Author Archives: codablo

Rise and Fall of Wasan

Since most would not be able to read my report on Sangaku (Artfully done tablet of Geometry) I thought I would do a blog post on the rise and fall of wasan (Japanese Mathematics) for you all to enjoy. Never mind it’s a topic I know a lot about now and have done a bunch of research on the topic.

The Japanese didn’t really have their unique math until about the year 1627 when Jink ̄o-ki was published. This was the first Japanese mathematics book published. The Jink ̄o-ki was a Japanese publication that explained how to use the soroban (Japanese name for the abacus) to do things like calculate pi, and would provide other math instruction and problems. Until then, much of the learning and study in math came from the classics of China, with heavy emphasis on The Nine Chapters and Cheng’s Treatise. I’ll explain a bit as to why there was such a long delay in developing mathematics.

Leading up to the late 1500s, most uses for math in Japan was to levy taxes on the land and for basic arithmetic for business transactions. The government of the time actually created The Department of Arithmetic Intelligence to go to each landowner and measure the property so the owner would know how much tax to pay.

These math specialists only knew just enough geometry to get the area of the land and calculate the tax required. The government saw math as a means to an end for acquiring money. This meant that math was a tool used by the government and only a special few were educated in mathematics as deemed necessary. But since there was no one to teach them, they had to rely on the Nine Chapters to be their teacher.

Around the early 1600s things began to change. A new set of rulers named the Tokugawa family took over all of Japan, uniting all the land under one government. Taxes were no longer tied to the amount of land owned and The Department of Arithmetic Intelligence was no more. This, in turn, led the farmers to no longer know how much land they had and, as consequence, the amount of food they could produce.

The Tokugawas also brought about another important change, the closing of the Japanese boarder. Iemitsu Tokugawa outlawed Christianity and closed the boarders. The problem was that a growing number of converts started a community together and began to band together. At the same time the Spaniards attempted to compete for converts and in a bold attempt to be the only missionaries in Japan told the Tokugawa family that the other nation’s missionaries were trying to create an army to conquer Japan. The Spaniards’ plan backfired and all missionaries were put to death along with those that would not give up on Christianity.

With the closing of the boarders and all the enemies of the government crushed, a period of peace was created called the Edo period in Japan that lasted until 1868 when boarders were opened again. It was during this period that the Japanese culture became its own and flourished. Everything from haiku poems to flower arranging to tea ceremonies was created during this time. By the end of the Edo period a gentleman was expected to know “medicine, poetry, the tea ceremony, music, the hand drum, the noh dance, etiquette, the appreciation of craft work, arithmetic and calculation . . . not to mention literary composition, reading and writing.” (Hidetoshi)

During the time of Great Peace the samurai became the new noble men of Japan. No longer needed as warriors, many were given government jobs to help ease them into normal lives. As consequence, the men became some of the more educated citizens. That being said, the pay they received for working for the government was terrible. Most samurai had to pick up 2nd jobs; many of them become traveling schoolteachers.

The stage was now ripe for an explosion of learning. We had farmers that needed to learn math, we had samurai that needed second jobs, and a place for it all to happen, the local shrine or Buddhist temple. Since there were no school buildings, most lessons happened at the shrines and temples that dotted the land. This encouraged more people to gather together for religious, educational, social functions. Over the next century the Japanese people would have the highest literacy rate of all the nations and become one of the most educated.

During this time, the people began to make sangaku, which is basically an artistically made wooden tablet containing a geometric problem and most of the time the solution. These tablets would adorn the temples and shrines showing off the newest knowledge learned. However, these tablets also had a deeper meaning. These sangaku became a way of thanking the gods and spirits for the new knowledge.

Many of the sangaku that have been found focused on finding lengths, areas of various shapes, and even volumes. The sangaku found below is one example of finding a length. The problem asks to find the diameter of the north circle inside of the fan. The problem is setup so that the entire area of the fan is a third of a circle and you can assume you know the diameter of the south circle. The answer ends up being (sqrt(3072) + 62 )/193 times the diameter of the south circle.


