- You have a homework due next time.
- I've assigned a new homework, due next Thursday.
- Last time we talked about spirals -- Fibonacci and Golden. What questions
do you have?
- We discovered that the Fibonacci spiral leads to rectangles that tend toward a common side ratio of about 1.618....
- The Golden Rectangle, defined by the Greeks, has a side ratio of
$\frac{1+\sqrt{5}}{2} \approx 1.618$ called the golden ratio.While the Fibonacci spirals were created by attaching squares, the golden rectangle was created by removing squares:
Both had roughly the same effect, however.
- Now, back to Pascal's triangle. Let's look a little further into it. In
particular, we note that it contains the Fibonacci numbers:
- Use the hexagonal paper to create a version of the Pascal triangle.
- Now let's find those Fibonaccis....
- So we find all kinds of numbers in Pascal. But what's Pascal's triangle all
about? Why was it created?
- First of all, p. 128 gives us a bit of the history of "Pascal's"
triangle, studied as far back as the 5th-3rd century BC in India --
that we know of!
- Pascal used "his" triangle to compute probabilities. It turns out
that it's very useful for counting things. In your text we learn that
Blaise Pascal and Pierre de Fermat were working on "Number Theory"
together (p. 139).
- Your textbook demonstrates that Pascal's triangle is useful for
counting the ways of combining terms in the expression
$(a+b)^n$
So, for example,
$(a+b)^3=1a^3+3a^2b+3ab^2+1b^3$
(see p. 128, the binomial theorem)
You'll notice that the coefficients of the terms are the numbers from one row in Pascal's triangle.
- First of all, p. 128 gives us a bit of the history of "Pascal's"
triangle, studied as far back as the 5th-3rd century BC in India --
that we know of!
- Now let's take another look at that simple card game from last time: I
want to give Aaron a chance to win $2.
- Now let's look at an example of how we can use Pascal. Consider the
concept of simple graphs, which I'll present in the context of Facebook. A
simple graph is made up of
- vertices (or points, which we will think of as people), and
- edges (which are also called arcs, connecting two points -- which we will think of as friendships).
Now, if we don't care about "who's who", then we can see all the different types of Facebooks we can have:
- Let's look at some simple cases: 2 or 3 individuals
- All simple graphs with four vertices.
- All simple graphs with five vertices.
- Now, if we add names to the "vertices", then we have more
Facebooks. If everyone's friends with everyone else, then the graphs we
can get will look like this:
Now how many different Facebook configurations are there for, say
- $K_2$?
- $K_3$?
- $K_4$?
