Qiandu, Yangma, and the Bienao

We talked about ancient Chinese mathematics in class the other day.  Among those we discussed was Liu Hui who lived in the third century and made numerous contributions to mathematics.  Our main topic of conversation was the cube he made from three different parts that he called the qiandu, yangma and bienao.  The qiandu is half of the volume of a cube, yangma is 1/3 and bienao is 1/6, so together they make up the whole volume of a cube.  So with the finished product in front of us (but without tracing or measuring from them!), we were told to make the nets of the shapes.  A net of a three dimensional object is a flat drawing of all the faces of the shape correctly positioned so that when you cut it out you can fold it into the 3D shape.  Before I started the nets I drew the above sketch to get an idea of how they would fit together.  I decided to make my final cube 2 inches on each side which made the diagonal of each face the square root of 8 (approximately 2.8) and the diagonal of the cube the square root of 12 (approximately 3.4).






After analyzing which sides would need to fold together I made my nets.  Here's a rough sketch of each shape with the measurements.  Here's a link to Jennifer's ready made nets (with a better picture).








So with my sketches, a ruler, scissors, tape and coral paper I made my nets and then folded them up to make my three shapes!  Not the prettiest, but they still make a cube!

Left to Right: Bienao (1/6), yangma (1/3), and qiandu (1/2).

As I played around with them I decided I didn't want to use the qiandu anymore, but I wanted to make two new shapes.  I decided I didn't want the yangma and bienao to form another qiandu like in the original, so I positioned them kind of diagonally (pictured).  

Once I got started on the nets for the two new pieces, I realized they were the exact same as the yangma and bienao.  I still made them though and played around with the ways that I could put them together.  This was interesting to me because I realized I couldn't put two complete faces of the same shape together.  In other words I had to make two qiandus out of a yangma and bienao.  I couldn't make any other formation from them that would still form a complete cube.  

Overall I think it's interesting that all the way back in the third century Liu Hui was thinking about volume and how to divide it in such a way that gave a unique combination of the shapes.  (You can check out a proof and more pictures here from Sherlock Holmes in Babylon: And Other Tales of Mathematical History.)  Plus, I very much appreciate the kind of puzzle approach to putting it together.  It makes me want to make more cubes made up of more complicated pieces, although I'm not sure if I would be able to manage making shapes that have such nice divisions of the volume like he did!