Animatrix 2003

# Animatrix 2003

The animatrix is a collection of short stories with elements of the matrix. Quit artistic, each has its own animation style. All based on stories by the Wachowski siblings. This one is `Kid's Story' directed by Shinichiro Watanabe. A school scene with some math content about linear difference equations. I just entered what I saw on the board into google and found that the text has been taken from these mathpages of Kevin Brown who is one of the pioneers in math blogging. The movie stops after "For example, the sequence tha...", but complete in the mathpages as "It can be shown that this same summation applies to any (convergent) linear recurring sequence with the initial values 0,0,...0,1,... (assuring the characteristic polynomial has distinct roots). For example, the sequence that satisfies the 3rd order relation sn = 3 sn-1-5sn-2+7 sn-3 consists of the values 0,0,1,3,4,4,13,47... and the characteristic polynomial is f(x)=x3-3x2+5x-7. " This text is a bit inappropriate for a general high school. The concept of solving general linear recursion difference equations of arbitrary order is mostly done in college and also then not in this forbidding way (but it is great for the story and similarly funny like the cubicle work in the software company, in which Nemo worked). The math could work in high school like for the very special linear recursion, the Fibonacci sequence Fn might appear. When I was in high school, our Math teacher Roland Staerk challenged the class to find a formula this famous sequence 1,1,2,3,5,8,13,21,... We tried but nobody could find the Binet formula, a formula that can also be found without linear algebra: with an Ansatz Fn=xn, the Fibonacci relation Fn+2=Fn+1 + Fn becomes xn+2=xn+1 + xn=0. The reason for the name characteristic polynomial is that this produces the characteristic polynomial x2-x - 1=0, which has the golden ratio φ and its inverse 1/φ as solutions. The general solution is therefore Fn = A φn + B/φn which is the Binet formula by fixing the constants with n=0 and n=1.