Crystals in cubes

My most popular model in recent years has to be my chemistry model kit. 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.

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).

Some important concepts in chemistry are 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, chemistry 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:


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:


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:


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

Cutting cubes

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 came up with a simple symmetric lock as shown in the CP.



But that's not all! 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.



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.