Last Time | Next Time |
Today we're going to review the golden rectangle and Pascal's triangle; then investigate Pascal's a little more, to find another number pattern within. Then we'll take another glance back at the Fibonacci spiral, and a look forward to the world of fractals, which we'll be encountering soon.
Note that the value is only approximately 1.618: is an irrational number, like , and its decimal representation goes on forever and never repeats:
This will allows us to do is to understand how this Chinese culture of the 13th century wrote its numbers (Chinese Bamboo Counting Rods). How did these people write 15? 28? 30?
This is a famous bit of mathematics that some of you may recognize, and we'll study it more on its own right soon.
Interestingly enough, there's a mistake in one row of the Chinese version of this table.
Here's an updated hexagonal version of Pascal's triangle. This one is particularly useful for showing the method for generating all the Fibonacci numbers from the table:
The claim is that there is some relationship between Pascal's triangle and the binary numbers. Let's use the simplified version below to discover this relationship.
So why does this happen? There's a nice way to think about this relationship. We need to understand a concept of sets, however. This concept is very important, and is actually the reason that Pascal was interested in it.
Here's the question Pascal's triangle answer: for a set of objects, say four objects, how many ways can one create subsets of the objects? What kinds of subsets are there, and how many are there of each type? Rather than objects, think of students.
If there are two cars and four students, in how many different ways can we put students into the cars?
Here's how a 10th century AD manuscript described Pingala's approach to the constructing the triangle:
"Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, ..."
(from the marvellous book The Universal History of Numbers, by Georges Ifrah). |
Turns out, that if you do it over and over and over again, you'll get a rectangle that gets more and more golden.
Let's try: we'll use a piece of grid paper (looking "tall"), and a 3x1 rectangle (start it running the tall-way at the 5th row down, 8th cell over). For the next two steps, go "right", "up".
Now, let's figure out what the recurrence relation is for the dimensions of our rectangles, and then figure out if we're getting golden....
"A paper" is something you've no doubt encountered before: it's the long sheets we occasionly use (usually "A4" paper), which is far more common in Europe.
"A paper" is constructed so that, if folded in half (do we say the long way or the short way?), you get a sheet which has exactly the same shape -- that is, the ratio of its side lengths is the same as the original sheet.
Let's see what the dimensions of A paper must be so that this requirement is fulfilled....
Call the long dimension of the A1 paper above , and the short dimension 1.
As you can see there's "a world within a world". This is a fractal, as is this spiraling version of the Fibonacci spiral: