The post explains, with animation, a mysterious fact about bacteria, using simple math. Along the way, we look at the way that capitalism ignores a whole class of cures, including cures that would spread themselves, because they are too hard to profit from. Occupy Math thinks letting people die to make more money deeply misunderstands what is actually worthwhile, something discussed more below. Here is the mysterious behavior that is the starting point. If you treat bacteria with a drug, they may become resistant to that drug, forcing you to use a different drug. Bacteria challenged by the second drug can lose their resistance to the first drug — much faster than if they are not challenged at all. This observation led to some theories about how one type of drug resistance was hostile to the other. That is not what is going on, and the truth is actually either very simple or a little weird, depending on how cool you are with mathematical ideas.
There is a problem with bacteria becoming resistant to antibiotics. This ability relies on conjugation, the bacterial analog to sex. Bacteria do not keep all their genes in their nucleus. They also have little circles of DNA called plasmids. Bacteria — even different types of bacteria — can exchange plasmids, like kids with collectible cards. When the bacteria reproduce, they duplicate their plasmids and assign them randomly to their two daughter cells. This means that the number of each type of plasmid changes randomly as various bacteria live, die, and divide. This is a situation described by a mathematical concept called a Markov chain.
Occupy Math’s 200th image of the week is an animation within the 252-Mandelbrot set. Forgive the zoom at the end — the action was slowing down!
Occupy Math returns this week to the task of finding interesting views in the Mandelbrot set. In this post we will use search masks, which is a grey-scale images, to say what sort of fractal we want to see. This is much easier than setting lots of numerical parameters and makes searching for specific types of things much easier. Occupy Math’s editor makes the following analogy. Suppose you are in a huge bakery and want pie, but there are cookies and baguettes and cakes and Napoleons and other things in all directions. A search mask is like a pair of glasses that can only see pie. There are lots of examples below the fold; the next paragraph is pretty mathy and can be skipped if you just want to see the search controls and the fractals they find.
The Mandelbrot set is a subset of the plane that is found by a simple algorithm. Treat a point in the plane as a complex number. Use that point as the starting number in a sequence. To get the next number in the sequence, square the current number and then also add in the number you started with. If this number never gets more than a distance of 2 away from (0,0), also called the origin, then the point is in the Mandelbrot set (the black parts of the picture, above). Otherwise, we choose the color for that point based on how many steps it took to first get more than 2 away from the origin. That number of steps is called the escape number for the point. The topic of this post is a really simple technique for (partly) controlling the search for cool pictures inside the Mandelbrot set.
This is a special Occupy Math on the epidemic and preparedness. Let’s start with a few relevant facts — or at least what we think are facts at the moment.
- A person with the virus sheds virus particles for 3 to 4 days before they get symptoms. “I feel fine” means nothing.
- Soap and mechanical agitation (washing your hands properly) break apart viral particles. Hand sanitizer does not do this nearly as well.
- There is currently a group in the media that is trying to pretend that COVID-19 is just a bad cold. They are contradicting every public health official in the entire world. This is a deeply serious thing.
Occupy Math has published in epidemiology — and even won a prize for best paper at a conference with one of his epidemiology publications — but he has specialized in providing tools for real epidemiologists. Everything here is Occupy Math’s best guess. Please, please follow the directives of health professionals where you live.
A view of a peninsula in the sea of the conjugate 323-Mandelbrot set.
One problem with math education is that access is very uneven. In theory, the internet can level things out a bit. This post is the next in Occupy Math’s series of activities that parents and teachers can use for enrichment and enhancement. For free.
Here is an example of a “guess-the-next-number” puzzle — “what is the next number in the sequence 2,5,8,11,14,?” The answer is 17. The student should figure out that the terms of the sequence increase by three every time, and 14+3=17. This post is about constructing this sort of puzzle with some notes on how to make harder and easier puzzles. Puzzles like this are good arenas to practice math skills. They can be structured as contests which is motivational, and with the information in this post, they can be tuned to your student’s needs.
A conjugate 323-Mandelbrot fractal from the river delta region.
An end-of-semester ritual in most university classes is filling out evaluations for your professor. One of the justifications for this is the assertion that students are the university’s customers. This reasoning is badly flawed, even though the students — or their parents — are shelling out serious money to pay for university. The key difference is this. The quality of customer service when you are buying a new coat or having a meal in a restaurant has almost no consequences for the customer in the future. Even if the winter coat is flawed, purchasing another one is not difficult, and it is often possible to return the bad coat. You cannot return a flawed education. In fact, you probably will not notice the flaws until much later. A better model would be that of the relationship between a doctor or dentist and their patient — trust from the student and professionalism by the professor are required. Treating evaluation of the quality of university instruction based solely on student evaluations turns out to do damage to both the students and the professors.
A dual Julia flower garden fractal!
This week Occupy Math returns to the topic of a post on both digital evolution and computer art: Evolution can do math that people can’t. In this post we will look at shaping the digital art with a technique inspired by the human immune system. These artificial immune systems are another nature-inspired algorithmic technology, like digital evolution. Julie Greensmith taught Occupy Math about these techniques.
Normally an immune system fights disease. The way you use artificial immune systems is to declare something you do not like to be a disease. This is, of course, a bit arbitrary and much simpler than a real biological immune system. The action principle of the artificial immune system is negative selection, the process of removing things we do not like. This post is about some recent work where we used negative selection to pick which sorts of images our computer art system finds by removing the others. One of the pictures generated by the original code, before we added the artificial immune system, appears at the top of the post.