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.
It’s a little subtle, but this is a Mandelbrot set view, rendered using a new convergence condition — a diamond shaped box that the fractal iterator must escape.
The frenetic little bubble animated above is a symbot, Occupy Math’s own name for a type of super-simple robot. These robots exist only in the computer; we don’t actually make physical versions. There is an interesting book about this type of robot and its more complex cousins: Vehicles: Experiments in Synthetic Psychology by Valentino Braitenberg. These robots have sensors and wheels; the input to the sensors controls how fast the wheels turn. The interesting thing is the number of different behaviors that you can get out of even really simple robots. The robot above can sense the flashing light and is trying to approach it. It lacks the ability to slow down, so it’s learned to run the light over repeatedly. Think Labrador puppy.
The week’s image is an animation of 48 different two parameter Julia sets. One of the parameters is changed by small amounts causing the fractal to morph. Like it?
The factorial of a number is what you get when you multiply that number and those smaller than it (down to one) together. That means that five factorial is 5x4x3x2x1=120. The mathematical notation for factorial is to use an exclamation point: 5!=120. Occupy Math was teaching a course that used factorials to count things and one of the sharper students kept getting problems wrong. Occupy Math wrote “5” on the board and asked “what number is that?” The student replied “five”. Occupy Math added an exclamation point to get “5!” and again asked the student what the number was. The student replied “FIVE!” This was a third-year university student — hence this educational post. This week’s Occupy Math looks at what factorials do (e.g.: they count things). Factorials also provide an example of something that grows faster than exponentially.
This week a deep zoom into the cubic Mandelbrot set. Way too much detail!
Occupy Math is going to look at a simple piece of math that is ignored or, worse, abused by researchers in many fields. It amounts to an example of ignorance of statistics that leads to publishing results that are bogus and so impossible to replicate. This problem is called the replication crisis because many important results seem to disappear when other researchers try and reproduce them. Occasionally this is the result of actual fraud — but more often ignorance of simple facts about statistics can let you publish a paper whose results cannot be replicated (because its results are actually wrong) without even noticing you’re doing it. There is also a separate problem — it is very difficult to completely describe an experiment, which means that the people trying to reproduce your results may not be doing quite the same experiment. That last is a big problem, but not what Occupy Math is looking at today.
The core message of today’s post is that peer-reviewed results in a top journal are sometimes wrong because we don’t teach statistics properly.