This is a fourth-power Mandelbrot set view — the neon jungle.
Occupy Math has already taken a shot at explaining what fractals are. He has tried to supply engaging holiday fractals. On of the least popular posts is on the incredible complexity of the Mandelbrot Set and there is a post on making family trees of fractals (and other things). What’s left? This week Occupy Math is going to turn up the weird to eleven and use fractal algorithms as lenses — a different type of lens from the one shown at the top of the post. The only thing you really need to know about fractals to get a sense of what is going on is that a fractal is based on an algorithm that moves a point around in a complex way until it is captured. The details of the algorithm and the conditions for “capture” give you the shape and then you also need a coloring algorithm. Today’s post is all about a really odd way to color fractals.
This is a shot from the galactic cluster part of the Mandelbrot set using wedge convergence.
One problem that Occupy Math has in teaching his first year courses is that many of the students have been trained, by their high school experience, to believe that a math class is a game that is scored in points with the goal of a grade. If it weren’t actually true in some of their high school classes, it would be nonsensical, and the whole notion is counter-productive to the goal of getting an education. Every year, Occupy Math has some students who are trying to do just enough work to pass the course, no more. Many of them flunk because a math course builds technique upon technique. What appears to be the correct level of effort for a D near the beginning of the term is actually preparation for an F or an F–. These grade-management tools from high school are also used as fear-management tools; instead of engaging with math, the student tries to scam a passing grade and so avoids the math. This has all sorts of bad downstream effects. In this week’s Occupy Math, we want to look at the issues of effort, fear, and effective teaching and learning.
This week we have a generalized Julia set with the interior of the set colored using a shade of yellow and the cosine coloring algorithm. This yields a pleasant, metallic appearance.
Today’s post looks at the following problem. Color a plane (an infinite flat surface) so that any two points that are one unit apart are also different colors. The picture above is an example of such a coloring, with two caveats. The black borders are there to help you see what is going on (remove them to get the actual solution) and you have to continue the pattern indefinitely. The goal is to use as few colors as possible. This smallest number of colors that meet the goal is called the chromatic number of the plane. The formal name of this problem is the Hadwiger-Nelson problem. This problem is famous, in part, because much of the progress on it has been made by amateur mathematicians. The professors ended up needing a lot of help on this one. We also still don’t know the final answer to this problem. Occupy Math will go over what we do know.
This is a context shot on a meandering fractal river in the Mandelbrot set. It uses a new type of convergence called wedge convergences that deems a point to have diverged when they move into cones with a vertex at one of (1,1) (1,-1) (-1,1) and (-1,-1).
Have you ever been in a conversation that made no sense at all until a key fact showed up, often more or less by accident? In this post Occupy Math is going to reveal the central goal of the field of mathematics. If you don’t know what math is striving for, if our motives are obscure, then understanding collapses — like the arch shown above would if you removed the green keystone. The keystone of math is the search for patterns, commonalities that unite diverse topics and situations. For many people, math is a chore or a terror with much drudgery and ever-present fear of being judged. Occupy Math hopes that by presenting the following perspective on the purpose of math, math will become less scary for those of you who are not so sure about us.
This week’s image is a deep zoom into the Mandelbrot set using a diamond-shaped convergence set. The positive and the negative space are interesting on this one.
This week in Occupy Math, we proudly announce a book published by Dr. Andrew McEachern, GAME THEORY: A Classical Introduction, Mathematical Games, and the Tournament. Game theory is a formal structure for studying and resolving conflict and encouraging cooperation that rephrases cooperation and conflict as a game. Andrew developed and taught a course in game theory taken by advanced students from many programs while he was at Queens University. This book is a text based on the course he taught and it is part of an effort to bring textbook prices under control. The book introduces the classical analysis used in game theory — his exposition of The Lady or the Tiger is wonderful — but Andrew also introduces material outside of the standard game theory fare. These include the math behind the fraction teaching game that Dr. McEachern and Occupy Math are developing and techniques for designing fair, balanced tournaments for anything from Prisoner’s Dilemma to Basketball. The book is a text for a course for non-majors that nevertheless has a solid mathematical foundation. We now ask Dr. McEachern a few questions about his book.