In this post we look at several ways traditional textbook publishers are poisoning math instruction — charging insane prices for books, writing lectures for professors (bad ones), and doing a bad job of generating problems for practice, homework, and examinations. We begin with the price issue. A while back, Occupy Math announced he had written a calculus book, Fast Start Calculus for Integrated Physics. The material from the book is being republished as the three books pictured at the top of the post. This is part of Occupy Math’s war on outrageous textbook prices. These books are distributed at low cost to a university and zero cost to students at participating universities.
This post is about some of Occupy Math’s current research. The picture at the left represents a 200×200 cell “ecology”. Each cell in the grid is occupied by one type of simulated critter. The critters compete with one another to take over cells (the rules for this are lower down in the post). The simulation is run for 1,000 seasons and one of the 40,000 creatures is mutated each season to create a new type of critter. We start with ten types of critters, but mutation sometimes drives the number of types of critters into the hundreds by season 1,000. The picture above is the state of the simulation, with different colors representing different critters, in the thousandth season. This project is generating a diverse collection of these small, complex artificial ecologies.
Occupy Math works with digital evolution on a number of projects including evolving parameters that generate interesting fractals. The advantage to doing this with a computer, assuming you can come up with an automatic function that at least sort of measures “this looks good”, is that you can sort through billions of fractals per hour. One of these is shown at the top of the page. The disadvantage is that people are much better than any of the automatic functions we have found so far at spotting cool fractals. If we use people, though, they burn out way before looking a even a paltry million fractals. This is the phenomenon of user fatigue. This post is about a way to let computers and people collaborate on a project, drawing on the strengths of each. Computers can evaluate huge numbers of fractals to find ones that might look good. Humans have a much better ability to judge which fractals are actually beautiful.
The moment none of you knew you were waiting for is here: Occupy Math is making fractals available to all — not just pictures but an app, Newt, for making your own. Our mascot, Newt, is part of one of the fractals (except for the eyes) and is named after the type of fractal he is — a Newton’s method fractal. One of the perennial topics in Occupy Math is fractals. There are a lot of different types of fractals. One of the less well-explored types is the Newton’s method fractal. Occupy Math Productions is now asking for beta-testers for a new Android app that lets you make your own Newton’s method fractals. If you are interested in beta-testing this app, send a request to firstname.lastname@example.org and include your gmail address (they are easy to get if you do not have one already). We are looking for one hundred beta-testers and would really like to hear from you — if you have an android phone or device. Tablets are fine.
This post is the second half of Occupy Math’s report on the last academic year. It focuses on the four senior research projects Occupy Math supervised this year. The “senior research project” is a double-credit course that lets students dip their toes in the pool of research to see if it works for them. One of the students took the course in the fall, three in the winter semester, the semester that just wrapped up. Two of the projects were on very different ways to generate maps, one was on the generation of a new class of mathematical puzzles, and the last was a project in bioinformatics that sought to tune the parameters of an algorithm that generates features for identifying different type of genetic sequences. Three of the students were computer science majors, something that will be happening a lot more as the computer science program at Occupy Math’s home university is expanding. Two of the students are going to start masters degree programs in the fall, one in math, one in computer science.
In the past Occupy Math has used evolution to find cellular automata, maps made with cellular automata, shape changing virtual robots, and control nets for little robots that pick things up. Occupy Math has also had several posts on fractals. This week, we evolve fractals! The hard part of this is getting the computer to have an idea what fractal will look good, and that needs some math.
This week, Occupy Math continues the series What do mathematicians do all day? This post is Part III, which means there must have been a Part II. This week we report on some research on drawing maps with evolution, something Occupy Math did a couple posts ago. This week’s post uses a completely different technology and produces a very different type of map. An example appears at the top of the post. The red room is where the whole thing starts, the blue bars are corridors. Evolution is trying to lay down a lot of rooms — optional extras like doors and monsters are added later. The interesting thing about this post is that the evolved object can generate arbitrarily large maps.
An earlier post in Occupy Math on making automatic dungeon maps talked about using digital evolution to automatically generate maps for use in fantasy role-playing games. In this post, we focus on generating individual tiles that can be used to make larger maps, but with symmetry. The picture at the top of the post shows all the possible ways to pick up a square and put it down so it covers the same area; these are the symmetries of the square and they are the symmetries we will use to construct interesting map tiles.
In an earlier post, Occupy Math introduced a very simple type of virtual robot, a symbot. Symbots have two eyes and two wheels, and their behavior is governed by how hard each eye stimulates each wheel, together with a default level of wheel drive. Occupy Math sets the symbots a task and then uses digital evolution to locate sets of five control numbers that let the symbots do the job. In this post, the symbots are given a harder task, shown in the animation above: pick up all the dots. In the animation a dot changes color when it is picked up. The post also looks at the issues of clustering the robots into groups with different behaviors.
A charge that math teachers have to answer all the time is that their discipline is not useful for anything in the real world. In this post, Occupy Math will use mathematics to show that capricious or chaotic government help gets you a worse economy than no help at all. As Occupy Math has noted before, it is untrue that math does not apply to the real world. In fact, convincing people that math is useless is a tool of oppression. Sun Tsu observed that “To subdue the enemy without fighting is the acme of skill.” If you are sapped of your will to even look at or understand quantitative evidence, then you are half-way to helpless in this complex modern world. In this post, Occupy Math will report on one of the papers he submitted to the 2017 IEEE Congress on Evolutionary Computation that shows how math can be used to make simple, informative models of economic policy.
One of the big results is that unreliable government subsidies are worse than no subsidies.