A deep zoom into the cubic Mandelbrot with color and more color.
In this week’s post we look at a game for practicing recognition of multiples of a number. It can be used in class by a teacher or it might be a good game to play with your kid by way of some math practice. The game prepares a student for multiplication by practicing how to recognize which numbers are multiples of five. The numbers 5, 10, 15, and so on generate scores when they come up in the course of the game. There are multiple versions of the game (for other numbers besides five), but the demonstrations are based on recognizing multiples of five. The game is a little like dominos and it’s a little like a crossword puzzle.
A Julia set with an added spin component. Whee!
Michel Mayor was one of two scientists awarded the Nobel prize in physics for finding the first earth-like planet outside of our solar system. This is one of the more exciting parts of discovering thousands of new planets, including the seven-planet Trappist system, forty light years away, diagrammed above. He gave an interview which has been widely distributed in which he said we will not colonize another planet as a way to deal with Earth becoming uninhabitable. The reason to write a blog about it is that what he is saying is both critically important and almost certainly wrong. The problem, as it often is, is a lack of nuance that can be restored by looking at the basic math. In this case the key phrase is “we will colonize” — the idea is problematic.
Lurking deep in the Mandelbrot set…
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.
Golden medusa in the fifth order Mandelbrot set.
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.
This week we have a combination of Newton’s method and Julia iterators that yield a type of object called a feather snake. The pallet is chosen to yield a number of contrasting regions.
This week on Occupy Math, we have a new puzzle to help make arithmetic practice go down more smoothly. Look at the array of numbers above. To play the puzzle you take groups of three numbers, either in a row or in an corner-shaped group of three squares. Examples appear below the fold. The numbers in the example above are chosen so that the groups of three numbers, when added, include all the numbers from zero to twenty seven. The goal? Gotta find them all. Warning: if you write down a 4×4 array of digits at random, then your odds of being able to get all the numbers are bad. Occupy Math used a design trick to create that example.