Sangaku

In my exploration of the Internet in search of non-Western approaches to mathematics, I was delighted to discover a tradition in medieval Japan known as Sangaku, based on a form of traditional Japanese mathematics known as wa-san.

As I discovered from a number of sources, including here at Wikipedia , because Japan was cut off from Western societies during the period known as the Edo Period, the Japanese developed their own system to deal with the analysis of shapes and numbers. Citizens ranging from samurai, farmers and laborers would devise difficult mathematical problems and represent them in colorful wood block cuttings that they would hang from the temple walls. These served as both offerings to the gods, as well as challenges to other temple goers. An example is the picture above from 1847 appearing in the city of Uchiko.

There is apparently an excellent book, called the Sacred Geometry of Japanese Temples, which discusses the history of Sangaku and traditional Japanese mathematics in greater detail. I found this excellent set of geometry problems here (but note that the links to the answers no longer work!).

Let's start with the most straightforward one:

Q1) In the diagram below, three orange squares are drawn inside of a green triangle. If the radius of the blue circle in the upper left of the triangle is equal to 3, what are the radii of the other two blue circles?

Q2) In the diagram below, the three circles are tangent to each other and to the line beneath them. If the radius of the orange circle is 10, and the radius of the blue circle is 6, what is the radius of the red circle?

The Quadratic Formula

The quadratic formula is one of the most important expressions taught to math students, for reasons that extend much farther than solving quadratic equations. As it turns out, as also discussed in Rational Algebraic Expressions, second degree polynomials, or polynomials that have an x-squared in them, appear throughout science and Nature as a result of solving differential equations using a special technique called a transform. This fact is actually responsible for the appearance of oscillating systems in Nature, such as water waves, sound waves and light, as the quadratic formula is an excellent opportunity to introduce the concept of imaginary and complex numbers.

So let us revisit what the Quadratic Formula, shown above, actually is: it is a formula that helps us to factor a second-degree polynomial (a polynomial with an x-squared in it) when the factors are not readily apparent (which they very rarely are in real world situations). If you remember, when we are faced with an algebraic equation that looks like this,

x² + 5x + 6 = 0 (Equation 1)

we can sometimes solve this equation by factoring the second degree polynomial on the left hand side, and then setting each of these factors equal to zero. So, when we factor the polynomial above, we can rewrite Equation (1) as

(x+3)(x+2) = 0

And it turns out that this is only true when x+3 = 0, or x+2 = 0.

The reason this works is that if any product of two numbers is equal to zero, then either of those numbers, or both of those numbers must be zero, in order for the product to be zero. Any non-zero number times another non-zero number can never be zero, regardless of how small or negative either of them are.

And so if (x + 3) times (x+2) is equal to zero (which is the same thing as x + 5 x + 6 = 0), then either x +3 = 0 or x + 2 = 0. There can be no other way. And so, in this case, x = -3, or x = -2, which are called the roots of the polynomial. To see that this works, we can then plug in these values of x into Equation 1 and we will see that these values of x do indeed solve the original equation.


But how can we solve this problem if we cannot factor our polynomial into two factors? This is when the quadratic formula is used. We won't go through the derivation of the QF, which utilizes a principle called completing the square, but it is good to convince yourself that it works by looking at derivations like this, this and this. Once you convince yourself that the quadratic formula works, we can begin to understand how it works and what it can tell us about an equation we are trying to solve.

GRAPHING SECOND DEGREE POLYNOMIALS

Just as understanding the equation of a line, y = mx + b could enable us to graph a line on a coordinate axis using the slope 'm' and the y-intercept 'b' (more here), similarly, we can look at the general equation of a second degree polynomial, ax^2 + bx + c, and figure out what this equation would look like if we graphed it. By doing so, we can understand a lot about the particular equation we are solving.

If you recall from the posts regarding the equation of a line, it is helpful to understand functions by treating them in their most general form. In the case of a line, the function we considered was

y(x) = ax + b

and we learned that for different values of a and b, the graph of the function y(x) would look different. Because the highest power of x in the function y(x) is one, the equation of a line is an example of a FIRST DEGREE POLYNOMIAL. Because there are two parameters that we can control to make our line look different (a and b), we say that the equation of a line has two DEGREES OF FREEDOM. Any line in two dimensions can be graphed using the equation above by controlling the values of a and b.

