple but interesting geometric shapes. I guess that I still had the k-dron in mind, because I decided to try making another cube bisection. I realized that bisecting a cube to two equal pieces has to involve the central point inside the cube and some symmetry about that point. Based on this, a simple way to get the pieces involved was to draw lines from the central point out to the corners of the cube. Alternating between connecting to an upper corner or a lower corner gave a simple saddle point geometry which wasn't hard to find the references for because of the √2:√3/2:√3/2 triangles.

The lock on the bottom of the model took a bit of playing around with. At first I wanted to use half of the cube width, but it ended up too weak. Eventually, Roman Remme (also in Lyon) came up with a simple symmetric lock as shown in the CP.

Pointed saddle cube bisection

Pointed saddle cube bisection

But that's not all! On my flight back home, I realized that there was another way to make a saddle point in a cube. Instead of having edges directly to the centre of the cube, I could have a flat square in the centre with equilateral triangles connecting to the corners. Here is the resulting CP:

The references are 15 degrees from the corners, which can be folded quite easily as mentioned in my last post. The lock for the bottom of the cube was the same as before. From the side, the result reminds me of the 元宝 (yuanbao), an old form of Chinese currency.

Flat saddle cube bisection

Flat saddle cube bisection

Because both models were based off the same geometric shape, I decided to call the first a pointed saddle cube bisection and the second a flat saddle cube bisection. The best part of these models is being able to take the two pieces apart and fit them together. Here is a video from my Instagram page: