Where are the Beautiful Julia Sets?

 

In this post we are going to look at a trick for figuring out what a whole type of fractals look like, before doing all the calculations needed to generate them. Occupy Math has posted on fractals before. The clue we will use to search for beautiful fractals is, itself, a very special fractal called the Mandelbrot set. Here is the core fact for today’s post. The Mandelbrot set lives in the complex numbers, which form a plane. Given that, the way the Mandelbrot set looks near a particular place (or number) in the plane of complex numbers is the way the Julia set based on that number looks all over. Another way to say this is that the Mandelbrot set indexes Julia sets.

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Euler’s Map Theorem: a Surprising Activity

Occupy Math’s editor tells him that activity posts must say what age they are for. Because this is a nifty activity that uses graph theory, it has a broad age range. A sharp kid who can count can always do this activity, but the official age range is 7th-12th grade.

Occupy Math has posted about graph theory before. The post “Cookies or calculus?” makes a case that graph theory would be better than calculus as a first math class in university. The post Map Coloring and Conflict Resolution shows an interesting application of coloring graphs. The post on an unsolved mystery used graph theory as well. In this post we look at another property of graphs — diagrammed as maps — that is a good activity for building math skills. It’s also a magic trick that can intrigue students.

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Fancy Math Talk for Move Half Way

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.

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Prime Numbers and Teaching Fractions

Primes

This week’s post is a follow-up to a post objecting to the way fractions are taught post from a while back. This is also Occupy Math’s third post about prime numbers. It should be much more down to earth than the first and, unlike the second, there are no insects or implications for ecology. This post explains why students who know about prime numbers will find it easier to do arithmetic with fractions. Being able to find the prime factors of numbers gives those numbers character: prime factors are like personality traits. Another way to say it is that numbers with a prime factor in common form a sort of a tribe with common characteristics. Also, there are some games you can play with prime numbers near the end of the post.

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Secrets of the Guild: How Occupy Math found one.

The phrase secrets of the guild was used by Bertie Wooster to describe the reason that his manservant, Jeeves, could not divulge the ingredients of his morning pick-me-up. This post is intended to give pieces that could be used for discovery learning about basket curves, a modified type of petal curve. Occupy Math has looked at petal curves before, to make flowers. This post highlights a “secret” in the form of a not-completely-obvious pattern that permits students to make complex and interesting pictures.

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The Cornucopia of Mathematics

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.

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Occupy Math’s Third Set of Holiday Ornaments

Occupy Math, with a third collection of fractal holiday ornaments, has established that such ornaments are a tradition on Occupy Math. If you would like the first or second collections of holiday ornaments, please follow the links. This year’s collection is made from generalized Julia sets (again), but with a very different focus. To make actual ornaments, print on stiff paper and add a hanger or glue the ornament to a styrofoam or wooden disk with a hangar.

 

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There are different sizes of infinity!

infinity

In this edition of Occupy Math we are going to look at a famous mathematical concept, the Cantor diagonal argument. This argument logically demonstrates that there are at least two different sizes of infinity. It also uses a useful logical technique called proof by contradiction which sounds much more contentious than it actually is. The existence of multiple sizes of infinity was discovered by Georg Cantor in 1891, a period in which math was getting not so much real as surreal, where it has stayed ever since.

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More math evolution is good at!

CorridorsIn Evolution Can Do Math That People Can’t, Occupy Math showed how to use digital evolution to create a self-limiting type of cellular automaton (apoptotic cellular automata) that are pretty to look at. These cellular automata also have their own mathematical version of “genetic material” that could be used in breeding experiments — one of which appears in the post. This work is a combination of mathematics with theoretical biology. In this post, Occupy Math will look at Cavern Automata which are used for gamining maps. A picture made with one of these automata appears at the top of the post. The rule that specifies how to make pictures like the one above was evolved, just like the rules for the pretty pictures in the earlier post.

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