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.
Occupy Math is going to look at how math helps us design and play games.
Occupy Math has been writing articles for a new journal Game and Puzzle Design which seeks to bridge the gap between professional game designers and academic game researchers. You can order issues here. This is a unique niche and fills a real need. Professional game designers know how to make a game fun and interesting; the academic game designers can apply math and algorithms to solving problems that the professional game designers find during the design process. The academics also study the general space of all games and try and figure out how to classify games. A key point for Occupy Math is this.
Making better games helps us discover new math. Games can build interest in math.
With all the commotion in the news lately, a lot of people are saying “Oh, yeah? Prove it!” (often in an angry voice). This helped Occupy Math to select his topic for this week which is about proof and interpretation. In math, a proof is a series of connected logical statements that draw a line between some things that you assume are true and something else that you are trying to demonstrate is true. The Pythagorean theorem is a good example: “If you have a right triangle then the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longer side.” You are assuming that you have a right triangle and, if you do, the sometimes helpful fact about the side lengths holds.
In this week’s Occupy Math we look at a math problem that can be explained in less than ten minutes — that still stumps every mathematician who has ever looked at it. This problem is also a marvelous place for students to explore and look for patterns (some problems for students are near the end of the post). The problem is based on this rule: “If a number is odd, triple it and add one, but if it is even, divide it in half”. Not a hard rule, but it is the basis of an unsolved problem: “if you start with any positive whole number, do you eventually get to the number 1?” This question has several names. One of the most used is the Collatz conjecture (follow the link for many other names including the very common “3n+1 problem”). A number of prominent mathematicians, including the incredible Paul Erdős, have expressed the opinion that this problem is too hard for us in our current mathematical infancy. The theory also floated briefly, after multiple math departments were consumed by this problem, that it was created by the KGB (the Soviet Union’s spy agency) to stymie mathematics research in the west.
How can such a simple problem be so incredibly hard?