This week on Occupy Math, we look at Venn diagrams. Most students encounter these early in their education and they seem pretty simple. In this post we not only look at the math, but humor and the use of Venn diagrams as a creative prop. We will also see that there are still graduate research level questions about Venn diagrams. A Venn diagram takes different sets of objects and diagrams them as circles — or other shapes — with objects in both sets appearing in the intersections of the circles. Optionally, the area of different parts of the diagram can give you information about the size of the set of things in that part of the diagram. The example at the top of the post is about the “positive whole numbers”. This is called the universal set, which is fancy talk for everything the diagram covers. The special sets shown in the diagram are the even numbers and the prime numbers. Since all the primes except 2 are odd, the only thing in the intersection of the even and prime numbers is 2, which the diagram shows.
Thousands of years ago, on the shores of the Aegean Sea, a man named Eratosthenes wrote out the numbers 2, 3, 4, 5, 6, … History does not record where he stopped. He skipped 1 because it is strange and special. He circled two and crossed off every other number. The next number not yet crossed off was three. He circled three and crossed off every third number. He kept at this, and the circled numbers were the primes: 2, 3, 5, 7, 11, 13, 17, 19, and so on. His technique for finding prime numbers is called the Sieve of Eratosthenes and it represents the beginning of our species’ fascination with strange patterns in sequences of numbers. There is an enormous repository of knowledge about sequences of numbers that, like the primes, mean something, even if we do not always know what they mean yet. That is the topic of today’s post.
This week’s post is about the Online Encyclopedia of Integer Sequences (OEIS). This website is an incredibly powerful tool for research mathematicians, but it can also be very helpful to students or people trying to solve simple counting problems. It is a searchable list of counting sequences. The picture at the top of the post is a pictorial demonstration that the number of ways to fill a 2-wide rectangle with dominos can be done one way with one domino, two ways with two dominos, three ways with three dominos, and five ways with four dominos. The sequence 1,2,3,5 are the first four terms of the counting sequence for the problem of filling a two-wide rectangle with dominos. If we type “1,2,3,5” into the search bar of the online encyclopedia we get:
A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.The cool thing is that the Fibonacci numbers are the counting sequence for the dominos-in-a-2-wide rectangle problem. There are eight ways to fill a 2×5 rectangle with dominos, which is the next term in the Fibonacci sequence. The “A000045” is the serial number of the sequence in the encyclopedia. But that’s not all we get! Click through the link, clear the search bar, and type in 1,2,3,5, and scroll for a while. The encyclopedia lists a huge number of other things that the Fibonacci numbers count. After that list are references that source the information and relevant web links.
Welcome to the first Occupy Math Extra, updating the original release of NEWT: fractals for everyone!. The android app Occupy Math has been developing has now gone to early access status on the Google play store. You can play with and design your own Newton’s method fractals.
We can still change things. Suggestions and comments can be made in the app or to email@example.com.
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics
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.
This post is about a mathematical notion called self-organized criticality and its relationship to a recent strike by Amazon workers. As we will see, this strike, on Amazon Prime day, was a small “avalanche” (in a sense defined later in the post) that is probably a harbinger of worse problems at Amazon warehouses. Amazon has created a digital work supervision system that tries to maximize productivity by automatically tracking the workers and telling them when they are not working hard enough. The theory of self-organized criticality suggests that, initially, using this system will permit them to find the most productive workers. It will also cause a workplace that is very stressful with high turnover. This is the claim that Amazon workers are now making to explain why they are striking. This situation forces workers to achieve “high productivity”, but also stresses them to breaking point. In the rest of the post we explain self-organized criticality — which actually appears in many systems. Hopefully this will let you recognize it and perhaps avoid doing it to other people.
This post is themed “look at the pretty pictures”, with an attempt to make summer appropriate pictures. It is a look at some of the things that math can do. The secret for making Mandelbrot sets look like bouquets of leaves is featured. This is something that Occupy Math discovered by accident, a nice example of when a code bug is really a feature. The Mandelbrot set is an excellent source of complex pictures. The Ghost Mandelbrot set shows how you can switch up the details and get new fractals. In this post we will apply quadrant convergence to the Mandelbrot set and use a green color scheme to get leafy-looking fractals.
Have you ever tried to enlarge a picture and lost detail? Maybe individual pixels became big single-color squares? The skull to the left is made of circles, ellipses, and rectangles, glued together as a series of tests. For a point in the plane, Occupy Math’s computer checks which of these geometric objects a point is inside of or outside of and, from that, deduces a shade: black or white. This may seem to be a lot of trouble to go to for a sorta-okay picture of a skull. The reason for going to that much trouble is that membership in the black parts of the skull, since we are asking about numerical points (x,y), has a resolution of trillions of pixels by trillions of pixels. This means that if we warp space, we will not get pixelization errors or other distortions. In other words, this skull image is far more malleable than usual images stored as pixels. In this post we are going to apply space-warps to the basic skull image. Having explained about the skulls, it is worth noting that Occupy Math created all this stuff while playing around but the math used to make many different skulls turned out to have applications in statistics. Following the fun parts of your work not only relieves stress, it can have big payoffs.
A combinatorial explosion is a situation in which, as you increase the number of things in a situation, the number of possible configurations of those things increases incredibly. Look at the flowers in the picture and consider how they are arranged. The number of arrangements that could have happened is beyond belief and the arrangement in the picture will not occur again before the heat death of the universe, even if flowers sprout in that field again, every spring. Phrased that way, this may sound a little intimidating, but combinatorial explosions can be useful, which is what this post is about. Combinatorial explosions are one of the more interesting outcomes of enumerative combinatorics.
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.
One of the things that Occupy Math works on is automatic content generation, using algorithms to generate content for games. Occupy Math’s colleague Julian Togelius posed the following question: “Can we find a space of parameters where all or almost all the ways we fill in the parameters make a good game?” In this case “parameters” are numbers like the board size, number of playing pieces, or numbers that specify rules of a game. This is a very difficult challenge and the other people in the room softened it to finding spaces where there were lots of useful elements for games. In this post we look at a new use for Voronoi tilings in generating game content.