Polyhedra! -Jordan

This lab is all about polyhedrons and investigating their properties. This is the only chapter that does not use the computer to view data. This instead requires you to use models to help you visualize and and think about the observations that are made.

Some of the questions that are explored include finding the diameter of each of the regular solids if the edge length is one and, “what is the diameter of each of the regular solids if the faces each have area 1?.”

A reason why this topic is of interest to me because it does involve geometry which is a subject that I enjoy. Also the fact that this lab does not use the computer means to me that it will be somewhat hands on which interests me a well.

Iteration of linear functions was the first lab we covered.  Towards the end of the lab, I felt like this was a precursor to fractal geometry.  After all, the Mandelbrot set is based on an iteration of a quadratic function with the use of complex numbers.  Chapter 14 deals with iteration of quadratic functions.  This is one step closer to fractal geometry…..and a precursor to chaos theory, both of which are very interesting to me.

Early definitions in the chapter are:

• fixed point:  given a function $f(x)$,  a point $u$ is a fixed point of  $f$ if $f(u)=u$.
•  attractor:  a fixed point is an attractor when all nearby points move towards it under iteration.
• repeller:  a fixed point is a repeller when all nearby points move away from it under iteration

Here’s a type of question from the chapter:

Given a function $f(x)=ax(x-1)$, how do various values of $a$ affect fixed points, attractors, repellers, and zeroes.  What about changing initial values?

Marc and Prime Numbers

A lab that I found interesting in the textbook was the Prime Numbers lab. Prime numbers are numbers that cannot be divided by any number other than one and itself. Because any number can be written as a product of prime numbers, prime numbers are basically the building blocks of arithmetic. Even though one cannot be divided by any number other than itself, it is not considered a prime number due to mere convenience because every number would have one as one of its prime multipliers which would be tedious. Yay, simplicity!

One question in this lab involves finding a proof that there are infinitely many prime numbers. This has always been a complicated question in the realm of mathematics!

Because prime numbers are such a fundamental and important group of numbers, I find it interesting to learn more about them for it always leads to learning more about the natural numbers in general. I can’t wait to see what labs others are interested in on Wednesday!