Sadly, wasan (Japanese Mathematics) was one of the few things that didn’t survive the Japanese Renaissance, which is why many of the records of wasan and sangaku are only now being discovered. At the end of the Edo period a new government was formed that outlaws wasan from being taught. It turns out that wasan lacked Calculus but more importantly, was different than the rest of the world. With the opening of the boarders, the government needed to adopt Western Mathematics to be able to communicate with all the new trade partners that were being re-established. To that end, a law was created that outlawed wasan and Western math was forced in the schools. Anyone that still taught wasan had his teaching license stripped and imprisoned.

Reference:

Hidetoshi, F., & Rothman, T. (2008). Sacred Mathematics. Princeton, New Jersey: Princeton University Press.

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What Does Being Correct Mean?

In class, we were discussing the Parallel Postulate by Euclid. Basically it says that if you draw a straight line on top of two other lines so that they intersect, and if the angles on the same side of the first line are less than 2 right angles (180o), the two lines will intersect at some point on the same side.

Image: 6054, via Wikimedia Commons.

It’s weird learning about proving something that feels so elementary that I assumed it was just true by definition. I mean I can just look at the picture and it certainly looks like it should be correct just by careful inspection. But I guess that doesn’t really prove it without a shadow of doubt. What if what I was looking at was 179.999o and I just said they would never touch even though they would intersect given enough space. Granted, I would assume it was 180o so I would be correct based on the assumption being true.

When I look at this problem, I can’t help but reflect on the lessons, experiences, and “truths” that have instilled within me from previous mentors and teachers. It becomes very hard to try and think about other approaches or ideas other than “duh that true”.

What allowed me to think about this Postulate was learning about how other people through out history thought about the Parallel Postulate and created their own “new math”; their own pseudogeometry; their own imaginary geometry. Here I am unable to think “outside the lines”, but these other people created whole new systems from looking at the problem from a different angle. I have no problems creating weird parallels with my jokes and puns but can’t seem to do the same thing with math. (Yes, I love bad puns).

Poincare and Lobachevski were both people that worked in this pseudogeometry, which is now called hyperbolic geometry. (The former or “normal” geometry is considered “Euclidean Geometry”). In hyperbolic geometry it’s possible to have lines that would normally intersect in Euclidian space be considered parallel and non-intersecting in hyperbolic space. I think looking at the picture below will really help. I know it wasn’t until I built a hyperbolic plane by hand that it really sunk in for me. ( Make your own at http://www.math.tamu.edu/~frank.sottile/research/subject/stories/hyperbolic_football/index.html )

A hyperbolic triangle. Public domain, via Wikimedia Commons.

Reflecting on the on hyperbolic plane I began to try to remember a time when what the instructor was teaching conflicted with something I already knew. As I thought I remembered something an art teacher told me about vanishing points. So imagine you’re standing on some railroad tracks that stretch straight forward for miles. As you look down the tracks, as you would if you were actually a train, at some point the individual components would become one whole line. Instead of seeing the left rail, the right rail, and everything else you would see a railroad track. At that point, the left and right rails have effectively become one, unable to tell them apart. Now what would happen if a train went down those rails that look like they became one? The train becomes smaller, or at least, it looks like the train is shrinking. At the time I could only think about how the teacher lost her mind. It wasn’t until I looked down a straight road that I realized how right she was.

After thinking about how perspective is everything I began to wonder what other things are different than they appear? I asked a friend, and she mentioned she actually had to unlearn some thing to be able to Fence (as in the sport) correctly. She told me that she had to change the way she extended her arm in order to be able to obtain the longest reach possible.

It turns out that a straight line with your arm is not the best way to have the longest reach. In all my learning, I had been taught that you to get the longest linear distance with line segments are to put each line segment end to end along the same axis. But in fencing, doing just that with your arm is not the longest. Why is fencing different?

When hold the sword in your hand, it seems that your muscles tighten to hold the load and your arm up. By tightening your muscles, you shorten your reach by as much as 2 inches for some people. When your muscles are relaxed, the joints can loosen allowing more space between the bones, which lengthens your arm. So by relaxing your arm a bit so it’s not parallel with the ground, your sword can reach just a little bit further.

Is math wrong when it comes to the physics of people and fencing? Absolutely not! In my case, it’s the model that the math was used on that was wrong. I assumed the arm was a rigid object with hinges at the shoulder, elbow, and wrist. Since I had modeled the arm in this fashion any math done to the model would never take into account the possibility of expansion of the hinges. Assumptions are the downfall of many people.