Last week’s Occupy Math spoke a bit about Arabic numerals and declared that zero, a deep and subtle innovation, would be the topic of a future post. In this week’s post we look at zero, the empty set, and the ways these objects affect the way we do math. The starting point for appreciating zero is Roman numerals.
Roman numerals have no zero. They also can’t be used on numbers bigger than a million.
Roman numerals are still used all the time. The twenty-second Superbowl game is “Superbowl XXII”. A feature of Roman numerals is that they are not positional notation, but instead each numeral has a fixed value; you add all the numerals present to get the number represented — for instance, XXVIII is two tens, one five and three ones, for 28. Because it could be confusing to have too many of the same numeral in a sequence, they invented subtractive notation, putting the number to be subtracted first — IV is five minus one, for 4, instead of the hard-to-read IIII, for instance; IX is ten minus one, for 9, instead of VIIII; XL is fifty minus ten, for 40, instead of XXXX. As Roman civilization got more sophisticated, they started inventing really odd notational hacks. Fractions were a nightmare. So what do you do instead?
My student, Mandy Saunders, has been testing the fractalizer for me and found, not so much as a bug, as a missing feature. The quadrant/leafy option for convergence leads to flat colors under the “Shady” option. I added an angular stretch parameter to the Shady command and the following fractal is the result. Thanks, Amanda.
One of the foundational works of Western (or, for that matter, human) civilization is Euclid’s Elements which lays out the foundations of geometry and parts of algebra. In the West, the Elements was lost, along with many other texts, in the wake of the final disintegration of the Roman Empire in the fifth century. In the East, the Elements was preserved by the Byzantines and passed on by them to scholars in the Islamic Empire who translated it and other mathematical and scientific works into Arabic. When the Muslims conquered and held most of Spain, they brought these books along with them, and cultural interchange between medieval Islamic Spain and medieval Christian Europe led to the retranslation into Latin of these priceless works, long lost to the West. Quoting wikipedia:
Though known in Byzantium, the Elements was lost to Western Europe until c. 1120, when the English monk Adelard of Bath translated it into Latin from an Arabic translation.
In fact, much of what Western Europe knows about the “the glory that was Greece and the grandeur that was Rome” came to us through cultural links between the West and the Islamic Empire and its successor states.
This week, the image of the week fractal is a Newton’s method fractal for a 20th degree polynomial. The roots are generated in symmetric pairs about the complex axis. Attractors for the first twelve are mapped to cosine-colored shades, the last eight to black. Points that dont converge in 12 iterations are mapped to a gold cosine-colored shade. The result is displayed below. This type of fractal is called a Newton filgree fractal.
Recently, Occupy Math looked at (and mocked) the idea that most students only need arithmetic in high school. This week’s post tries to provide an interesting look at compound interest which is one of these things, like toxicology or nuclear physics, where you are better off having a working knowledge of at least the major points. Its also an example of something beyond arithmetic that would be good to cover in high school. Strike that: substantial human misery could be curtailed if we covered this more completely in high school. Knowing enough to avoid abusing credit cards is a key skill that would not be that hard to teach, yet the subject is slighted in most schools! This issue is one of the key points supporting Occupy Math’s slogan that Math is the Right of all Free People.
Diffuse, inadequate treatment of finance in education endangers students.
This is a Newton’s method fractal for a polynomial with 13 complex roots. The roots were randomly generated, but Occupy Math zoomed in to find an area with interesting structure. As with the other colored images of the week, this one is rendered to 2000×2000 pixel resolution. Enjoy!
Last week’s blog looked at the presentations that Occupy Math made at the World Congress on Evolutionary Computation – brief summaries that tried to give a sense of what the papers were about. This week Occupy Math takes an in-depth look at the techniques inspired by Darwin’s theory of evolution that are the core of his research. The key point? Evolution can solve math problems in ways humans cannot. Sometimes this means evolution is just a faster technique, but other times evolution pulls off stunts that human effort cannot match. Some people might find this concerning but
Occupy Math studies evolution as a tool for amplifying human capabilities.
Heading off in a new direction (that previews the main post at the end of the week). This image is generated by a cellular automaton rule that was evolved to know its own boundaries. The cool thing is that 39 one-byte integers completely specify this bad boy. If you have the rendering software.
One of the big problems with science and math is that the practitioners are terrible at explaining what they do. Occupy Math will use this post as a way to work on his skills at communicating what on earth he does outside of the classroom. This week in Vancouver is the 2016 IEEE World Congress on Computational Intelligence and Occupy Math is presenting several papers. This week’s post summarizes what they are about. Hopefully this will help students who might like a career in computer science, mathematics, or statistics see what they are getting into.
This week Occupy Math is switching the Monday feature to colored pictures – and ones that are way too detailed for a coloring page. This one is a 2000×2000 generalized quadratic Julia set – scan around it at full resolution to see, well, a lot of detail.
This image was made with a tool called the fractalizer. Let Occupy Math know if you would like to be on the beta-test list.