Similarly, we can arrive at any parabola seen in Figure ## by changing the values of a, b and c in equation (2). Just as there was a rationale to how different values of a and b would affect the type of line we would get (the amount of slope, whether the slope was up or down, and where the line would intercept the y-axis), so too there is a systematic rationale to understanding how to make a particular parabola by changing the values of a, b and c.

One of the big leaps that happens when considering the parameters of a second-degree polynomial (a, b and c), compared to a first-degree polynomial (m and b), is that the shape of our graph -- in this case a parabola -- depends not just directly on the parameters themselves, but rather on the relationships of the parameters. In the case of a parabola, the value of 'a' does tell us the direction our parabola will open (up or down) and the value of 'c' will tell us the y-intercept of the parabola (the place where it will cross the y-axis). But these characteristics themselves do not provide enough information to know what the entire parabola will look like.

The first thing to notice about all of the parabolas in Figure 2 is that we can draw a vertical line through each of them about which the graph is symmetric -- if we fold the parabola along this line, it will fold back onto itself. This line is therefore called the axis of symmetry, and is arrived at by the equation:

x = -b/2a

It is important to notice that since this equation is x equalling a constant value, it represents the equation of a vertical line. Once we draw this line, we can begin to see where our parabola will be located on the coordinate axis, even though we still don't know exactly what it will look like and where it will be. What we can see, however, for each of the parabolas in Figure 2 is that the value that the parabola takes at the axis of symmetry is going to be the biggest or smallest value that the parabola ever takes (depending on if it is going up or down). As you will learn, a parabola can only change directions once, and does so at the axis of symmetry. Since the axis of symmetry only gives us an x-coordinate, we need to plug in that x-coordinate to our original quadratic equation to determine what the value of y is at the axis of symmetry. The x-y coordinates of the parabola at that point is called the VERTEX. Once we find the vertex, we can position our parabola vertically along the axis of symmetry.

THE ROOTS

Once we know where the vertex is, and if our parabola opens up or down, the next step is to figure out how wide or narrow our parabola is going to be. There are a few ways to do this, but the first way we will consider is to figure out the places where our parabola crosses the x-axis, which are referred to as the roots of our polynomial.

As mentioned above, in cases where we cannot factor our polynomial we will use the quadratic formula to determine the roots. If you notice from the QF written above, there are actually two solutions for the roots due to the +/- that appears before the square root. So in fact, the roots of a polynomial ax^2 + bx + c are given by


Though this may look rather complicated, we can break it down to make it much easier to grasp.

If you notice, the roots each have the term -b/2a in them, and then either add or subtract the same quantity from -b/2a: . So, the roots will be equally spaced around the value -b/2a, which they should be because we said that parabolas are symmetric around the line x = -b/2a.

Games that Teach: Algebra v. Cockroaches

THE EQUATION OF A LINE

If you ask most people who have been through high school mathematics what the equation of a line is, a surprising number can spit out "Y equals m x plus b", or y = mx + b. Many of them can even tell you that the "m" stands for the SLOPE, and the "b" stands for the Y-INTERCEPT, and that the number that we pick for each of these parameters controls the appearance of the graph of the line that we get.

But I also don't think many people understand why this equation is so important. Part of the answer is that this particular equation ISN'T that important. Though there are many examples of processes that behave like a line, most often we need much more complicated equations or functions to describe real world phenomena. But what IS important about the equation of the line are the techniques we will use to analyze equations of polynomials of which "mx + b" is an example. In fact, analyzing y = mx + b is one of the simplest cases of a function that we can analyze, which is why it is discussed so often.

To understand how to analyze our equation of a line, and how changing the values of m and b affect the shape of our line, I have been using the game below called Algebra v. Cockroaches, where students need to fill in the equation of the line that the cockroaches are traveling on to be able to exterminate them.

y = mx + b

Games that Teach: Gyroball

The following game, Gyroball, is an excellent exercise in spatial reasoning on a 3D coordinate plane, as well as a keen test of eye-hand coordination. In using it with different students and people, I am fascinated by how different people initially approach it. The key to Gyroball is patient control, which is something that is hard for many people.

I am also continually fascinated by how much people, especially young people, like to play games. And different students like different types of games -- understanding why and how would be interesting to understand learning and cognitive differences amongst students.

Your task for this post is to complete the first 5 levels of the game, and answer the questions in the form beneath the game. To do so, you might need a review of 3-d coordinate geometry.

