This image is a two Julia, fourth power newton hybrid image with quadrant convergence, which is what causes all the feathered snakes.
This week we examine cooperation and conflict again, revisiting a topic from an earlier Occupy Math entitled “It’s not them or us“. President Donald Trump has recently floated the idea of rejoining the Trans-Pacific Partnership. This was a free trade treaty that the President destroyed — after years of negotiations — just after he took office. The pact was enormously advantageous for the United States: it opened agricultural markets in Japan to US products, for example. Europe stepped up and inked a deal with Japan to supply those food-stuffs, costing American farmers billions. These are, of course, people that voted for the President disproportionately. Largely due to Japan working on it overtime, the remaining countries in the proposed partnership negotiated a version of the agreement (without the US) which was recently signed.
Since 100 is a big, round number, Occupy Math has prepared a 50-frame animation of a transect of cubic Julia space. This is a six-parameter Julia set with one of those size parameters changing from -0.02 to 0.200 in variable increments that are at least attempting to keep the amount of change interesting, if not level.
This week’s post looks at a mathematical structure, the Voronoi diagram, that lets you generate some pretty cool images. An example appears at the beginning of the post. Occupy Math will also show you how to modify Voronoi diagrams to get even neater pictures. The idea is simple: you start by picking special points inside a square (or any shape) and then color the square based on which special point the point you are coloring is closest to. Regions that are the same color are all the points that are closest to the same one of the special points.
In the picture above Occupy Math is assuming the square wraps around top-to-bottom and left-to-right when computing “closest”. This means that if you use the image as a background it will tile correctly. Notice that a lavender tile of the diagram appears at all four corners of the square.
Part of the exploration of the Quartic Mandelbrot – peacock tree tower!
Since Occupy Math’s readers liked the last humor post, we will try again.
☆ The problem with math puns is that calculus jokes are all derivative, trigonometry jokes are too graphic, algebra jokes are usually formulaic, and arithmetic jokes are pretty basic. But I guess the occasional statistics joke is an outlier. (with thanks to Elizabeth K.)
☆ I see you have graph paper. You must be plotting something.
Occupy Math has not done a Newton’s method fractal in some time, so here is the butterfly starship.
Occupy Math will start with a quote from the American Declaration of Independence:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed.
This part of the Declaration is saying that the reason we have rules (laws) and government institutions to enforce them is so the rights of all citizens will be respected by all other citizens. Corruption is any of a large number of practices that set aside protection of the rights of individuals to create an advantage for those engaging in the corruption. In this post, Occupy Math will make the case that corruption is obviously disadvantageous to a vast majority — possibly to everyone — using logical argumentation, one of the great branches of mathematics.
This post is not as clearly mathematical as most that appear in Occupy Math, so it may be worth noting that mathematics, in spite of all the focus on arithmetic and algebra, is primarily about the creation and manipulation of formal systems of rules. There is a reason that “math major” is one of the two best majors for getting into law school. It also means that corruption is more obvious to the mathematically inclined. With that excuse, let us continue on to the corruption!
This is a deep zoom into the cubic mandelbrot using leaf convergence. It has an odd, finished character.
In a number of Occupy Posts, we’ve looked at fractals. A long time ago, in the Goldilocks information post, we looked at the problem of having too much or too little information. Today’s post reveals one of Occupy Math’s secrets: how to let the computer look for interesting fractals on its own. The word “interesting” is chosen carefully because the fractals located this way are not beautiful or elegant (yet). They are just interesting in a very specific way. The use of this is to turn a berjillion fractals, most of which are not that good, into a short list that a human can select from or even brush up a bit. This is an example of a type of computer code where the computer is a computational collaborator.