This post is on a really easy method to solve some hard problems, including locating a pretty fractal. It is called the bisection method, but you may know it as a high-low game. One person thinks of a number and the other guesses. The person who knows the number replies “high” or “low”. A good strategy is for the guesser to move half way between their most recent high and low guesses. Bisection is fancy math talk for “move half way”. The thing is that this method can solve hard problems and do COOL things. Read on to find out.
Occupy Math is a working researcher and, at least the way he does research, this means that people come to him with problems. In this week’s post we will talk about the type of problems that come up and what the impact of math on the process of solving problems is. The three examples we will use are the design of a wood burning stove, a 3D ultrasound scanner, and a weird abstract computing problem solved with ideas from a science-fiction novel. In all three examples, the point will be that stirring in math to the process helps. The solution became either much simpler or much more useful because math was present and accounted for.
Occupy Math has talked before about complex systems. Few human systems are more complex than the economy of a large country like the United States of America. This post reacts to the recent proposed tax bill in the United States — a very complex collection of modifications to a super-complicated system. Since the US economy is pretty much beyond human comprehension, Occupy Math will start with a much simpler complex system. This relatively simple example builds on an earlier post Evolution Can Do Math That People Can’t. That post was about evolving cellular automata. The way cellular automata work is that we start with a row of zeros, except the middle three numbers in the row are 1,2,1. The numbers can go from 0 to 7. We sum blocks of five adjacent numbers — getting a sum from 0 to 35 — and look up the new numbers using the table of 36 numbers:
So, for example, if the five numbers in a block sum 15, the new value for the number in the center of the block is “3” because the 15th number (starting counting at zero in the list above) is “3”. We apply the rule over and over to generate rows of new numbers — mapping the numbers to colors to draw a picture. The color map is zero;white, one;red; two;green, three;blue, four;yellow, five;violet; six;cyan; and seven;black. The picture for the rule above is:
Here is a simple experiment in complex systems: change one number in the list of numbers that forms the rule for the cellular automata and look at the picture that the new rule draws. There are 36 examples below the fold.
In this post, Occupy Math tackles the thorny issue of diversity from mathematical perspective. Many of us take it as given that diversity is a good thing — while some of us think it is a plot to destroy our culture — but there is a mathematical perspective from which diversity is intrinsically valuable. When a crisis comes, it is difficult to predict what qualities are needed to deal with it. This means that a diverse population hedges its bets while a homogeneous society is placing an all-or-nothing bet on their survival. This is why the genetic diversity of an endangered species is of such concern to conservation, but similar principles apply to everything from problem solving to human society. This even shows up in our corporate culture where believing in a “one true way” of thinking about or solving problems can be a barrier to progress.
There is an inaccurate stereotype of millennials as lazy, entitled, self-absorbed lumps who live with their parents. When Occupy Math’s editor graduated from Bard College, it took her a couple of internships and several years to find a good position. Her grandmother could not understand why finding a job took so long and was concerned that lack of zeal might be the problem. What’s actually going on is that good jobs are far rarer than they used to be and affordable housing is also really hard to find, hence living with their parents. This inaccurate stereotype is an example of a smooshing error. This is an error where you combine a whole bunch of diverse factors (job availability, housing prices, the impact of automation and the internet) into one thing in your head and then map it onto your own experience, drawing a wildly incorrect conclusion. In the late 1940s (when the above-mentioned grandmother was hunting for a job), inability to find a position probably would have required laziness. It was a different world then.
In his awesome Discworld Series, Terry Pratchett often included twisted, hysterical versions of modern science and scientific myths. One of my favorites among these is the quantum weather butterfly. In this post we are going to talk about the famous butterfly effect and the odd notion of deterministic chaos as well as one of the biggest human misunderstandings of an obvious mathematical truth.
When people figure out how bad something is, they usually try and figure out how much changing one thing changes another and then use that to calculate the estimated impact. If you raise the price of a car model by $500 and sales drop 0.5% then you estimate a $1,000 dollar increase would drop sales about 1%. If several things are changing, you check the impact of each one individually and add up the results. In Occupy Math’s calculus+physics class we actually work out (as part of error analysis in physics) when this type of estimate works. The answer is when the factors either don’t interact or when the changes in all factors are very small. Most changes are pretty small so that means this “calculate them individually” approach works pretty well. In this week’s post we look at situations where this method of estimating impact fails catastrophically.
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.
Humans have a natural ability to do logic predicated on learning rules from their environment. Training in mathematics improves your logic — and makes you more effective in an argument (at least the polite sort). It’s often humorous when somebody does not follow those rules. In the show Parks and Recreation, actress Amy Poehler plays the director of the Parks and Recreation department in a small town in Indiana. At one point a citizen, at a town meeting, says this. “I found a sandwich on a bench in one of your parks! (pause) Why wasn’t there any mayonnaise on it?” Something that sounds like a complaint about littering turns in a completely unexpected direction. Occupy Math starts with this because the fact that this condiment twist in the citizen’s complaint was funny means that there is hope in a moderately awful situation.
This week’s Occupy Math is venturing again into the sociology and politics of education in the service of addressing a pressing problem. There are places where Occupy Math managed to wedge in a little math — and there are tips on self-defense for those of you who might pay for educational opportunities. The basic thesis of this post is that trying to run education like a business degrades education and also fails at the normal goals of business. The basis of the problem is that education is critical, people really want it, which means the demand for it is inelastic (doesn’t change much with price), which in turn means price can get totally out of control. This also makes education a fertile ground for con-artists, who are always willing to exploit people who really want something.