Time Series

The Virtual Bead Loom

Although mathematics is often taught to be and thought to be solely the product of European culture, nothing could be farther from the truth. It IS true that Western European culture is responsible for much of the symbolic and logical structure of modern mathematics, and being fluent in this often arcane tongue is what is sometimes as thought of as being GOOD at mathematics. But the beauty of mathematics is that it tends to deal with concepts that are UNIVERSAL to the human experience, such as our understanding of shapes, patterns and quantities.

As mentioned briefly in the post Induction from Patterns, when I was in Guatemala recently I discovered that geometric patterns were an important part of Mayan civilization. I later learned that Guatemalan women have actually passed down their understanding of geometry to their daughters through the act of weaving. The patterns below are a picture I took in a street market in the town of Santiago de Atitlan along the shores of Lago Atitlan.




There is an excellent resource here called Culturally Situated Design Tools from the work of Ron Eglash at RPI which provide interactive environments to explore the connection between mathematics and art in indigenous cultures. There is a great overview of these tools in his paper here. The persistent theme across cultures is the presence of repeating geometrical patterns with different types of symmetry and regularity. There are tools to explore cornrow patterns in African American culture, beadwork in Native American cultures, and pyramid design and adornment in Mayan culture. There is little information on how to incorporate more advanced mathematics, particularly high school concepts, to the use of these tools, however.

Rational Algebraic Expressions

Many people have heard that mathematics and music have a lot to do with each other, but often are unaware how. Although I am not a musician, I HAVE gravitated strongly to the areas of mathematics which deal most heavily with music, particularly the fields of SIGNAL PROCESSING and DIFFERENTIAL EQUATIONS. Though a thorough understanding both of these subjects involve mathematics much further along than that taught in high school, I have been finding a number of topics in high school mathematics where it is appropriate to introduce math's relationship with music.

A student of mine has recently been working on what are called RATIONAL ALGEBRAIC EXPRESSIONs, or when two polynomials are divided by each other. As you may remember, a RATIONAL NUMBER is a number that can be expressed as a FRACTION. So numbers like 3 (or 3/1), 3/4, 22/7 are all rational numbers. If you remember, the word RATIO and RATION are names for fraction, which is why these are called rational numbers.

You may remember that some fractions, when expressed as decimals, involve an endlessly repeating sequence. For instance 1/3 or one-third is equal to 0.33333... and so would need an infinite number of numbers to express exactly. Another example is above, 22/7 which can be represented as 3.1414141414... and is often used as an approximation to PI. But because these numbers can be expressed as a fraction, the decimal sequence will always a nice pattern to it. In the case of 1/3, the number "3" over and over again; in the case of 22/7, the numbers "14" over and over again.

Alternatively, there are numbers which, when expressed as a decimal, are also endless sequences but which CANNOT be represented in a repeatable pattern. The most prominent examples are the square roots of numbers that are not PERFECT SQUARES. So numbers like the square root of 25 can be represented by 5 or 5/1, but a number like the square root of 23 could only be represented in decimal form if we kept an infinite number of decimal places. And so these numbers ARE NOT expressible as a fraction, and are called IRRATIONAL NUMBERS.

An excellent exercise to test your ability to identify rational and irrational numbers, as well as perfect squares is Number Cop, by Hans Software, which I have embedded below . . . Use the drop-down menu to decide which types of numbers you would like to test your knowledge of.

From Rational Numbers to Rational Algebraic Expressions

Similarly, a RATIONAL ALGEBRAIC EXPRESSION is an algebraic formula that is represented as a fraction. So, for example, the following are rational algebraic expressions:


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The techniques that are stressed in teaching RAE's are finding what are called the ZEROS, or the places where the RAE is equal to ZERO, and the POLES, or the places where the denominator is equal to ZERO. Though it may seem like a completely random and useless subject to introduce, it turns out that understanding expressions like these, and the meanings of ZEROS and POLES, is essential to understanding many physical systems from bridges to car suspensions to musical amplifiers. We'll consider an example below from music and electronics, known as a BANDPASS FILTER, once we look at how to analyze expressions like the ones above.

So let's say that we have a typical RAE like:


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If we were able to factor the numerator and denominator, we would be able to determine the values of s that make this expression equal to zero (by finding the ROOTS of the numerator) or which make this expression infinitely large (by finding the places where the denominator is equal to zero. If we were to graph an expression like the one above, we would in general get a graph that looks like:

FILTERS

In mathematical and engineering lingo, a FILTER is an object that selects certain frequencies to pass, while blocking others. (To learn more about frequencies, click here). The most common implementation might be a radio tuner. In general, in any city every radio station is being broadcast all the time to every place in its range. So, the antenna in your car is actually picking up every station that is being broadcast in your area -- if you tried to listen to all of them at the same time, it would likely sound like a whole bunch of static.

