A rainbow palette cubic Mandelbrot view, in case you were color-deficient!
Dr. Hilda Geiringer was an Austrian mathematician whose life is closer to the plot of Indiana Jones and the Last Crusade than almost any other mathematician. She was the first woman to serve as a university instructor in applied mathematics in Germany and, completely against the current of the times, was up for promotion to extraordinary professor when the Nazi laws kicking anyone Jewish out of any worthwhile job kicked in. She fled through Turkey, eventually coming to the United States where her general excellence triumphed — partially — over the open discrimination against women in mathematics that was usual at the time. She was extraordinary and made remarkable contributions.
This is a complex spiral from inside the quintic Mandelbrot set.
Today we introduce a new activity, factor grids. Suppose we want to place the numbers 6, 10, 14, 15, 21, and 35 in a 3×2 grid, as shown below. The goal is to get the highest score, where we get points for the largest common factors between any two adjacent grids, added up. One of the best scoring solutions — worth 32 points — is shown below. The grid on the right shows the scores for numbers in adjacent grids in red and then adds them up. Fifteen and thirty five, for example, have five as a greatest common factor.
An instance of this puzzle requires a grid size and a list of numbers. Occupy Math has written code to find all possible solutions and then mines that for the best solution. This is a “brute force” approach, but for the grids in this post, with sizes up to 4×3, it works perfectly well.
A critical part of the question of the status of mathematics as a discovery or an invention is the fallacy of false duality. By asking if mathematics was invented or discovered, we imply that these are the two possibilities are what is available. There is an unending debate on this topic. When a debate does not end, this is often because the debate started with a false premise — in this case that mathematics is either invented or discovered. The second starting point for this post is a wonderful article, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, that establishes that mathematics is naturally the language of physics, the laws of which are clearly discovered.
This week, a very busy Newton’s method fractal.
The whole numbers, the first numbers we learn, are infinite. If you take a number and add one to it, you get a new and bigger number — but not always. Keep counting up from one and you can get to 213 — but start at one o’clock in the afternoon and you never get to two-hundred and thirteen o’clock. In this post we are going to look at a type of number where we cannot keep going forever. Instead, as we keep adding one, the numbers cycle around, like the numbers on a clock. It turns out that there is an infinite family of these number systems, and some of them obey almost all the normal rules of arithmetic.
When you are first learning to add and multiply, you might make tables. Memorizing the multiplication tables is somewhat controversial in math education (Occupy Math comes down firmly in favor of memorizing your one-digit multiplication facts). At the top of the post are addition and multiplication tables, but not the normal ones. This post is about a whole collection of number systems based on the integers, each of which has only a finite number of numbers. The addition and multiplication tables above have five numbers, for example. As we will see, these number systems show up in several applications. They obey many of the same laws of arithmetic as the integers. Mathematicians think of these number systems as smaller images of the integers that preserve some of their properties.
This post is the next in our series on what mathematicians do with their time. The other posts in the series are indexed at the end of this post. Since everyone knows about the teaching mathematicians do, the series focuses on research. Today we are going to look at the walking triangle representation for optimizing functions. If you have finished a calculus course, it’s likely you already have optimized a function; this means finding the place where the function is highest (maximizing) or lowest (minimizing). We call the high points and the low points the optima of the function. To demonstrate the walking triangles, we will be maximizing the function shown at the top of the post. This function is really easy to maximize — you just head uphill.
A view into a Newton’s method fractal with the character of a layered egg.
A post on calculus and its higher level forms recently appeared in Occupy Math as part of the Islands of Mathematics series. One key piece of feedback from Occupy Math’s editors was substantial confusion about complex numbers. Since complex numbers are key to a number of the types of fractals that Occupy Math likes, including the one at the top of the post, there are now several reasons to have a post which explains complex and imaginary numbers.
There is also a rather startling mathematical technique to be explained: if, in a system you are using, some object does not exist, then declare it to exist by fiat, and see what impact that has. Here is a key point the post tries to make. The object that “does not exist” that we are going to investigate is the square root of a negative number. Many people “know” that negative numbers do not have square roots, but this is false. Something much narrower is true — those square roots only exist if we enlarge our concept of what a number is. Interestingly, the new notion of number actually lets us explain laws of nature and do engineering tasks that are either impossible or much more difficult without these new numbers.