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!

Archimedes' Stomachion

Doing Math:

I love puzzles.  Love love love them.  I will spend an entire day putting together a giant jigsaw puzzles and be perfectly content.  So when I saw that one of our assignment options was trying to put together Archimedes' Stomachion I was very excited!  The Stomachion is basically a little 14 piece puzzle that you try to fit into a square.  I did this puzzle on Geogebra which you can check out here:  Stomachion on Geogebra.  Make sure your window is zoomed to show all the pieces and then you can rotate them and fit them in the square!

When I started to experiment with fitting the pieces in, I tried to be methodical and fit in the biggest pieces first.  With the big pieces in first, I could then see which smaller pieces could fit in between them.  One of the main goals was to find out which sides of the different pieces would fit together perfectly since this would make the most of the space.  At times I found it slightly difficult to spin and move the pieces exactly how I wanted them which made me wish I had physical pieces in front of me.

 This was one of my first attempts.  I tried to make bigger shapes out of the smaller pieces, like the yellow blue and green pieces at the top formed a triangle. This seemed like progress because it matched those ugly edges of the yellow piece up with others to make a shape that was pretty much the same as the triangle made from the long purple and green pieces.  Of course the two lighter blue triangles wouldn't fit, so I knew I wasn't maximizing the space as well as I needed to.

Next decent attempt.  Again I noticed certain pieces would make bigger shapes, like the rectangle in the bottom right which I kept from the first attempt.  Also the square in the top right corner.  However, I abandoned my first thoughts of putting in the biggest pieces first, which resulted in overlap from the red piece and vacant space around the small purple triangle.

This was my best and last attempt.  If there was a way to flip over that last purple triangle, it would have fit.  The right half of the square seems great.  The fact that it forms a rectangle and could be cut into multiple different triangles makes me think that it is correct.  However the left half doesn't have those same intracacies.  Unfortunately, I could not find another way to arrange it to make it fit.


Overall this was a very fun puzzle, despite being a little frustrating at times.  I always enjoy when math and puzzles connect.  Interestingly enough, after I worked on this puzzle (for a long time) I was browsing through the new book I got for class (The Math Book by Clifford Pickover) and it had a page talking about Archimedes and the Stomachion.  It even showed this solution, although it's not possible to make it this one in the Geogebra file without being able to reverse some pieces.  I like how every piece is a part of at least two bigger triangles. Plus there is a line splitting the square into two rectangles and another line on the diagonal forming two triangles.  I think it's so interesting how these strange pieces make up bigger shapes within the square.  So although I didn't find one of the "17,152 solutions" it was very fun, and interesting to examine the details of a final solution.

What is Math?

Yesterday we had a brief discussion in class about what math is.  For a bunch of math majors we came up with an explanation centered on the notion that it is a quantifiable way to explain physical phenomenon but also includes ways to predict imaginary situations.  The main thing I took away from this discussion was that our definition is probably very different from most of the general population.  I believe most people would consider math to be pure computations, glossing over its ability to apply to a vast amount of situations.  So when the follow up question "what was the first math" was asked I thought of a different question.. I started to wonder what the first people who studied math thought it was.  When did they realize this was a new subject worth studying?  I imagine early civilizations didn't think of bartering at a market and traveling long distances in terms of math like we might, so it's hard to say what they thought the first math was.

Whatever the first math was, there are points in the history of math that stand out as adding great benefit to the current study of math or even the current state of the world.  I'm personally intrigued by the mathematics that went into planning overseas journeys of discovery.  I can't fathom figuring out how many supplies it would take to last until the ship docked again, in addition to how much weight a ship would carry if it was to return with goods.  Also how to calculate times and distances of these journeys.  Without all of that careful planning that no doubt took a lot of mathematical calculations we might not be in America right now. :)

In terms of the study of mathematics, Euclid made an enormous contribution by defining the components of geometry.  Not only did his definitions give us the foundation to discuss geometry, but he gave us a way to prove concepts so every mathematician could easily communicate.

I also think the invention of a definite monetary system was a significant development.  Although it doesn't really add the to the academic side of math, it is probably the most widely used application of math and occurs every second of the day. The idea that math is only for people who actively study it is discounting it a great deal.  Even though exchanging money seems pretty basic it provides ways to talk about fractions, decimals, operations, even exponents for interest rates.  And because money plays a crucial role in people's lives I think it's an important point when talking about math.

History is so rich that I find it very interesting to pull out the details concerning mathematics.  Then analyzing it to see how those experiences tied into the minds and cultures of the past to influence the concept of what math was at that time intrigues me greatly.  Knowing that math has changed a lot since its conception makes me wonder where it could go from here!