This week we look back at the great-tree region of the fourth order Mandelbrot set under quadrant convergence.
Please don’t be intimidated by the calculations below! Occupy Math needs them to make a point — part of which is that lines two through six are completely not needed. This post was written directly after grading a midterm in a first-year university calculus class. What that grading taught Occupy Math is that the students have been deceived about the very nature of math!
One of the questions on the midterm was “Find a sixth degree polynomial with exactly two roots”. Occupy Math thought that this would be a freebie question. There are many correct answers, one of which is the first line of the calculations above. About 20 of the 100+ students at the exam took this correct answer and drove it right off a cliff by trying to multiply it out. Occupy Math has multiplied it out to show how much work that is! Here’s the punch line: not one of the students got the arithmetic correct. As far as Occupy Math can tell, they have been indoctrinated to believe they have not answered a question until they have done a good deal of calculation? They feel that math is not math without the busywork!
This images is a second-fifth order hybrid Mandelbrot set. The short version? It’s never done this before.
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.
This is a view from the cubic Mandelbrot set using quadrant convergence. Its another example of an abrupt change in the colors creating an island.
In this post, Occupy Math tackles the thorny issue of diversity from mathematical perspective. Many of us take it as given that diversity is a good thing — while some of us think it is a plot to destroy our culture — but there is a mathematical perspective from which diversity is intrinsically valuable. When a crisis comes, it is difficult to predict what qualities are needed to deal with it. This means that a diverse population hedges its bets while a homogeneous society is placing an all-or-nothing bet on their survival. This is why the genetic diversity of an endangered species is of such concern to conservation, but similar principles apply to everything from problem solving to human society. This even shows up in our corporate culture where believing in a “one true way” of thinking about or solving problems can be a barrier to progress.
There is an inaccurate stereotype of millennials as lazy, entitled, self-absorbed lumps who live with their parents. When Occupy Math’s editor graduated from Bard College, it took her a couple of internships and several years to find a good position. Her grandmother could not understand why finding a job took so long and was concerned that lack of zeal might be the problem. What’s actually going on is that good jobs are far rarer than they used to be and affordable housing is also really hard to find, hence living with their parents. This inaccurate stereotype is an example of a smooshing error. This is an error where you combine a whole bunch of diverse factors (job availability, housing prices, the impact of automation and the internet) into one thing in your head and then map it onto your own experience, drawing a wildly incorrect conclusion. In the late 1940s (when the above-mentioned grandmother was hunting for a job), inability to find a position probably would have required laziness. It was a different world then.
This week, the Newtons method fractal of a 27-root polynomial with complex coefficients. The input is random but the algorithm is not!
In 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.
This week the image is drawn from the 2-3-2 Mandelbrot set, a fractal generated by applying the squared and cubed Mandelbrot iterators in the pattern 2-3-2-2-3-2-2-3-2-etc. Some parts of this set are pretty irregular, others look like slightly different versions of the ordinary squared set. This image is a Minibrot with the classical squared shape but a moderately odd surrounding texture.