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.
Occupy Math’s image of the week: a Julia set with marching whatnots over flames!
Suppose that you have a friend helping you rake the yard. If you take three hours to finish the job by yourself, and your friend takes two hours by himself, then, if you work together efficiently, how long does it take for you to finish the job working together? The answer is one hour and twelve minutes. This post is about a tool called the harmonic mean that lets you do calculations like that. The harmonic mean also shows up in some of the laws of physics, which is interesting. You may remember that “mean” is the highfalutin’ mathematical term for average. We will also talk about the sense in which the harmonic mean is an average, even though it does not act like the averaging you are used to.
A complex triple Julia set with a chain of beads with spirals.
This post is themed “look at the pretty pictures”, with an attempt to make summer appropriate pictures. It is a look at some of the things that math can do. The secret for making Mandelbrot sets look like bouquets of leaves is featured. This is something that Occupy Math discovered by accident, a nice example of when a code bug is really a feature. The Mandelbrot set is an excellent source of complex pictures. The Ghost Mandelbrot set shows how you can switch up the details and get new fractals. In this post we will apply quadrant convergence to the Mandelbrot set and use a green color scheme to get leafy-looking fractals.
Occupy Math’s image of the week. A triple cubic Julia set, with Medusa like characteristics.
A number carries a certain authority. We are bombarded with statistics about what people want, what they need, and what they will tolerate. Mathematical models are used to decide what to build, what to build first, and whether to build anything at all. Most of these planning results are summarized as a cost. There is a problem with this: the context and meaning of the numbers are often missing. The cost of a new subway line is not accompanied by a careful statement of the cost of not building it — partly because the cost of not building something is usually not a number. In fact the cost usually has dozens of components. These include reduced traffic, improved commute times, change in property values near stations, changes in both car and pedestrian traffic patterns, impact on local businesses, and on and on.
A dragon of the fifth power Mandelbrot set.
Have you ever tried to enlarge a picture and lost detail? Maybe individual pixels became big single-color squares? The skull to the left is made of circles, ellipses, and rectangles, glued together as a series of tests. For a point in the plane, Occupy Math’s computer checks which of these geometric objects a point is inside of or outside of and, from that, deduces a shade: black or white. This may seem to be a lot of trouble to go to for a sorta-okay picture of a skull. The reason for going to that much trouble is that membership in the black parts of the skull, since we are asking about numerical points (x,y), has a resolution of trillions of pixels by trillions of pixels. This means that if we warp space, we will not get pixelization errors or other distortions. In other words, this skull image is far more malleable than usual images stored as pixels. In this post we are going to apply space-warps to the basic skull image. Having explained about the skulls, it is worth noting that Occupy Math created all this stuff while playing around but the math used to make many different skulls turned out to have applications in statistics. Following the fun parts of your work not only relieves stress, it can have big payoffs.
A triple cubic Julia set: a triplet spinner fractal.