Newton’s method fractal with fifteen roots!
Newton’s method fractal with fifteen roots!
In an earlier post, Occupy Math introduced a very simple type of virtual robot, a symbot. Symbots have two eyes and two wheels, and their behavior is governed by how hard each eye stimulates each wheel, together with a default level of wheel drive. Occupy Math sets the symbots a task and then uses digital evolution to locate sets of five control numbers that let the symbots do the job. In this post, the symbots are given a harder task, shown in the animation above: pick up all the dots. In the animation a dot changes color when it is picked up. The post also looks at the issues of clustering the robots into groups with different behaviors.
From the Newton’s method collection, Occupy Math presents a fractal heart!
This is another in Occupy Math’s series of activity posts. A bouquet puzzle presents some bouquets of flowers, together with their price, and then presents another bouquet with an unknown price. These puzzles are intended for 4th through 8th grade, though they can be fun at any point after a student masters addition. This post starts with an example of a bouquet puzzle. These puzzles also prepare students for abstract reasoning and for solving simultaneous linear equations, an important skill that appears from business to calculus. In this post we will give several examples of bouquet puzzles and also give a technique for creating your own puzzles that yields puzzles that can be solved.
Many of Occupy Math’s images-of-the-week have used the Newton’s method technique to make the fractal, but Occupy Math zooms in on an interesting bit. This weeks image of the week is the full fractal with a view window that includes all fifteen polynomial roots used to generate the image. Notice how different parts of the fractal space have different appearances!
This post introduces some of the laws of counting which are the foundation of a field of math with the cool name enumerative combinatorics. Pictured above are the twelve different ways to put five squares together, joining along full faces of squares, to make connected shapes. The fact that there are twelve of these is an eternal truth, but a pretty minor eternal truth. These objects are called pentominoes (five-square dominos) and they were first counted by exhaustion (keep trying to find new ones for a long time). The proof that this is all the pentominoes is tedious, but this is all a nice introduction to the idea that counting things is a field of mathematics. In this post we will introduce some of the simplest — and most widely used — laws of counting.
A white monster fractal from the Newton’s method gallery.
Today’s post is about a type of machine learning algorithm called clustering. The key point is this — clustering and related algorithms create systems with powerful predictive power limited by the quality of the information used to train them. They can do good or evil and, if we feed them the wrong starting information, they can do evil by accident.
Clustering is any of a large number of algorithms that take data and attempt to group it into collections so that each collection is made of similar things and different collections are made of different things. This sounds pretty straightforward, but it is not. The picture at the top shows data points that measure three sorts of irises (the flower): petal lengths, sepal lengths, and other measurements. The panel on the left shows the result of clustering, the one on the right shows the true species classification: the algorithm messed up! Clustering is not only hard to do well, it is a core technology in data science. That means it will probably affect your life.
This week, a Newton’s method fractal that shows what can happen when a couple of roots are close together. Spider temple?
F. N. David is the professional version of Florence Nightingale David’s name. Her parents were friends of the famous nurse and named their daughter after her. This name proved prophetic. An earlier Occupy Math chronicles how the earlier Florence Nightingale saved thousands of times more lives by inventing and applying new sorts of statistical analysis than she did by her nursing efforts in the Crimean war. F. N. David became a professor of statistics when this was almost impossible to do as a woman. Her success seems to have included luck, force of personality, and a deep well of talent.