When you set the dial of a radio to a particular station (106.1 on FM is my fave in the Bay Area), what you are doing is telling your radio antenna to only listen to a particular frequency, as the number of a radio station corresponds to the frequency of wave that is carrying that station's signal. So, when I set my radio dial to 106.1, I am telling a circuit inside of my radio to FILTER out every other frequency that it hears besides 106.1, and so my speakers only broadcast the music that is coming from that station.

The name of such a circuit is a BANDPASS FILTER, since I am able to set a particular range of frequencies, or FREQUENCY BAND, that are allowed to PASS. (BAND-PASS, get it?). Usually, this is implemented by a circuit as shown below; the mathematics of such a circuit turn out to be controlled by a RATIONAL ALGEBRAIC EXPRESSION.


The diagram above is an example of what is called an RLC circuit. The R, L and C are the names given to electronic components called a resistor, an inductor and a capacitor. When desigining a circuit like the one above, we are able to specify which frequencies are allowed to pass by changing the values of the resistor, inductor and capacitor. If we were to analyze this circuit using techniques specified HERE, we would find that the equation governing its behavior is given by the rational algebraic expression:



For the purposes of this lesson, the most important thing to notice here is that this expression has a polynomial on the top and the bottom of our fraction, and that the coefficients of our polynomials is goverened by the values of the resistor R, inductor L and capacitor C.

Algorithms

One of the most interesting and elegant fields of mathematics, particularly as it applies to modern computers and technology, is the study of algorithms. An algorithm is a systematic technique we can use to solve a particular problem, or even better, a whole class of problems. The study of algorithms is often concerned with finding the fastest or least expensive system that will solve our problem satisfactorily over and over again.

In some sense, an algorithm is a little bit like a recipe. Let's say that you wanted to makes some chocolate chip cookies from scratch -- the recipe is the step by step system that will take you from raw ingredients to a batch of cookies. In a sense, the recipe is your solution to the problem of how to make a cookie given a bunch of flour, butter, sugar, chocolate chips, pans and an oven. A recipe might be defined as "good" for a wide variety of reasons: if it uses the cheapest ingredients, it makes the best cookies, or it takes the least amount of time. And a good recipe is also one that we can use to make any amount of cookies we want, not just a pre-set number.

The study of algorithms, however, is not the study of how to FOLLOW the recipe. It is the study of how to WRITE the recipe given a type of problem, and the ingredients and constraints to solve the problem. To be able to solve the problem I posed in the post about Induction from Patterns, you would have needed to develop an algorithm to deal with the cases where the number of diamonds grew too large.

If you ever interview with a software company, they are likely to ask you questions that test your ability to develop algorithms. At a place like Google or Microsoft, they often ask less sophisticated algorithm questions to non-technical applicants to test their ability to think logically. These problems are often in the forms of puzzles or games.

One of my co-workers Mr. Ankamah recently showed me a Flash rendition of one such problem, known as the Missionaries and Cannibals. This is an excellent game since it requires somewhat non-linear and non-obvious techniques to accomplish the task. Also, it is analagous to techniques that are used in computer algorithms that perform very complicated tasks as efficiently as possible.

It is interesting that one of my students, of Mexican heritage, actually pointed out that a particular version of this game that I found (but can't anymore) is really racist, and he refused to play. In this rendition, the cannibals are depicted as dark black people, and the missionaries are white. This game, he remarked, made it seem like black people are savages and white people are helpless spiritual victims. I was very impressed with his insight, and didn't make him play that version.

But I found THIS version, which makes the cannibals look like alien monsters. He told me that this version was racist against monsters but since, I argued, there are no such things as monsters, I did not deem this argument as persuasive, and made him play anyway.

To begin, try the game at the link above, and answer these questions:

RIVER CROSSING PROBLEMS

We will consider in this section three examples of river crossing problems, including the Missionary and Cannibals problems mentioned above.

An often overlooked part of solving problems like these is developing a system to write down the various steps that you are trying to solve the problem, and to begin to find a pattern in the way the steps are executed. In the case of finding the triangles in the Latin American bracelets in Induction from Patterns, we called this process of extending our solution to a single case to the general case, induction. To see how this works, let's start with a different type of river crossing problem, namely the jumping frogs.


http://www.cut-the-knot.org/SimpleGames/FrogsAndToads.shtml

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