The Law of Averages says that if you repeat a random event — like rolling a die or dealing a hand of cards — many times, then the outcomes you observe will be close to the average behavior. A six-sided die, for example, will roll similar numbers of 1s and 6s. This knowledge can cause a problem — people think there is some sort of auditor or physical field enforcing this law. Since there is no enforcement of the law of averages, people have a false intuition, called the gambler’s fallacy that, when a random proccess has wandered away from the average value, it is “due” for outcomes that will bring it back to the average. This is not true — if something unusual happens with dice or coin-flips, then the events downstream of it do not react in any way. They just continue as usual. This belief that the law of averages is somehow enforced leads to a number of questionable behaviours, which this post examines.
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.
In this post we look at depictions of snowflakes, a winter-appropriate topic. This post was inspired by Kele Lampe: thanks! The post continues an earlier one on making fractal faces. The issue of the day is that snowflakes have six-fold symmetry. Searching the internet shows that many people know this, but it is not difficult to find completely unnatural snowflakes, labeled as snowflakes. A few examples appear below. This leads to an interesting point. A snowflake-like object that does not have six-fold symmetry, or occasionally two-fold, three-fold, or twelve-fold symmetry is just plain wrong, but a snowflake-like object that does have six-fold symmetry is not, necessarily, a shape that could really be a snowflake.
In today’s post, Occupy Math will show you a family of puzzles that help you sharpen your logic skills. These puzzles are Occupy Math’s expanded version of some wonderful exercises developed by Peter Harrison called Bovine Math. Dr. Harrison’s exercises are a bridge from arithmetic to algebra, trying to ease the mental transition from concrete numbers to the abstraction of having variables. If you are a teacher unfamiliar with bovine math, definitely follow the link. Occupy Math notices that these wonderful exercises could be formalized as graph theory and found a family of puzzles. This is not Occupy Math’s first foray into educational puzzles that use graph theory. The puzzles in this post are for grades 2 and up, unless you learn to add before that. The larger the puzzle, the harder it is.
This week we have a Newton’s method fractal with a lot of gold lace.
A view in the quintic Mandelbrot set.
This post contains an activity you can do with a calculator, a bit of a magic trick. The post is also about a very special number called the golden ratio. The golden ratio is not as famous as pi or e, but it keeps showing up again and again in multiple contexts. The spiral above is made by choosing quarter-circles that cover two earlier quarter circles, along their sides. After going through the activity, Occupy Math will reveal how this spiral involved the golden ratio.
A Newton-Julia flower with large borders, to achieve a cartoon like appearanace.
This week we look at a mixed type Julia set with a cubic iterator followed by a quadratic iterator. Note the interesting central structure.
This week we take a Newton’s method fractal, with fifteen roots, and show how it forms in an animation. Sorry for the small size — animation uses multiple pictures so it makes big images.