This post is the second on the game FRAX and spends some time explaining how the game works and where it came from. The first post was on Occupy Math’s sister blog Dan and Andrew’s Game Place. FRAX seems, based on our initial testing, to be a fun game in addition to giving the players practice with fraction arithmetic. To get the rules to FRAX (and if you’re interested in testing the game), click the link! We are giving away FRAX sets to people who will help us with the play testing. FRAX is a card game, not a computer game, though we have some thoughts in that direction.
This week in Occupy Math, we proudly announce a book published by Dr. Eun-Youn Kim, On the Design of Game-Playing Agents. This book gathers, summarizes, and extends work on how much the way you choose to represent software agents in your code changes their behavior. This is a research project that has lasted a decade and is still going strong. The original publication suggested there are problems with thousands of published papers (they did not control for the factor “how were the agents coded?”) and, in the intervening decade, Dr. Kim, Occupy Math, and a dozen collaborators have drilled deeper into this discovery, finding many more factors that need control. Frankly, while this is high-impact work, its also a wonderful illustration of the way mathematicians end up cleaning up after other researchers and putting their work on a firmer foundation. To be clear — this is a huge clean-up and it addresses some serious problems.
Occupy Math will set some context by explaining the big story that Dr. Kim’s work covers. The original discovery was that experiments that were modeling cooperation completely changed their outcome when we changed the type of code learning to cooperate. One encoding, called a finite state machine, learned to cooperate quickly. Another, called an artificial neural net, almost never cooperated. A large number of published papers did not consider the choice of agent encoding at all — they just picked an encoding. Initial reaction to the work was somewhat extreme — denial and some attempts to ignore the work. After that the good researchers took Dr. Kim’s result on board and the smart ones noticed that it meant they got to publish their work again in a new, improved form. In the rest of the post, an interview with Dr. Kim and a few thoughts on the impact of the work.
This week’s post takes a look at the most usual way that you can become a professional mathematician: going to graduate school. The first university degree in mathematics gets you ready for a lot, working for a bank, starting up the ladder of the actuarial profession, or becoming a codebreaker for the military. Because the advanced reasoning skills are useful, you can usually find employment as a system administrator or as a manager of some sort. Someone who has been trained in mathematics gets an edge in many careers. This post is about moving on to the level where your job is to invent new mathematics.
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.
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.
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.
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 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.
Occupy Math often tries to find click-baity titles for his posts. This week is not an exception, but it is unusual in that a phrase that Occupy Math heard more than ten times at the IEEE 2017 Congress on Evolutionary Computation in Donostia-San Sebastian Spain last week. In other words, the curse of dimensionality is a real thing. This week’s post looks at the very strange behavior of normal-seeming objects when we create higher-dimensional versions of them. This strangeness often corresponds to problems getting much harder as we increase the dimension, hence the use of the word “curse”.
Big issue – your intuition is shaped by two-dimensional and three-dimensional objects and is just wrong in higher dimensions.
This week Occupy Math looks at Newton’s method. If you have a nice smooth formula, like almost everything you get in a calculus class, then Newton’s method starts with a guess at a number that makes the formula equal to zero when you plug it in. Applying Newton’s method improves the guess. If you perform the calculation that Newton’s method specifies over and over, then the guess gets better and better. The place where a formula is zero is often the solution to some problem and so Newton’s method is very useful in science and engineering. Returning to a common theme in Occupy Math, the standard education about Newton’s method does not cover the fact that it is also an artist’s tool.
Newton’s method is a key scientific tool and also a super-powered paintbrush.